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L-function for the standard tensor product representation of GSp{2) X GSp(2)
Jiang, Dihua, Ph.D.
The Ohio State University, 1994
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
L-function for the Standard Tensor Product Representation of GSp{2) X GSp{2)
dissertation
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Dihua Jiang, B.S., M.S.
*****
The Ohio State University
1994
Dissertation Committee:
Prof. Stephen Rallis Approved by
Prof. Avner Ash Prof. David Ginzburg Adviser Department of Mathematics to my wife Ying and my son Fan
n A cknowledgements
I wish to express my gratitude and indebtedness to my advisor and my teacher,
Professor Stephen Rallis, for his bringing me into the mathematics area of Auto- morphic Representations and L-functions, for his suggestion and formulation of the research project for my Ph. D. Dissertation, and for sharing his ideas and insights during my implementation of the project. His guidance, advice, trust, patience, and supports are greatly appreciated.
I am grateful to Prof. D. Ginzburg for helpful conversations during my working with the project and for his careful reading this dissertation. I would like to thank
Prof. A. Ash, Prof. C. Rader, and Prof. R. Stanton for their frequent help. My thanks also go to Prof. A. Ash, Prof. .1. Hsia, Prof. M. Rosen (Brown University),
Prof. K. Rubin, Prof. A. Silverberg, and Prof. R. Stanton for their wonderful topic courses on Number Theory, Algebraic Geometry, and Representation Theory.
Finally, I would like to thank my family for their love, patience, and understanding throughout this time.
m V i t a
Oct 17, 1958 ...... Born in Wenzhou, P. R. China.
1982 ...... B.Sc., Zhejiang Normal University, P. R. China.
1987 ...... M.Sc., Department of Mathematics, East China Normal University, Shanghai, P. R. China.
1989-1994 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio.
R epresentation T h e o r y a n d N u m b e r T h e o r y
Major Field: Mathematics
IV T a b l e o f C o n t e n t s
ACKNOWLEDGEMENTS ...... iii
VITA ...... iv
CHAPTER PAGE
I Introduction ...... 1
II L-function for the Standard Tensor Product Representation of GSp{2) x GSp{2) ...... 9
2.1 P relim in a ries ...... 9 2.1.1 The Group of Symplectic Similitudes ...... 9 2.1.2 Root Systems and Representations ...... 11 2.1.3 The Negligibility of O rbits ...... 12 2.2 Rankin-Selberg Global Zeta Integral ...... 14 2.2.1 The Whittaker functions on GSp{2) ...... 15 2.2.2 The global zeta-integral ...... 19 2.3 Unramified Local Zeta Integrals ...... 25 2.3.1 Reduction ...... 26 2.3.2 Computation of 29 2.3.3 The Unramified Local Zeta Integral ...... 37 2.4 Local Langlands Factor of Degree 16 for GSp{2) x GSp{2) . . 40 2.4.1 Kostant-Rallis A lgorithm ...... 41 2.4.2 Symmetric Space (S'p(4),5p(2) x Sp{2))...... 43 2.4.3 Branching formula for Sp{2) 4- SL{2) x S L { 2 )...... 47
2.4.4 Multiplicity Z((mi,m 2 ; ni,n 2 )) ...... 49 2.4.5 Orbit Decomposition on {B x B)\K e...... 52
2.4.6 Homogeneity di((m i,m 2 ;ni, 7 1 2 )) ...... 61 2.4.7 Application of Kostant-Rallis’ Algorithm ...... 85 2.4.8 The Proof of Theorem 2 .3 .1 4 ...... 89 2.5 The Fundamental Id e n tity ...... 90 2.6 Notations Used in Chapter II ...... 92
III The Local Theory of Rankin-Selberg Convolution ...... 93
3.1 Som e E s t im a t e s ...... 94 3.1.1 Estimates of Whittaker Functions: ...... 94 3.1.2 Estimates of Some Integrals ...... 99 3.1.3 Estimate of the Partial ‘Fourier’ TVansform...... 105 3.2 Nonvanishing of Local Zeta I n te g r a ls ...... 112 3.2.1 Nonarchimedean C a se s: ...... 112 3.2.2 Archimedean C a se s: ...... 120 3.3 Absolutely Convergence of Local Zeta Integrals ...... 126 3.3.1 Nonarchimedean C a se s: ...... 127 3.3.2 Archimedean C a se s: ...... 132 3.4 Meromorphic Continuation of Local Zeta integrals 136 3.4.1 Nonarchimedean Case: 136 3.5 Notations Used in Chapter I II ...... 146
IV The Poles of Eisenstein Series Attached to Sp{n) ...... 147
4.1 P^-Orbital Decomposition on \ S p {n )...... 150 4.2 The P"-constant term of Eisenstein Series ...... 155 4.3 Local Analyses of Intertwining Operators ...... 162 4.3.1 Intertwining Operator ^(g): general sections ...... 163 4.3.2 Intertwining Operator general sections ...... 166 4.3.3 Intertwining Operator ^(s): general sections ...... 174 4.3.4 Intertwining Operators U^^(s), % (s), and U^^{s): spherical section ...... 182 4.4 Poles of Eisenstein Series: n < 3 ...... 188 4.4.1 Proof of Theorem 4.4.4 for the case of s ...... 194 4.4.2 Proof of Theorem 4.4.4 for the case of s = |: ...... 196 4.5 Poles of Eisenstein S e r i e s ...... 207 4.6 P roof o f Two L e m m a s ...... 216 4.7 Notations Used in Chapter I V ...... 227
VI V Degenerate Principal Series Representations of Sp{n) over a p-adic Field 228
5.1 Basic F a c t s ...... 229 5.2 Computations of Hecke Operators ...... 236 5.2.1 Hecke Operator A ...... 236 5.2.2 Hecke Operator W ...... 255 5.3 P roof of th e M ain T h eo rem ...... 258
5.3.1 The Dimension of ifomsp( 4 )(/ 3 (s),/ 3 (—s)): ...... 258 5.3.2 The Proof of the Main Theorem : ...... 261 5.4 N otations U sed in Chapter V ...... 269
BIBLIOGRAPHY...... 270
Vll C H A PT E R I
Introduction
Let F be a number field and A its adele ring. Let GSp{n) be the group of symplectic
similitudes of rank n and GSp{n,A) the adele group of Sp{n). We denote G :=
GSp{4:) and H := (GSp{2) x GSp{2))°. It is clear that H can be embedded into G via the doubling method. Let ttj and TTg be irreducible automorphic representations of GSp{2,A) with trivial central characters in the sense of [11]. Let E^{g,s]fs) be an Eisenstein series associated to a section /« in the degenerate principal series representation I^is) of G(A) induced from a standard maximal parabolic subgroup
P3 of non-Siegel type (Chapter II or Chapter IV). For any cusp forms (/jj G Tj,
<^2 € 7T2, we set our Rankin-Selberg global convolution (integral) as follows:
, El{{g-i,g2),s-,fs)(l>\{g\)(l)2{92)dg\dg.2 ( 1.1) JC(A)H(F)\H(A) where C = C4 is the center of GSp{A).
The first main result proved here is the following basic identity:
z (> .K < h j) = « f ' ) . j j (1.2) where L'^’(.s', tti ®'ïï2iP\ <8 >Pi) is the Langlands L-function for the standard tensor product representation p\ Cg) p\ of GSp{2) x GSp{2), which is of degree 16, s' =
1 d^ls) is the normalizing factor of our Eisenstein series, and Zv{s, W\ , W^, /s) for u £ 5 are ramified local integrals. Note that those integrals (both local and global) require that the representations tti and TTg have nonzero Whittaher models. We assume from now on that the representations tti and 7T2 are generic.
This basic identity is the starting point of my Ph.D. dissertation research project under the guidance of Professor Stephen Rallis. According to this project, the degree
16 Langlands L-function Cg) 1^2,pi 0 pi) should enjoy following properties:
(1) (Langlands Conjecture). L^{s\Tr\ (S) it2, pi pi) converges absolutely for
Re{s) large and continues to a meromorphic function on the whole complex
plane which has only finitely many poles and satisfies certain functional equa
tion relating its value at s to its value at 1 — .s.
(2) L'^'(.s',7Ti 0 7T2,/3i 0 Pi) is holomorphic for Re(.s) > 1 and has an at most double
pole at .s = 1. If £^(s', 7Ti 0 7T2,pi 0 pi) achieves the double pole at .s = 1, then
the irreducible automorphic representation 7r2 should be equal to the contragre-
dient representation Tr^ of t t j .
(3) For a generic cusp form tt over GSp(2, A) with a trivial central character, the
L-function £^(s',7t 0 ir'^,pi 0 pi) achieves the double pole at .s = 1 if and
only if the image Û 2(7t) under the theta lifting for the dual reductive pair
(G5'p(2), G0(2,2)) is a nonzero cusp form over G 0(2,2; A).
The significance of these properties of the L-function L‘^(.s', tti 0 7T2, pi 0 pi ) is clear.
For instance, (2) is an analogue of .Jacquet and Shalika’s results for G£{n) in [35] and implies important applications in the theory of automorphic forms; (3) is an analogue of Waldspurger’s description of Shimura liftings and is predicted by Rallis’ theory of ‘towers of 0-series liftings’. Both (2) and (3) will imply that the degree
10 Langlands L-function L^{s',ir,Ad) is holomorphic and nonvanishing at s = 1.
Finally the doubleness of the pole at .s = 1 and the lower boundary line of the domain of holomorphicity of the L-function tti (g) 7T2,/3i (E> Pi) are predicted by means of the seesaw identity for our seesaw pair of similitude groups, which are
(C?5p(4),GO(3,3)) and {{GSp{2) x G5p(2))°, (GO(3,3) x GO(3,3))°), in the sense of Kudla [41] and Harris and Kudla [25], and Langlands’ principle of functorality for the natural embedding: GSp{2, C) (jL(4, C).
The strategy to prove that the L-function L^{s', tti (g) TTg, p\ (g) pi ) eventually pos sesses those three predicted properties is:
First of all, the global zeta integral, following from the general theory of Eisenstein series, converges absolutely for Re{s) large and continues to a meromorphic function on the whole complex plane which has poles at the points where the Eisenstein se ries has and satisfies certain functional equation. Property (1) will be proved after establishing the local theory for Z„(s, VFi, W-z^fs) as in [68 ] and [69].
For the determination of the location and the order of the poles of the L-function
7Ti (gi 7T2,pi eg) pi), the theory, which is analogue to that developed by Kudla and Rallis [43], for the family of Eisenstein series E^{g,s;fs) should be established.
From Kudla and Rallis’ theory on the first term identity of Siegel-Weil formula, the
Eisenstein series E^{g,s]fs) has a possible double pole at .s = 1 for these sections fs in /^(.s) determined by the regularized theta integral of the dual reductive pair
{GSp{é),G0{3,Z)). For our application, the existence of the double pole at .s = 1 of
the Eisenstein series -s;/«) should be proved for general sections fa € -faC-s).
Finally, for the applications to theta correspondences, i.e.. Property (3), it becomes
critical to understand the residue representation of the Eisenstein series E^{g,s-,fs) at s = 1. As the theory of Kudla and Rallis goes, we should have the ‘first term iden tity’ which says the double residue representation of our non-Siegel Eisenstein series
E^{g,s;fa) at .s = 1 should physically be identified with the residue representation of the Siegel Eisenstein series at s = ^ of 5'p(4) inside the space of square integrable automorphic forms of G5p(4, A). To this end, we should have enough information on the local degenerate principal series representation of 5'p(4), especially, on the irreducible quotient representations of as [44] and [51].
This Ph.D. dissertation consists mainly four parts: Chapter II, III, IV, and V, including complete or relatively complete results I have obtained during my working with the research project designed by my advisor. Professor Stephen Rallis. We shall describe those results in some detail below.
In Chapter II, we shall establish (in section 2.3) the global integral Z{s, (/)■[,
Z{s,
and the section which is eulerian by the uniqueness of Whittaker models
[65]. The unramified computation consists of two parts:
(1) the computation of the local zeta integral with the unramified integrating data (section 2.3), and
(2) the computation of the predicted Langlands local L-factor (section 2.4).
Following Casselman and Shalika’s formula for the unramified Whittaker functions, the computation of the unramified local integral 2^„(s, W°, Wij*, f°) is reduced to that of an local integral /(u j, 02; f°, %6) (section 2.3), which can be eventually reformulated as certain ‘Partial Fourier Coefficient’ or a special case of degenerate Jacquet integrals
[71]. The most technical part of this Chapter is the computation of the Langlands local L-factor Lv{s', ttu 7T2„,/Oi 0 p\) (section 2.4). The computation is based on the algorithm of Kostant and Rallis [40], which gives an explicit spectral decomposition of the space of harmonic polynomial functions over a symmetric space, which is in our case (5p(4),5p(2) x 5p(2)), the multiplicity one branch formula for (5p(2),5p(l) x
5p(l)), and the Borel-Weil-Bott theorem.
The local theory of Rankin-Selberg convolution 2’„(.s, Wi, Wi,/*) is developed in
Chapter III. Following the standard estimates of Whittaker functions on the splitting torus, which is analogue of the estimates made in [33], [35], and [ 68 ], the technical part is the estimates of the local integral /(ui,a 2;/s ,’/’) for a general section fs and a additive character »/> (subsection 3.1.3). In nonarchimedean cases, the theory is completed. In other words, we have proved that the local integral 2'„(.s, W], W 2, /,) converges absolutely for Re{s) large and continues a meromorphic function on the complex plane, and after choosing an appropriate data (W,, W 2,jf,), the integral
■2„(.s, Wi, W2,/s) can be made to be one. In archimedean cases, however, the theory is not yet completed. We have proved: (i) the local integral -Zv(s, Wi, W2, fs) converges absolutely for Re{.s) large with ,
W2 being any -finite Whittaker functions on G5'p(2, A), and fa any smooth
section in (Theorem 3.3.5); and
(ii) nonvanishingness of 2„{s, Wi, W2, /«) for a suitable choice of A-finite Whittaker
functions and W2 and a smooth section /* (Theorem 3.2.4).
One of the subtle points is that the general theory of Eisenstein series assumes the
/f-finiteness of the section /, at archimedean local places. In order to apply the local
theory to the L-function ®ir2,Pi ^ pi), we thus have to prove either one of following statements:
(1) The local integral Z„{s, Wi, W2, fs) continues to a meromorphic function on the
complex plane with A-finite data {Wi,W2, fs), and for a given value .sq there
exists a iC-finite data (Wi, W2, ) so that the local integral 2 „(.so, Wi, W 2, fg^ )
does not vanish.
(2) The local integral Zv{s, Wi^Wz, fs) continues to a meromorphic function on the
complex plane with -finite Wi, Wz and smooth section fs, and for fixed W^,
W2 (which can be /f-finite), the local integral 2 «(.s, Wi, W2, /«) is a continuous
functional over the space of smooth sections /* (which is Frechet space in a
natural topology).
I do not have a complete proof for Statement ( 1) or (2) as yet.
Chapter IV goes generally for Sp{n), symplectic group of rank n. We are con cerned with a family of non-Siegel type Eisenstein series fg). We shall determine the location and the order of possible poles of /,) for general
holomorphic sections /, in the (global) degenerate principal series represen
tation of Sp{n) (Theorem 4.0.4), which is an analogue of the weak form of Kudla
and Rallis’ Theorem of this type [43]. The ideas and the methods used here were
developed by Piatetski-Shapiro and Rallis [54] and [55], and Kudla and Rallis [43]
and [44]. With an inductive formula (Theorem 4.2.3), the information about the poles
of the constant term El^_-i pn{g,s\fs) of the Eisenstein series jE"_,( 5t,.s;/s) along the
maximal parabolic subgroup P ” will be given by that of Eisenstein series of lower
ranked group Sp{n — 1) and that of the relevant intertwining operators. Following
the inductive formula, the most technical part in our proof of Theorem 4.0.4 will be reduced to the special case of n = 3 and >s = | (section 4.4). For general n, we have proved that the Eisenstein series -s;A) always achieves a double pole at s = and a simple poles at s = | and s = At other points on the positive part of the critical strip, the Eisenstein will achieve double poles if does not vanish at s = 0. The existence of the double pole at s = of the Eisenstein series P"_i(<7,-s;/s) was predicted by means of Kudla and Rallis’ first term identity
[43].
Finally, Chapter V is devoted to the study of the quotient representations of the degenerate principal series representation f"_^(s) of Sp{n, P„) over the p-adic field F„.
The methods used here was basically developed by Casselman (general formulation
[14]), Gustafson (Siegel parabolic case [23]), and Jantzen (a special maximal parabolic case). Theoretically, the dimension of the space ffom 5p(„)(/^*_^(.s),/"_i(—.s)) and 8
the information about the irreducible quotient representations of should be
given by explicit computations of certain Hecke operators associated to the parahori
subgroup 7p;_j acting on the subspace 7”-i of consisting of sections
fixed by the parahori subgroup • Our computations are made for general n.
However, the proof is eventually reduced to find the a ‘nicer’ basis in the Hecke
module , which is much more complicated than the basis in [23]. We
have proved the main result only for the case n = 4, which is: 73(5) has a unique
irreducible quotient representation if
F(s) := (1 - g-'-=)(l - + q“){l + ) # 0 . (1.3)
This partial result is sufficient for our application to the degree 16 L-function. It should be mentioned that the polynomial F{s) will give a criteria for irreducibility of the degenerate principal series representation 73(a) of 5'p(4), which is an analogue of
Gustafson’s result.
The archimedean version of such result has not yet been completed. This, together with all other predicted results, will be worked out after this Ph.D. dissertation. C H A PTER II
L-function for the Standard Tensor Product Representation of GSp(2) x GSp{2)
2.1 Preliminaries
We shall first recall some basic facts on representation theory of algebraic groups for our later use. For the general theory, we prefer T. A. Springer [67]. We only consider two algebraic groups H and G of symplectic similitudes, which relates to each other via the doubling method. We will study the ff-orbit decomposition of some flag variety constructed from G and the negligibility of those H-orbits. General discussion of the doubling method can be found in S. Rallis’ IMG paper [57].
2.1.1 The Group of Symplectic Similitudes
Let (V, ( , )) be a 4-dimensional non-degenerate symplectic vector space over a field
F with characteristic zero and GSp{V) the group of similitudes of (V, ( , )), i.e.,
GSp{V) = {
Let (W, ( , )) be the doubling symplectic space of (V, ( , )), i.e. W = © V~, where 10
V'*' = {(u,0) V E V} and V~ = {(0, v) v 6 F}, and the symplectic form is defined
by
((ui,U2),(vi,U2)) = (wi,vi) - («2,^ 2) for ui,U 2,Ui,U2 e V.
Then (IF, ( , )) is an 8 -dimensional non-degenerate symplectic vector space over the
field F. We denote by G = G5p(W) be the group of similitudes of (TF, ( , )).
In (V, ( , )), choose a symplectic basis {ei, 62, e,, so that the underlying vector
space V is identified with F ‘^ (row vectors) and the form ( , ) corresponds to the
matrix J2 =(^ q )• ^ben in (W, ( , )), we have a typical symplectic basis
{(ei, 0), (62, 0), (0, —61), (0, —62), (e'l, 0), (e^, 0), (0, e'l), (0, e^)} (2.1)
under which the underlying vector space W is identified with F® (row vectors) and the
form ( , ) of W corresponds to the matrix J 4 = ^ ^ Under the chosen basis,
the group of symplectic similitudes can be embedded into a general linear group, i.e.,
GSp{V) = GSp{2) = {g e GL{é) : gj^^g = s{g)J2} and GSp{W) = G5p(4) = {g E
GL{8 ) : gJ^g = s{g)J4}, and the action of GSp{V), GSp{W) on V, W corresponds
to that of GSp(2), GSp(4) on F ‘*, F® to the right, respectively.
Let = (GS-^g) X Ggp(2))= = {(g„g2 ) e G5'p(2) x G^p(2) : a(gi) = ^(^[ 2 )}.
Then the reductive group If can be embedding into the reductive group G in a canonical way: i : H G, («1,^ 2) • *(<7i,<72) = (^1^1,^ 2(72)- In other words,
\ - C " D' )
From now on we will identify the group H with its image i{H) in G. 11
Let Lo = F(ej,0) 0 F(eg,0) © F(0, e^). Then Lo is a three dimensional isotropic subspace in W. Let P3 = Staho(Lo)- Then P3 is a maximal parabolic subgrou^j of
G, whose Levi decomposition is P 3 = with
= (GL(3) X G5p(l)) ( a Q 0 0 \ 0 0 x2 e GSp(4) : a! = (X1X4 — X2Xa)‘a \ a G GL(3)} = { 0 0 a' 0 \ 0 ®3 0 X4 / and / /a X w y \ 0 1 y' 0 eGSp(4) :x,y,V ,V 6M (3,l)}. 0 0 /3 0 V 0 0 æ' I / where is the unipotent radical of P 3 . We take, as a maximal P-split torus T| C P 3,
T)(P) — {t — ; tg, (3 , ^4, tg, tg, ty, (g) — ^2^6 — ^3 ( 7 — ^4^8}) where h{- • •) indicates a diagonal matrix element in G under the basis. Let B4 be the standard Borel subgroup of G with T4 C P 4 C P3 and TV* be the unipotent radical of P 4 .
2.1.2 Root Systems and Representations
Let X*(T 4) be the group of characters of T 4 and £,■ be such a character that = ij.
Then 8 X*{T4) = : £1 + £5 = £2 + £e = £3 + £7 = £4 +8 }-£ •=i Let = $(G, T4) be the set of roots of T 4 in G, the set of positive roots of $ 0 determined by N ‘*, and A o the set of simple roots in Then we have the root 12
system for G
$G = {±(e,- ± £j), ±2£,-:i< j, i,j = 1,2,3,4}, (2.3)
= {(ei ±£j),2£i :i Ac; = {«1 = £i — £2, «2 = £2 — es, as = es — £4, «4 = 2£4}. (2.5) Proposition 2.1.1 For tfce complex group GSp{2,C), we have (a) The fundamental dominant weights are = £1 and A2 = £1 + £2, (b) For any dominant weight n\S\ + »2£g, u\ > n -2 > 0, there is an irreducible complex representation P(„,,,i 2) of GSp{2) with highest weight ii\e\ +n- 2£2 and any irreducible complex representation of GSp{2) is equivalent to the product of some power of the character s{g) (the factor of similitudes) and some /^(ni.nj)- Actually, / 5(„,,„2) is the representation of the derived group of G5'p(2), which is Sp{2), with highest weight ui£i + ng£ 2, n, > n-i > 0, and is trivial at the ‘similitude’ part, i.e. ^ dh GSp(4) : d is non-zero scalar}. Proposition 2.1.2 [7] The complex dual group of GSp{2) is GSp{2,C). 2.1.3 The Negligibility of Orbits In this subsection, we consider the if-orbital decomposition on the flag variety P ^\G and describe the negligibility of these ff-orbits. Those results will be used to construct 13 our Rankin-Selberg global integral in the next section. The notion of negligibility was introduced in [54]. Let C, = {all 3-dimensional isotropic subspaces L of W} and Lo = jF(e',,0) © F(eg,0) © i^(0,6j) as chosen in subsection 1.1. Fix once for all the following isomor phism from P3 \G onto C via g 1— )• Log. Let 7T^ be the projections from W onto V'^ or V~, respectively. Let L be any 3-dimensional isotropic subspace of W. Denote = LV\ and L' = 7T'*'(L), L" = ir~{L). Then it is easy to check that dimL""" -|- dimL" < 3, dim < 2, and diniL'-|- dimL" = dim L" -|- dimX+ = 3. Theorem 2.1.3 Let ti^{L) = dimL"*" and k~{L) = diinL". Then (k^{L), k~{L)) completely determines H —orbits of L € C; that is, for L ,M £ C, {k'^{L), k~{L)) = (K+(M), k~{M)) if and only if there is g E H such that Lg = M. Proof: It is evident that {L), k~(L)) is an invariant of the ff-orbits on L since [L{gug 2)V = and [L((/i,(72)]" = L~g2- Moreover, (k+(L),k"(L)) completely determines Lf-orbits because, for L ,M E C, we can use the same argument as in [54] to prove that there are g\,g-z G Sp{2) such that L(g], <72) = M. □ By straightforward calculation, the flag variety C has a decomposition of /f-orbits as P = ^(2,1) U /I(i,2) U -^(1,1) U U /^(o,i) U /I(o,o), (2.6) where £(,j) is the If-orbit with invariants («‘•'(Z), k“(L)) = {i,j), and the only non- negligible if-orbit in the sense of Piatetski-Shapiro and Rallis [54] is Z(o,o). The 14 unique nonnegligible ff-orbit is represented by a three-dimensional isotropic subspace L(o,o) = F(e',, e'l) ® ^(eg,e^) ® F(e 2,eg) or X(o,o) = Lo"fo where 1 \ 0 0 1 0 1 0 1 1 0 1 1 1 0 - 1 0 . -10 0 / It is easy to see that i^(o,o) is the diagonal embedding into W of a three-dimensional subspace L* = Fe[ ® Fe^ © Feg of V. Let Q = Stab//(L(o,o))- Then we deduce that Q = StabH(Lo7o) = {{91,92} € i î : <7iU* = <72^*, L*gi = L*, i = 1, 2}. (2.7) Let = StabG5p(2)(L*). Then is a maximal parabolic subgroup of GSp{2), which is of form, under the chosen basis {ei, eg, e(, el^}. ( a x z s \ 0 X\ s' X2 £GSp{2) : ad = X4X4 — X2X3 ÿ( 0}. 0 Q d 0 \ 0 X3 x' X4 J If g e GSp{2) such that g\i,* = 1, then e\g = e\ + ze^, i.e. g = %g(, (z), where Xgj, is the one-parameter subgroup of GSp{2) associated to the root 2ê] . Let Z-x = {X2ei(0 : t E F}- Then we obtain that Q = StabH(Lo 7o) = F"'^(Zg x I4) (2.8) where P^’^ is the diagonal embedding of Pf into Q, and P^joH corresponds to P(o,o) — — Lo'^oH. 2.2 Rankin-Selberg Global Zeta Integral We assume from now on that F be a number field and A its ring of adeles. Let F„ be the local field of F associated to the place v. When v is finite, we denote by 0„ the ring 15 of local integers in By automorphic representations of adelic groups we mean that in the sense of Borel and Jacquet’s Corvallis paper [11]. We shall establish a global zeta integral via the doubling method, which will be a Rankin-Selberg convolution of two cusp forms of GSp{2) against an Eisenstein series of G5'p(4). The location and the order of poles of such a family of Eisenstein series are explicitly determined in Chapter 3. Before doing so, we shall first study Whittaker functions on G5p(2, A). 2.2.1 The Whittaker functions on G5p(2) The properties of Whittaker functions studied here will play a critical role in our proof of the eulerian properties of our global zeta integral in the next subsection. For general description of Whittaker models, see, .J. Shalika [65] or S. Gelbart and F. Shahidi [21]. Let 7T be irreducible admissible automorphic cuspidal representation of GSp{2, A) with trivial central character. Let t/> be a generic character of the standard maximal unipotent subgroup N'^ of G5"p(2, A). Without loss of generality, we may assume that ?/> has form: / 1 X z w 0 0 T 0 ) = V’o(a: + y), (2.9) \0 0 -z 1 / where 0o(^} is a nontrivial additive unitary character on F \ A. We define, for < 5^ G tt, a function (j)* by the following integral 16 where is the one-parameter subgroup attached to the root 2e i, as defined in section 1. The ?/>-Whittaker function associated to = L m W A , (2.11) Then we call the representation tt ^-generic if there exists a function ^ such that ^ 0. In this case, we call ^ ^-generic. The idea to introduce the function (j>* is from Gelbart and Piatetski-Shapiro [20]. Let C2 be the center of GSp{2). Let ^ a b * \ Rx = { c d * G GL{3)}. (2.12) 0 1 y Then we have Lemma 2.2.1 (a) C2Z2 is a normal subgroup of P^; (c) (d) There is a ‘Fourier’ development for (f)*{g) i.e. r i s ) = z » ♦ ( * ) 0eU2(F)\GH2,F) where the group GL{2) is embedded into R\ as in (2.12) and U 2 is the standard unipotent subgroup of GL{ 2). 17 Proof: We can write any p G Pf in form: p — cmn, with c G C2, n E Ni, and / e 0 0 0 0 a 0 6 m = 0 0 10 y 0 c 0 rf (a) is true since Z-z is the center of the unipotent radical Ni of P^ and -I / e 0 0 0 \ / 1 0 z 0 \ / e 0 0 0 \ / 1 0 e"'z 0 \ 0 a 0 6 0 0 0 0 a 0 6 0 0 0 0 0 (2.13) 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 \ 0 c 0 d \ 0 0 0 1 / I 0 c 0 d ) \ 0 0 0 1 / To prove (b), we consider the projection 9 \ P^ R\ /' e 0 0 0 \ / 1 X 0 s \ 0 a 0 6 0 1 .s 0 c d 0 and 0 0 10 0 0 10 0 0 1 ^ 0 c 0 d y \ 0 0 — X 1 where e = ad — be. It is not difficult to check that 9 is surjective and ker{9) — CZ-z- Thus 9 induces an isomorphism CzZ-z \ Pf = R\. The left [C'2Z2](A)-invariance of (j)* is easy to check since the central character of t t is trivial, while the left P]{F)- invariance of ft* can be proved in the following way: For any p = cnm G P^{F), we have since Z-z is the center of N\. By means of identity (2.12) and the fact that f) is auto morphic, we conjugate the integrating variable 2 by m change the variable: e~^z z, and then we have We are now going to prove (d). Let us denote y>(r) = f>*{9~^{r)g), which is a cusp / 1 y 8 \ / 1 !/ ^ \ form on Jf2i(A), and let U 3 = { I 0 i x | G i2i} and 0 1 x 1= i/)o(;c -f- y). It \ooi/ \ 0 0 1/ is evident that tp is i/>-generic if f) is. Since the following isomorphisms of varieties CzN'^XPj = Us\R\ = Uz\GL{2), it is well known that the generic cusp form y can 18 be recovered by its ?/>-Whittaker functions, that is, ip{r) = ^ ( 2 .1 4 ) fieU2{F)\GL(2,F) where vAA — / (p(nr)il;{n)dn. Ju3(F)\U2(A) On the other hand, the restriction of 6 to the unipotent subgroup N'^ gives us a central extension of unipotent groups 1 ^2 —¥ Uz \ and as algebraic varieties = Z2 ><■ U3. Therefore we have jV^(A) = Za(A) x [/ 3(A) and N\F) = ^z(F) x Uz{F), and for ^ e [Uz{F) \ Rx{F)] U2{F) \ GL{2, F)), JU3(F)\U3(A) = f (j)*{e~^{nfi)g)^[n)dn Ju3(F)\U3(X) = f f (n/))(y)^(z^-^ (n)dzdn Ju3(F)\U3(A)Jz2(F)\Z2(A) = w * m - Now let r = 1, we obtain the ‘Fourier’ expansion for the function (^*, i.e. Aid) = y (i) = v’V'(^) H^U2(F)\GL(2,F) E /36l/2(F)\C,X(2,F) The proof is finished. □ 19 2.2.2 The global zeta-integral The global integral we axe going to establish is a Rankin-Selberg convolution of two cusp forms of GSp{2) against the Eisenstein series of G5"p(4). We state the basic properties of such Eisenstein series, the proof of which will be given in Chapter IV. Let P3 be the maximal parabolic subgroup of G = G5p(4) as in subsection 2.1.1 and a(p) = det(a). The modulus character S = 5p* will be J(p) = |a(p)|®|xiX 4 - X2Xz\~^. (2.15) Let 7^4 = be the maximal compact subgroup of G(A) with the standard maximal compact subgroup of G(P,) in the usual sense for each place v of F. Let 5a{p) : = S^{p). Then J, can be extended to be a character of P 3 (A) which is trivial over 7^(P)1V^(A). We denote by 7^(s) = degenerate iirincipal series representation of G(A), which consists of smooth functions /(•, s) : G(A) -> C satisfying the following condition: f{pg,s) = ^i(p) G(A). The group action is given by right translation. By smoothness of the function / we mean that the function / is locally constant as a function over the non-archimedean local variables and smooth in the usual sense as a function over the archimedean local variables. A section f{g,s) is called holomorphic or entire if the section f{g,s), as a function of one complex variable .s, is holomorphic or entire. We also assume that sections f{g,s) is right A-finite. This implies that 7j(s) is, in fact, a representation of (goo, Koo) X G(A/), where goo is the Lie algebra of Goo and A / indicates the finite adeles. 20 We define, for any section fs = /(-,s) E /K s), an Eisenstein series as follows: E3{g,s;fs)= fsiig)- (2.16) 'reP*{F)\G(F) The normalization E3 *{g, s\ fa) of the Eisenstein series E^{g, .s; fg) by the normalizing factor df<{s) is defined as ^'*((7, a ;/.) = (2.17) where d^(.s) = n«g.sCu(2.s + 2)Cu(s + l)(«(s + 2)(«(.s + 3) and the finite setS is determined by the ramification of the section fa, see Chapter IV. We state below the basic properties of E3 *{g,s-, fg), which will be proved in section 4.5, Chapter IV. Theorem 2.2.2 For any holomorphic section f{-,s) G /^(.s), we have (a) E 3{g,s,] fa) converges absolutely for Re(s) large and has a meromorphic con tinuation to the whole complex plane with finitely many poles, and satisfies a functional equation which relates its value at s to its value at —s [47] and [ 2]. (b) The normalized Eisenstein series E 3 *{g,s,', fg) is holomorphic except fors in Xz = {±1,±2, ±3}, has at most a simple pole at s = - 2 , - 3 and at m,ost a double pole at s = —1, and achieves a simple pole at .s = 2,3 and a double pole at s = 1. Let 7Ti,7r2 be irreducible automorphic cuspidal representations of GSp{2,A) with trivial central characters. Then an element in an irreducible automorphic cuspidal 21 representation tt will be a cusp form [ 1 1 ] . Recall that H = {GSp{2) x GSp{2))°. We define, for (j)\ E TTi; E Tg, our (unnormalized) global integral as follows: Pi), s;fs)M9i)M92)dgidg2 (2.18) where C = C4 is the center of GSp{4:). The (normalized) global integral is defined as Z*{s,d>\,(l>2ifs) = da{s) • Z{s,(l>i,(j>2,fs). (2.19) The Haar measure dg\dg2 is canonically chosen as [25] and [24]. Theorem 2.2.3 (a) If (f)i,(j>2 are cusp forms in TT], TTg, resp., then the global integral Z*{s,^-[,(f) 2, fs) converges absolutely for all complex value s except possibly for those finitely many values of s at which the normalized Eisenstein series E^'*{g, s; fs) may achieve a pole, and satisfies a functional equation which relates the value s to the value —s. (b) If ft and I.e. Z*{s,])x, where T>{A) denotes C{A)N'^'^{A){Z 2 x l 4){A)\H{A) and N'^’^ is the diagonal embedding of the standard maximal unipotent subgroup N'^ of GSp{2) into H. 22 Proof: The first part of (a) follows since cusp forms are rapidly decreasing and Eisen stein series are slowly increasing when they are restricted on an appropriate Siegel domain, and the second part of (a) follows from the corresponding functional equa tion of the Eisenstein series. Fix from now on such a value of a where the integral is absolutely convergent. It is enough to prove statement (b) for the unnormalized integral Z(a, Unfolding the Eisenstein series, we have where ffj = OH is the stabilizer in H of 70 or Lojo- As in Piatetski-Shapiro and Rallis [54], the integral will vanish over the negligible ÆT-orbits Pf^oH. Therefore our global integral become: 2 {s,(j)^,(j)2,f) = f fi'ro{g\,g2))M9^)M92)dgidg2 (2.20) where Pf'ioH is the unique nonnegligible £T-orbit, whose stabilizer is P ^{F ){Z 2 x I4). Applying the function = L)A'.A,n,&xw,A,\«,A, here we have used in the first step of the computation the condition that <^2 is auto morphic and in the last step the fact that P( is in the stabilizer of 70. Continuing our computation, the last integral equals = /( 7 o ( <7i, <72 ) ) (<7i ) (fif 2 )d g x d g ^. The î/)-Whittaker function of ^2 is defined by Wlig-z) = [ Mug2yKu)dudgxdg2. Thus we obtain that = / y,(7.Wi,172))<(«7,)W^(«72X^7i((<72. (2.21) Multiplying both sides by the normalizing factor dg(.s), we achieve what we want to prove. □ 24 Now, if we choose the additive characters ?/> and and (j>\ G E TTg, and / G Iz{s) all decomposable, that is, ij) = and t/) = and / = and 02 = where (g)' denotes the restricted tensor product, then, by the uniqueness of the local and global WhittaJker models, and the uniqueness of the restricted tensor decomposition of see [65] and [21], we obtain the eulerian property of our Rankin-Selberg global integral. Corollary 2.2.4 The global integral can he decomposed into an Euler product: Z(@,0 i , 02,/ ) = n l,^,fv{7o{gu92),s)W}[l{gi)W}^^{g2)dgidg2, where 0 = ®u0 «, and 0 = (g)„0 „ are generic characters of the standard maximal unipotent subgroup N'^ of GSp{2,A). Here we have assumed that for each place v, the local integral in the infinite product converges absolutely for Re(.s) >> 0. The identity in the Corollary holds for Re{s) large and then for all value of s by meromorphic continuation. The absolute convergency and the meromorphic continuation of such local integrals will be given in Chapter III. Let us introduce a notation for the local zeta integral. For W\„ G W (7roo, 0), W2v e W (7T6, 0), and G we define 2v{s,Wu,W2v,fv) = [ fvilo{gi,g2),s)Wx„{gi)W2v{g2)dg^dg2. (2.22) JV(Fv) Then we have 2(.s, 0i, 02, / ) = ,  )- We call 2„(s, IFi„, /«) the (unnormalized) local zeta integral. The (normalized) local zeta integral is defined 25 as Z:(s, Wz., /«) := 4,c(a) - /«), (2.23) where d„^a{s) is the u-factor of (Iq{s) as in p. 20. 2.3 Unramified Local Zeta Integrals The ‘unrainified computation’ in the Rankin-Selberg method is to identify the local zeta integral with the local Langlands factor at any ‘unramified’ finite place v of F. In this section, we shall evaluate the normalized local zeta integral 2*{s, Wi„, W^u, /„) with unramified data [W\vi /«, 0u)- This will be done by computations of certain partial ‘Fourier transform’ of the unramified f° and application of Casselman and Shalika’s formula for unramified Whittaker functions [17]. Let Fu be the local completion of F at the local finite place v and the ring of integers in F„. Let = G5'p(4, d„) and K2,v = G 5p(2, 0„) be the chosen maximal compact subgroup in G5p(4, jP„) and GSp{2,Fv), respectively. Then (A'a,,, x A 2,«)° is the standard maximal compact subgroup in H{Fv) = {GSp{2) x GSp{2))°{F„)^ and (ATa.t) X A'2,«)‘’ ^ K 4,v under the canonical embedding of H{Fv) into G(F„). Let 7Ti„ and 7T2u be irreducible admissible representations of GSp{2,Fv) ( u-component of irreducible automorphic representations tti and 7T2, resp.) and and 00 are generic character of the standard maximal unipotent subgroup N'^ of GSp{2,Fv) induced canonically by 0 and 0 . The integral we are going to study is .2„(.s, Wi,,, W 2„,/„) as in (2.22). We will 26 define an unramified local finite place of F for our local zeta integral as follows: Definition 2.3.1 A local finite place v of F is called unramified if the following hold (a) The local component and tt 2,v of the automorphic representations tti and tt 2 are unramified, or of class one, or right K 2,v-spherical; (b) The local component of the nontrivial additive unitary character 'i!>o on F \A is unramified, that is, the conductor o/V’o.u is Ou ït is well known that almost all local finite places are unramified. At any unram ified place V, w;e can pick up unramified integration data (Wj°u, ^2,vi /u)> where f° is the unique normalized right 7'£!'4,„-spherical section in so that /°(1) = 1, and Whittaker functions W°„ and are right «-spherical and normalized so that W°„(l) = 1 and = 1. The local zeta integral with the unramified integration data is called the unramified local zeta integral. For simplification of notations, we will drop v from the subscripts of our notations if no serious confusion will occur. 2.3.1 Reduction We will reduce our computation of the unramified local zeta integral to that of certain ‘Fourier coefficient’ of the unramified section f°. To this end, we consider the Iwasawa decomposition of the reductive group H{F), that is, H(F) = (iV2 X N^){T2 X T2)°{K2 X K2T, 27 where Tg is the standard maximal split torus in GSp{2,F), so that N^T-zK-z is the Iwasawa decomposition of the reductive group GSp{2, F). Then the domain of our local integration has the following decomposition: V = [CN^'^iZz X 74 ) ] = [Zz\N^][C\(Tz x Tzy]{Kz x I For our chosen unramified data (^°, <^21 /«)) have the following Lemma 2.3.2 The (unnormalized) unramified local zeta integral (2.25) JC\(T2XT2)'> JZ2\N^ where Sh is the modulus character of the standard Borel subgroup of H (restricted to C\{TzX TzY). Proof: By the Iwasawa decomposition, the element {gi,gz) E [CN'^’^{Zz x l4)]\H can be written as (gi,gz) = {utik\,tzkz). Then we have that f°{'fo{utiki,tzkz)) = f°{'yo{ut-i,tz)), and W°{utiki) = ?/>(u)W°(ti) and W-^itzkz) = W^itz)- Thus the unnormalized, unramified local zeta integral Z(s, ) equals = [ ^i°(*i)W^ 2 (^2) [ / f°{'yo{uti,tz)fi(>iu)du]6jf{tx,tz)dtidtz. ./C\(T2XT2)° JZ2\N^ The lemma is proved. □ From this lemma, it suffices to compute the integral I{U,tz\f°,^l^)= I f°{'yo{utx,tz))iKu)du. (2.26) JZ2\N^ 28 This will be carried out in the next subsection. We conclude this subsection with results on Lemma 2.3.3 The modulus character = SGSp(2){tl)-5GSp(2){t2) = \a\'-^b'-'^\ ■ \c^d~^d'-\ = h{a,b,a',b') x h(c,d,c',d') e Tg x Tg. Proof; By the definition of the modulus character, we have = \det[Ad{ti,t2)\n]\ = ldef[Ad(ii)|nJ| • \det[Ad{t2)\n:,]\ = ^GSp(2){U) ■ SGSp{2)ih), where n is the standard maximal nilpotent subalgebra of the Lie algebra of H and Ha is that of the Lie algebra of GS'p(2). Then the result follows from some standard computations. □ Note that (Tg x Ta)° = {h{a,b,a',b') x h{c,d,c',d') : aa' = bb' = cc' = dd' ^ 0}. we can choose parameters so that C \ (Ta X 3a)° = {A(a6, u, 1) x A(cd, c, : abcd^Q). (2.27) From now on, we will fix this parameter system on C \ (Ta X 2a)° and have C orollary 2.3.4 //(ti,ia ) E C \ (Ta x Ta)°, then ^//( 2.3.2 Computation of V’) The computation of will be reduced to that of a partial ‘Fourier trans form’ of certain Schwartz-Bruhat function over F^. To this end, we need several lemmas. Let %«(<) denote a one-parameter subgroup of G corresponding to a root a and X{w,y,x,u,v) = - = 4 («)- (2.28) Lemma 2.3.5 Let f° he the spherical section chosen above. Then for u E Zg \ N'^ and (ti,<2) E C \ (Tg X Ta)”, we have /° ( 7o(uti, ia)) = -y , x, 1, l)h{ab, a, c"’a, c~'^d~^a, b ~^, 1, c, cd)). Proof: As mentioned before, for (ti, t-f) E C\{T-2 X Ta)®, we can rewrite it as, under the embedding H G, (ti, fa) = h{ab, a, cd, c, 1, c~^d~^a, c~^a). Then its conjugation 7o(*i,<2)7o"’ by 7o is X -n—4 (l)X-.,-e,(l)A(o&, o, c a,c d a,b ,l,c,cd)%_:,_,,(-l)x_,,_,X-l). Similarly, for (u, 1) G iV^ X I4, its conjugation 7o(u, 1)77^ by 70 is equal to f i x —w —z — wx z w \ 1 —y -w w' y 1 I x{-w,-y,x,0,0) 1 —X 1 w' y 1 —z — wx Z W 1 J = piw,x,y,z)x{-w,-y,x,0,0) (2.29) 30 with p{w,x,y,z) € P3 . Hence we have, since f° is spherical, that riioiutx,t2),s) = r(7 o (u , \ a) = -y , 1, l)h{ab, a, c~^a, c~^d~^a, 6" \ 1, c, cd)). Lemma 2.3.6 For convenience, we set x(w) = %(w, 0,0,0,0), %(w, y) = %(w, y, 0,0,0), X(x) = x (0 ,0, X, 0,0), • • •, x(i^) = x(0) 0,0,0, v), and so on. Then we have following identities: (a) h{t\ ,t' 2,t^,t^,t^,tQ,ty ,t = x(Wg^t«, Wg^x, titg^u, (b) X-iciiv) = h{l, 1, -y-\ 1,1,1, -y, l)x2cAy)K> for \v\ > 1/ (c) X-^i-<^A^) = /i(l,-x"’,l,-x~\l,-x,l,-x,l)xe 2+54(*)^x, for |x| > 1; (d) X -ei-,4(«) = /i(-u"\l,l,-u~ ’,-n, l,l,-u)Xï,+e4(î^)^u. for |u| > 1; (e) X-,:-„(w) = h{\, - v ~ \ 1 ,- V , -v , l)Xc2+=,(^)L, for |v| > 1; (f) x{w,y)xc2+^4i^) = x,2-:3(-wx)x=2+,4(:r)x(u;,y); (y) x {w)x 2 M = Xc2-cA-wy)x2Mx-2cA-w^y)x{wy, (h) x(u),%/,x)x.:+.Xu) = X.._% (-zw)Xe, -e, (-mu)xe, +,4 («)x("), 3:); (i) x(w, y, i)x=2+q(v) = Pi (u', y, a;, «)x(w, y, x); (3) x{w,y,x,u)Xe2+e3{v) = P2{w, y, X, v)x{w - xyv,y,x,u), 31 where ky,kx,ku,K G K4 and pi{w,y,x,v),p 2{w,y,x,v) 6 o-nd also = h*{P 2{w,y,x,v)) = 1. The proof of the lemma is straightforward. The integral as defined in (2.26) can be reduced as follows. Proposition 2.3.7 Let S2 = S^(h(ab,a,b~^,l) x h(cd,c,c~^d~^a,c~^a)) = |6'^c’d^|. Then the integral I{tj,t 2; fs,' J 2(|a |2|i)||c|"^}®'''^ / f°{x{w,y,x,abc~^d~^,ac~^))tl’o{a~^cdx + a~^c^y)dwdxdy. Jf ^ Proof: Let h^{a,b,c,d) = h{ab,a,c~^a,c~^d~^a,b~^,l,c,cd). Since 7o(uti,t2)7r' i* equal to x{w,y,x,l,l)hi{a,b,c,d)x- 6i-:t{-l-)x-c2-^i{-'^) and x{w, y, X, 1, l)-^i(a, b, c, d) = A,(a, b, c, d)x{ac'~^d~^w, ac~^y, ac~^d~^x, abc~^d~^,ac~^), we have /;( 7o(ut,,f 2)) = |o=6c-^|'+=|a|-:('+=) (2.30) • y, (x(oc"^d"^u),ac"^y, ac~^d~^x, abc~^d~^ ,ac~^)). Thus we can deduce our integral I{t\^t2\ /°, ?/>) as follows: / . /s (7o(wii,t2))V’(n)du JZ2\N^ = |a^6c“M®+®|art(»+3) • J^^f°{x{ac ^d ^w,ac ^y,ac V ^x,abc ^d \ a c ^))%l)o{xy)dwdxdy = f^(|o|#|6||c |-')'+ ' • / f°{x{w,y,x,abc~^d~^,ac~^)) 7jjo{a~^cdx + a~^c^y)dwdxdy, (2.31) Jf2 32 here we change the variables by ac »->• w; ac t-4- y; and ac ’x i-> x. □ Now we have to compute an integral of following type: I{u,v,a,^-,f°,il;)= f f°{x{w,y,x,u,v)yi}o{ax + ^y)dwdxdy. (2.32) Lemma 2.3.8 For the normalized unramified section f° in 7g^(.s) and the unramified additive unitary character %l>, the integral I{u,v, a, j3', f°,i/jo) equals Sf ^ f!ix{w,y,x))il;o{ax + fiy)dwdxdy i f |u|, |u| < 1; fps f°{x{w,y,x))'^oi-vax + v'^fiy)dwdxdy i f juj < 1, |u| > 1; fpi f°{x{‘W,y,x))TlJo{-uax + /3y)dwdxdy i f jnj > l,|u| < 1; fpa f°{x{w,y,x))il^o{uvax + v^(5y)dwdxdy i f juj > 1, juj > 1. Proof: We will evaluate the integral 7(u, v, a, fi; fa, t/’) case by case. (1) If |u| < 1 and |u| < 1, then there is nothing to do with because by definition X{w,y,x,u,v) = x{w,y,x)x{u,v) and x(u,u) G I<4- (2) If |u| < 1 and |u| > 1, then by Lemma 2.3.6 (e) and (i), fsix{w,y,x,u,v)) = fsix{w,y,x,v)) = - v ~ \ 1,1, -V, -V, l)x(-v~^w, v~^y, Hence the integral can be deduced as I{u,v,a,fi\f°,tl)) = I f°{x(w,y,x)fil;oi-vax+ v^fiy)dwdxdy, J 33 here the variables are changed by setting —v~^w i->- w,v~^y t-4- y, and —v~^x i-4 x. (3) If |w| > 1 and |u| < 1, then by Lemma 2.3.6 (d) and (h), fsix(^,y,x,u,v)) = f°{x{w,y,x,u)) = 1, 1, - u ~ \ - u , 1, 1, y, -U~^x)Xe2+C3{u)k{u)) = \u\~‘~^f°ixi-u~^w,y,-u-^x)xs2+sAu)) = \u\~"~^f°ix{-u~^w,y,-u~^x)). Hence the integral has form: I{u,v,a,^]f°,^l)) = f°{x{w,y,x))il)o{-uax + (iy)dwdxdy, here the variables are changed by setting —u~^w w,y y, and —u~^x i-)- x (4) If |u| > 1 and |n| > 1, then by Lemma 2.3.6 (d), (e), (h), (i), and (j), we deal with the variable v first and then with u and obtain that fsix{w,y,x,u,v)) = / s (^ (li 1,1, - w , -V , l)x(-v~'^fx,v~'^y, = v~'^y, -v~^x, «)%„+„(«)) = + v~"^yx, v~'^y, -v~^x, u)) Hence the integral is equal to I{u,v,a,fi-,f°,tl)) = / f°{x{w,y,x))dioiuvax + v^/Sy)dwdxdy, JF^ 34 here the variables are changed by setting (uv)~^w (->• w, v~'^y t-4- y, and {uv)~^x i-> x. □ According to the above reductions, we have to compute the ‘Fourier transform’ of function f°{x{w, y, x)) with respect to additive characters ?/>o(o:a; + fiy) for a, fS £ that is, the integral as follows: I{a ,/3 ):= f f°{x{w,y,x))^l)o{ax +/Sy)dwdxdy. (2.33) This is done in the following lemmas. Lemma 2.3.9 The integral I{a,^) can reduced to a product of several one-variable integrations as follows: l{a , 0 ) = / f°{x{w,y,x))%l)o{ctx-\-jdy)dwdxdy J = [1 + / • [ / ij)o{oix)dx + f |a;|“*“''^ï/>o(cvx)fZ.c ./|u>|>l V|z| ■{[. M M ^ y + [ . Proof: If I a; I < 1, then /(» ,/)) = I f°{xiw,y,x))rpo{ax + /3y)dwdxdy JF^,\x\<ï = I f°{x(w,y))tljo{ax + /3y)dwdxdy JF^,\x\<\ = I il)o{oix)dx I f°{x{w,y))%l}o{fiy)dwdxdy. ./|x |< l jf ^ If |æ| > 1, then by Lemma 2.3.6 (f), X{w,y,x) = h{\,-x~^,l,-x~^,l,-x,\,-x)x{-x~'^w,y)xe 2+cA^) = h[l, -a ;"’, 1, - a ; - \ 1, -ar, 1, -x)p{w, x)x{-x~'^w, y). 35 where p(w, x) = Xea-es (~'«^a;)Xe2 + £ 4 (®) belongs to P^. Hence we have f°{x{w, y, x)) = lxj~^~^f°(x(—x~^w,y)) and the integral I{a,^) equals I f (%(w, y, x))ipo(ax + fiy)dwdxdy JF^,\x\>\ = / \x\~^~^f°{x{-x~'^w,y))il)o{ocx + py)dwdxdy JF^,\x\>l = / \x\~^~'\l}a{ax)dx [ f°{x{w,y))xl^o{^y)dwdy, where we change the variables by —x~^w t-4- w,x t-)- x, and y y. Therefore we deduce that the integral l{a,0) is equal to ^ 3 f (x(w,y,z))^o(ai + /3y)dwdxdy = [f V’o(a!x)d.T+ / \x\~^~'^i/}o{oix)dx] f f°{x{w,y))il)o{fSy)dwdy. J|a:| In a similar way, we can integrate the other two variable w, y f°{x{w,y))MMdwdy = [ / , M M d y + I \y\~^~'^ipoi/^y)dy] [ f°{x{w))dw J\y\<^ J\y\>\ JF = [ / , M M d y + I \y\~'~'^MMdy][l dw+[ The proof is finished. □ Lemma 2.3.10 Let q be the cardinality of the residue class field of F . The one- variable integrals in the last lemma can be evaluated as follows: N 1 + /h > i = (1 - 36 0 i f |a| > 1, = { (1 - i f |q;| < 1 Proof: The computation to establish those formulas is standard and is omitted here. □ We are finally able to complete our computation of the integral 7 ( t],t2; / « ,’/’) as in (2.26) and state it in the following theorem. Theorem 2.3.11 Let m\ = ord„(a), m 2 = ordv{b), n\ = ordu{c), andri 2 = ovdv{d). LetS^ = 5fj{h{ab,a,b~^ ,1) X h{cd,c,c~^d~^a,c~^a)) = Ih^c^d^l andt = —{s + 1). The integral I{ti,t 2’, f°ii}) equals a product 0/ ; ^ 2) ^ +"2)(^ _ _ ç(2iii-mi+l)i j ^y mi+m2 Proof: By Proposition 2.3.7 , one has that fti^k) = ^ ^(|a|^ |6||c|"^)*‘''V (a 6c " 'd " ’, a c ~ \ o“ ’cd, a"V^; /° , ?/>o) Then, by Lemma 2.3.8 , the integral I{abc~^d~^ ,ac~^,a~^cd,a~^c^; f°,i/:o) can be re duced as follows: (1) If mi -F m-2 > Ml + ri2 and mi > ug, then I{abc~^d ~^, ac~^ ,a~^cd, a~^(?\ f°, i/>o) = I{a~^cd, a~^c^). 37 (2) If mi + m2 > «1 + ri2 and mi < «2, then I{abc~^d~^,ac~^,a~^cd,a~^c^-,f°,il;o) — d, a). (3) If mi + m2 < n-i +U 2 and mi > n2, then I{abc-^d-\ac-\a-^cd,a~^c‘-,f°,il}o) = a"^c'^). (4) If mi + m2 < «1 + ri2 and mi < «2, then I{abc-^d~\ac-^ , fl-’cd, 0 - 7 ^; / ; , 1A0) = g( 3n.+n2- 3m, ^ g). Now the theorem follows from Lemma 2.3.9 and Lemma 2.3.10. □ 2.3.3 The Unramified Local Zeta Integral With the preparation of last two subsections, we can complete our computation of the local zeta integral. To this end, we first recall the formula of Casselman-Shalika for the unramified Whittaker functions, which works for any quasi-split reductive groups over a p-adic field [17]. For our special case, such a formula can also be found in Bump’s survey [12]. Theorem 2.3.12 (Casselman-Shalika[17j) For the unramified Whittaker function W° € W (7T„), W°(h(ab,a,b~^,l)) vanishes if ord„{a) < 0 or œ'd„(b) < 0 and also W°(/i(a6, a, 6“\l)) = a, l))fr(ord„(a), ordu( 6)), (2.34) 38 where tr{m, n) is the trace of the representation p{m,n) evaluating at the conjugacy class of semisimple elements in GSp(2,C), the L-group of GSp{2)[7j, which is de termined by 7T„ via Satake isomorphism and /9(m,t») is the irreducible representation of GSp{2, C) with highest weight {m,n) = msi + n(ei + £2), m ,n > 0. Theorem 2.3.13 Let X = k\ = ordu{a), k -2 = ordv{b), l\ = ord^{c), and li = ordv{d), with ki,k 2,h ,h ^ 0. Then the (normalized) unramifled local zeta integral Z*v{s, W.^„,f°) = d^,G{s)Z^{s, f°) equals 1 (1 - X 2) ( I ^ X4) • * 1 + * 2 > < I + '2 > * I ^ / + £ {k„ h \ 2h - k , , ~ fc]+fc2 > ( i + l 2 ) + £ (2.36) * l+ fc 2 < ‘l + '2 V-*- ^ ) 2 l i > * i > + £ ( t „ b |2i, - ~ ~ il< fc ,+ fc 2 < i i + i 2 V-*- ) k\ Proof: The unnormalized local zeta-integral is JC\{T2XT2j° 39 Applying the formula of Casselman-Shalika for the case that t\ = h{ah, a, 1) and t'i = h{cd, c, c~^d~^a, c~^a), we obtain that the nonvanishing of c, c~^d~^a, c implies that 2/i — kijz > 0 and the nonvanishing of implies that k-[, A’2 > 0 , and also W ° M = = \c^d‘^a-^\^^Hr{2h - k,,h). Thus we obtain that Note that Cv{s) = (1 — . Applying these data and Theorem 2.3.12 to our (unnormalized) unramihed local zeta integral, we will get what we needed. □ We are now going to state our main theorem on the unrainified computation, which equates our local unramified zeta integral to the local Langlands X-factor. Theorem 2.3.14 Let s' = (.s -f- l)/2 . Let p\ he the irreducible representation of GSp{2, C) with highest weight e-i . Then 'Z»(a, W°„, W2„, f°) = X„(.s', wu CS) 7T2„, (m (g) pi )„) Note that the local Langlands L-factor X„(.s', 7Ti„ 0 7T2„, {pi 0 p\)v) is of degree 16. By the definition of local Langlands L-factor, 1 Xu(.S , TTiu CX) 7T2«, (Pi ^P\ }t)} — det[l - (pi 0 T2„)X] 40 where X = q~^' and Ti„, T2„ are representatives in seniisimple conjugacy class in ^GSp{2,Fv) determined by 7Ti„, respectively. Since an element in the maximal split torus T2 can be written as 22,(3, 24) = Æ(2i , (2, 2J"\2.j')A(1, l , 2i(3, 22(4) and pi(fe( 2i , 22, 23, 24)) = /9i(fe(2i , 22, 2J"\2.J^)), see Proposition 2.1.1, we can assume that pi is the irreducible representation of 5^(2, C) with highest weight £1 and T],„, T2,» are the associated semisimple conjugacy classes in 5^(2, F„). This theorem will be proved in the next section. 2.4 Local Langlands Factor of Degree 16 for GSp(2) x GSp{2) We are going to study the 16 degree local Langlands Z-factor 7ri„(8 )7r2„, (pi 0 pi ),,) by means of the representation theory of the complex dual group (G5p(2,C) x GSp{2,C))°. As explained in Proposition 2.1.1, it is actually reduced to the rep resentation theory of Sp{2, C) x Sp{2, C). The method we will apply is the Kostant- Rallis’ algorithm about the explicit spectral decomposition of the space of harmonic polynomial functions over a symmetric space described in [40]. We will recall first the algorithm of Kostant-Rallis in general, and then apply it to the computation of our local Langlands Z-factor. Let us mention that the normality of some nilpotent orbits is a critical sufficient condition to mahe the Kostant-Rallis algorithm work. However, it is usually not an easy job to check the normality of a nilpotent orbit in the case of symmetric spaces. Instead of checking the normality of the relevant nilpotent orbit, we will use branching formula to make the Kostant-Rallis algorithm work in our special case. 41 2.4.1 Kostant-Rallis Algorithm We will recall some notions from Kostant-Rallis [40] and J. Sekiguchi [61], and restate their main results as a theorem. Let g be a complex Lie algebra and Û a complex linear involution of g. Then we have as usual the following decomposition as linear spaces: g = t -f p, where t = {;c G g : û(x) = z} and p = {z G g : ff(x) = —z}. In this way, we obtain a (complex) symmetric pair (g, t) in terminology of Sekiguchi [61]. The subspace p is called the vector space associated to the symmetric pair. Let G be the adjoint group of g and li' the analytic subgroup of G corresponding to t. Let K& = G G : ff(ÿ) = ff} (the involution ^ of a Lie algebra will induce an involution of the adjoint group, which is denoted by the same notation Û). According to Lemma 1.1 in [61], one has that coincides with the normalizer of K in G. It follows from the decomposition g = t 4- p that p has a structure of Kg-module by the adjoint action. Naturally, we have the coadjoint representation of A'g on S the ring of all polynomial functions in p, which is defined as follows: for any g G Kg and f G S, and any z G p, ( Theorem 2.4.1 (Kostant-Rallis) Let z be a regular semisimple element in p and e a regular nilpotent element in p. Let Zo be any element in t such that [zo,e] = e. 42 Let r he the set of all equivalence classes of rational irreducible representations of Ke and V~! the space of the representation 7 for 7 G F. Let Og he the Ke-orhit of z in p and R{0.) the algebra of all everywhere defined rational functions on 0~. Let S he the ring of all polynomial functions on p. Then (a) ‘Separation of Variables’ s = «9^'0 n where is the suhalgehra of Kg-invariants in S and H is the space of all Ke-harmonic polynomials on p. (h) the restriction map ro. : / H- f\o. gives a Ke-isomorphism U -> R{Og). (c) ‘Multiplicity % = with multu{')) = = ^(7 ) 76r where Kg = StahKg{z), is the suhspace of all Kg-fixed vectors in Vy, and 'Hy is the isotropic suhspace of 7 , that is, the set of all functions in H which transform under K$ according to the representation 7 . (d) ‘Homogeneity’: The isotropic suhspace can he described as '(7) Tly = ^ 1 = 1 where TLy^i is, for i = 1,2, " 5/(7 ), an irreducible Kg-module in Hy consisting of hom.ogeneous functions of degree di('y), corresponding to a unique monotonie sequence of increasing nonnegative integers di{'y), i = 1, 2, • • •, /(7 ). 43 (e) For'j E F, there is axo-stable l{'y)-dimensional subspace V^(e) C C Vy such that Xo is diagonalizable on Vy{e) and the eigenvalues are exactly the nonnegative integers di{'y), z = 1, 2, • • •, l{'y), defined above. In general, for a regular element e 6 p, the subspace Vy{e) is contained in the subspace Vy ®, and those two subspaces may not be equal. It is not difficult to check that V^(e) = if the JF^-orbit of e is a normal algebraic variety [40]. 2.4.2 Symmetric Space (Sp{4), Sp{2) x Sp(2)) In this subsection, we consider the symplectic group G = Sp{4, C) and an involution / /2 0 0 0 \ _ 9 defined by conjugation of the element ?? = I “ o I G, that is, 9{g) = \ 0 0 0 ~/2 / T]gT}~^ for g E G. Then the subgroup Ke — {g E 5p(4) : 6{g) = g) is in form: X3 X4 0^ ) satisfying the conditions below: for X,-,î = 1,2,3,4, 0“ n 0^ n / -‘X3X1 + = 0; + '% - 2X4 = 0; and - ‘X3X2 + 'X1X4 = 1, and similarly for Yi, i = 1, 2,3,4. Proposition 2.4.2 There is an isomorphism between Ke and Sp{2) x Sp{2) given by Xi 0 X2 0 \ 0 Yi 0 -Y2 \ (Xs 0 X4 0 I 0 -V b 0 Yi / On the level of Lie algebras, correspondingly, the involution 9 gives following decomposition of sp(4) the complex Lie algebra of 5p(4) sp(4) = t + p, (2.36) 44 with [t,p] C p. Under the adjoint action, the vector space p has a ATg-modnle structure, which can be described as follows. Proposition 2.4.3 Let p\ be the ^-dimensional standard complex representation of Sp{2). Then one has the following isomorphism of Ke-modules: P = m®m- (2.37) Let ( 0 .4i 0 0 \ '^*0 0 - ° i I • Ai =diag(a,b)}. ( 2.38 ) 0 0 -/I] 0 / Then a is a maximal abelian subalgebra of sp(4) in p. This implies that our symmetric space (5p(4), Sp{2) x 5'p(2)) is of rank two. Since Ke is a semisimple algebraic group, according to the Invariant Theory, 5^® is a polynomial algebra generated by two homogeneous polynomials u\ and u-2 with degree 2 and 4, respectively. Then we can construct a fibration u : p -)• C'^, u{x) = {u\{x)^U2{x)) According to Kostant and Rallis, u is equidimensional, each fiber ^ G is A"g-stable and there is a unique maximal-dimensional ATg-orbit in each fiber which consists of regular elements of sp(4) in p. Note that the nilpotent variety (0) may not be normal as mentioned in [40]. Let us recall the identity for Poincare series 1 °° det( J - AX) ^ (2.39) 45 where A is an arbitrary square complex matrix, % is a sufficiently small complex variable, and Sym‘{A) is the 1-th symmetric power of A. Let X = Applying the identity to the matrix A = Pi)v{tiv,T2v), we have OO 7Ti„ Cgi 7T2„, (pi ®/ 0i)„) = Y^trace{Sym\{pi ® P\)v{t]„,T2v)))X^. (2.40) 1=0 By the separation of variables in Kostant-Rallis Theorem and the Ke- module struc ture on p, one can easily see that the right hand side of (2.40) is equal to OO OO '^dim{SymgKg)X‘ ■Y^trace{Sym}^{{px 0 /9i)„(ti„, T2„)))A'', /=0 1=0 where Sym^Kg denotes the 1-th symmetric power representation on the polynomial algebra 5^® and Syw}^ denotes the restriction of the 1-th symmetric power represen tation Sym^ to the algebra % of all RTg-harmonic polynomials over p. According to the Invariant Theory, say [56] or [62], one has OO 1 J2dim{Sym^^Kg)X‘ = (2-41) Hence we have Proposition 2.4.4 The local Langlands factor can be expressed as It is crucial to understand trace{Syin^y{{p\ 0 /9i)«(ti„,T 2v)))X‘. Let (mi, m2), (n i,n 2) be the complex representations of 5p(2) with highest weights mi£i -f 771,282 , ni£] + n 2S2, mi > m2 > 0 and ni > U2 > 0 integers. Then we denote by 46 {mi, m 2', Til, U2) the tensor product representation of (mi, m 2) and («1, 712), which is a representation of Ke- Applying the Kostant-Rallis’ algorithm to our special case, we obtain Proposition 2.4.5 The following identity holds 00 '^trace{Symif^{{pi where (mi,7772|7ii7i2) = trace{{mi,m2;ni,n 2){Tu,T2u)) and P{7ni,m2,ni,7i2;X) is a polynomial with non-negative integral coefficients and is in form: P{mi,m2,ni,U2',X) = (2.42) 1=0 and the teTrmpiX^ indicates that the irreducible representation (mi, 7722; 711, 712) of Ke occurs in the Ke-submodule ofK consisting of all homogeneous harmonic polynomdal functions of degree I with multiplicity pi. Following Kostant-Rallis’ algorithm, the polynomial P(77ii, 7712, 7%i, 772; X) is deter mined by the subspace {7ni,m2;7ii,n2){e) for any regular nilpotent element e G p in following way: If Xo is an element in t so that [xo, e] = e, then the Vs are eigenvalues of Xo in the subspace (mi, m 2; 77i, 7*2)(e) and the p/’s are the multiplicity of I, and also Yu-oPl = /((m i,m2;721, 722)) is the multiplicity of the representation (m i, 7722;721, 722) occurring in H. However, the subspace (mi,m2;72i, 722)(e) is abstractly defined in [40]. In contrast, the subspace (mi,m 2; 721, 722)^» is easier to deal with by means of Borel-Weil-Bott Theorem and the Invariant Theory. Our scheme to compute the polynomials P(772i, 7722, 721, 722;X) is: 47 (1) Compute the multiplicity /((mi,m 2;ni,U 2)) by means of the known Branching Formula; (2) Determine the eigenvalues and eigenspaces in (m j, m 2; nj, for an element Xo € t so that [xo, e] = e, by applying Borel-Weil-Bott Theorem and the Invari ant Theory; (3) Prove that (mi,m 2;u i,n 2)(e) = (mi,m 2; 711, 712)^^ and determine the polyno mial P(mi,m,2,n^,n 2',X) in terms of results in (1) and (2). 2.4.3 Branching formula for Sp(2) ^ SL(2) x SL(2) We restate first Branching Rule for so(277 4- 1) j. so(277) from [74] Let Va be the irreducible representation of the complex Lie algebra so(27z -f 1) with highest weight A = (Ai, A2, - " , A„) (with respect to the standard Cartan subalgebra), which are simultaneously integers or half-integers satisfying Ai > A 2 > • • • > A„ > 0, and the irreducible representation of the complex Lie algebra so(27i) with highest weight jj, = (/i],/i2,• • •,/7„) (with respect to the standard Cartan subalgebra), which are simultaneously integers or half-integers satisfying yui > fJ-2 > • •• > Theorem 2.4.6 [74] The following Multiplicity One Branching Rule holds Vx = ^ V , 48 as so{ 2n)-modules, where fi = (/ii,/X2, ■ • • are all dominant weights of so( 2n) such that -^1 > /^i > ^2 > ^2 > • • • > A„_i > //»_i > A„ > |/x„| and all numbers fn are integers or half-integers together with A,-, i = 1,2, • • •, n. Now ,we shall use Zhelobenko’s Branching Rule to prove Theorem 2.4.7 (Branching Rule for 5^(2) I {SL{2) x SL{2))) LetV^,^^,,^) be an irreducible representation of Sp{2,C) with highest weight (n i,n 2), ni,U 2 are integers satisfying rii > ri2 > 0. Let Vm, 0 %»; be an irreducible representation of SL{2,C) x SL{2, C) with highest weight (mi, m2), mi, m2 are integers satisfying mi > 0, m2 > 0. Then the following multiplicity one Branching Rule holds ,»2) = 0 Vh+h <0 Il Xz where all l \ , I2 are simultaneously integers or half-integers together with (ni +U2)/2, (ui - n-i)l2, and satisfy the following inequalities: 1^2! ^ (™i — U2)/2 < /i < (ui + 112)12. Proof: It suffices to consider the same branching rule at level of Lie algebras since the groups involved are simply connected. By accidental isomorphisms of low ranks: sp(2, C) = so(5, C) and sl(2, C) x sl(2, C) = so(4, C), the theorem follows. □ 49 2.4.4 Multiplicity /((mi,m 2 ;n i, 722)) Following Kostant-Rallis’ algorithm, the multiplicity /((m i,m 2; n j,712)) equals the dimension of the subspace (mi, m 2; U i, n 2){z) for any regular element z Ep and also equals dim(mi, m 2; Ui, 712)^^ for any regular semisimple element z in p. Proposition 2.4.8 Let z be a semisimple element in a with a ^ b as (2.39). Then the stabilizer of z in Kg is Kg = Stab^giz) = SL{2, C) x SL{2, C). Proof: Without loss of generality, we assume that z = ( '^ ^ with 1 0 0 -1 t 0 \ 0 -1 / Then the stabilizer Stable(-z) = {& E Kg : kzk~^ = z). The isomorphism follows from easy computations. □ Corollary 2.4.9 The element z with a ^ b is a regular semisimple element in p. Proof: For such a semisimple element z, we have dim 0 . = dimu~^(u(z)). By defini tion in [40], z is regular. □ Let V and W be irreducible representations of Sp{2, C). As we know, SL{2, C) x SL[2, C) Sp(2,C) is determined by ( a\ 0 02 0 0 61 0 —62 \f as 04“0 f X \f 03 A* O M4 ) «3 0 04 0 ^ 0 —bs 0 64 50 and 5'L(2, C) x SL{2, C) = Kg is determined by the diagonal embedding {SL{2, C) X SL{2, C)) A Sp{2, C) x Sp{2, C). Then by the multiplicity one Branching rule for Sp{2) J, SL{2) x SL{2)^ we have V|sL(2)xSL(2) = 0 1 ^ a n d W\sL(2)y.SL(2) = • j where Vi and Wj are irreducible representations of SL{2) x SL{2). Theorem 2.4.10 Let V and W he irreducible representations of $p{2,C). Then 1{V®W) = dim (y 0 = card{{Vi,Wj) : Vi = Wj as SL{2) x SL{2) — modules) (2.43) where (%-, Wj) are such that V (83 Wj are summands o fV 0 W for all possible i and j. Proof: By using the Branching rule and the fact that Vi = Vu ® Vu and Wj = Wji Wj2, where Vu, Wjk are irreducible representations of SL{2), we have y = 0 v:i 0 VS2 (8 (8 :y, 2. •J Thus the SL{2) x S'X(2)-invariant subspace of V (83 ly is {V (83 = ( 0 C83 Wj, (g) •<3 = 0 (%i 0 <8 (% (g, On other hand, for A; = 1,2, 0 otherwise 51 Therefore we prove that dim(y (g) = card{(Vi, Wj) : Vi = Wj as SL{2) x SL(2) — modules}. Corollary 2.4.11 Let ^6 the complex irreducible representation of Sp{2) with highest weight (m i,m 2 ), mi > m 2 > 0 and (m i,m 2 ; » i ,M2 ) = 0 Then /((m i, m 2 ; ni, M2 ) equals the number of the pairs (h^lz) of integers satisfying the following conditions: ( m j ^ ^ ^ ^ ly . (2.44) where U , /a are simultaneously integers or half-integers together with (7 M1 +/M2 )/ 2 , mii — Mi2)/2, (7ii +M 2)/2,aM c/ (Ml — M2)/2. Proof: Using the corresponding between irreducible representations and dominant weights, we will achieve the result above. □ Corollary 2.4.12 Let mi > m 2 >0; Mi > M2 > 0 6 e integers. We have (a) //( m i + m 2 ) > (Ml + M2 ) > (mi — m 2 ) > (mi — M2 ), then /((7Mi,m2;Mi,M2)) = ^(M i + M 2 - m i + m 2 + 2 ) ( m i - 712 + 1 ) ; (b) I f ( 7 M1 + m 2 ) > (Ml + M2 ) > (Ml — M2 ) > (mi — m 2 ), then /((m i, m 2 ; Ml,M2 )) = (m2 + l)(mi - m 2 + 1); 52 (c) If (ni + ng) > (mi + ma) > (mi — ma) > (ni — na), then /((mi,ma;ni,na)) = (ma + l)(ni - na + 1); (d) If [n\ + na) > (mi + ma) > (ni — na) > (mi — ma), then /((m i,m a;ni,na)) = ^(mi + ma - ni 4- »2 + 2)(mi - ma + 1); (e) If [nil — ma) > (ni + na) or (ni — na) > (mi + ma), then Z((mi,ma;ni,na)) = 0. Proof: It follows from solving the inequalities: 2.4.5 Orbit Decomposition on {B x B)\Ke Let B be the standard Borel subgroup of 5p(2, C). We will introduce a special subgroup R and study the orbit decomposition of {B x B)\Kg under the action of R, via the right translation. In particular, we will describe the all Æorbits in (B x B)\Kg with codimension less than two. Proposition 2.4.13 Let e g ^ m p and Xo = j in t with ; ° .)' Then 53 (a) e is a regular nilpotent element in p and Xo is in t such that [xo, e] = e. (b) The stabilizer StabKg(e) = Kg of e in Kg is / < 0 0 0 \ / <-> 0 0 0 \ w <-i 0 0 y < 0 0 { X : x't + wy = xt x' <-' — w «2 —X < ~y \ X y 0 t / \ —x' w 0 <-: / Proof: The stabilizer Kg = {Â; E Kg : kek~^ = e}. Part (b) follows the standard computation of matrices and proposition 6. Since Kg is a 6-dimensional subgroup of Kg, the dimension of the /Cg-orbit of e is equal to the dimension of the zero fiber u“^(0), both of which are 14. Therefore e is regular. □ Let S = {exp{tXo) : < E C}, a one-parameter subgroup in Kg, which is in form: S = {h 2 (e“‘, e~‘, e‘, e‘) x 1, e'^‘, 1)} where h 2 {- • •) denote the diagonal matrix in 5p(2). By setting s = e‘, we have S = { .^ = h 2{s \.s ^,s,s) X h 2{s ^,l,s^,l): s E C^}. (2.45) and 7/p(s) = .gP is a rational character of S where p is an integer. Proposition 2.4.14 (a) S C NfCg{Kg) and (b) SKg is a 7-dimensional subgroup of Kg, i.e. ( s~H 0 0 0 \ ( 0 0 0 \ SK! = { («0 ”* 0 0 y < 0 0 x't-t-wy = xt ’}. sz\ sx' st~^ —sw s^Z2 —s^x s^t —s^y \ .sx sy 0 .st —x' w 0 <“' ) 54 Proposition 2.4.15 Let X = he the one-parameter subgroup associated to the root —ai — «2 o-nd R = XSKg as algebraic variety. Then R can be written as ( s"‘< 0 0 0 \ / s~H-^ 0 0 0 \ , S“ *U) («<)“ ' 0 0 j 2/(00 x'jt+ syw = t~’xi . ‘ z \ æ'i s (“ * - s w I Z2 s^( -s^2/ " +yw = txg ^ »i s y 0 s t / \ X2 to 0 (“ * / and enjoys the following properties: (a) X S is two dimension subgroup of Kg and X S normalizes Kg, (b) R is an 8 -dimensional subgroup of Kg and including Kg as a normal subgroup, Proof: Since S normalizes X as subgroups in Kg, XS is a group. By the previous proposition, S normalizes Kg. It is easy to check that X commutes with the torus part of Kg and normalizes the unipotent part of Kg. Part (a) follows. Part (b) follows from part (a). □ Since R is 8-dimensional subgroup of the complex group Kg = Sp{2) x Sp{2) and the dimension of the flag variety X = {B x B) \ Kg is also 8, the restriction to R. of the action of Kg on the flag variety X has a (Zariski) open dense fZ-orbit. Hence R is a spherical subgroup of of Kg in the terminology of T. Matsuki [50]. Such a open dense i2-orbit is unique as the flag variety is irreducible and the number of all iî-orbits on the flag is finite [50]. By Invariant Theory, all of those codimension-one fZ-orbits are on the boundary of the unique open i?-orbit. We are going to find out the unique open /2-orbit and all codimension-one iî- orbits by means of the decomposition of Bruhat cells. Let B~ be the Borel subgroup opposite to the standard one. Then {B x B){B~ x B~) is the open cell in the Bruhat decomposition of 5p(2, C) x 5p(2, C) with respect to the standard Borel 55 subgroup B X B, and {B x B)(Z, x a;«,)(5“ x B~), {B x B){l4 x Ua^){B~ x B~), {B X B){ùJon X Ia){B~ X B “), and (5 x 5)(w«2 x Ia){B~ x 5 “) are ail codimension-one Bruhat cells, where i = 1,2, are the Weyl group elements corresponding to the simple reflections with respect to the simple roots a,, i = 1,2, respectively. Since ( 1 0 0 o \ / d o 0 0 \ 0 0 ° -a jx 0 0 A -j-' 0 0 0 6-'/ \ 0 0 0 1 / we know that the codimension of the Bruhat cell {B x B){g x h){B~ x B~) must be less than or equal to 1 if the codimension of the double coset {B x B){g x h)E, is lees than or equal to 1. Lemma 2.4.16 In the Bruhat cells of codimension < 1, R-orbits can be described as follows: (1) In {B X B){B~ X B~), R-orbits are of form [B x B){x-ai{c) x {d))R. where c,d E C and 0 if c ^ O andd ^ 0, codim{B x (c) x x-ai{d))R = 2 i f c = d = 0, 1 otherwise (2) In [B X B){l4 X Ua^){B~ x B ), R-orbits are {B x 5)(x-a,(a) x u!„^)R. for a G C and _ f 2 i f a = 0, codim{B x B)(x-ai(a) X oJai)R I 1 otherwiseise (S) In {B X B){l4 X Wa2){B~ x B~), R-orbits are {B x B){l4 x for a G C and co d M S X B){U X = { I 56 (4) In {B X B){uai X U){B~ x B~), R-orbits are {B x B){uai X X-ai(«))-R for a € C and codim^B X B )K , X x_., (a))R == | ^ ’4 “ . (5) In {B X B){oJa2 X /4 )(5 x B ), R-orbits are {B x 5)(u;a2X-ai (®) x / 4 ) iî for a E C and codim{B X B)(u,,x-«,{“) X / ,) « = | j ‘4 “,“ Proof: (1) Since the 2-dimensional torus in section of R in (B~ x B~) goes to {B x B ), we have {B X B){B- X 5 ") = Ue,d6c(-B X B){x-ai{c) X x-a,(d))iî. It is easy to check that codim[(5 X B){x-a,{o) x x-«i(d))/2] = dim[StabR((B x B)(x-«,(c) x x-«,(d)))] since dim Æ = dimX. On the other hand, Stabft((5 X JB)(x-«,(c) X x-a,(d))) = h{s~^t,{st)~^,st~^,st) X h{s~‘^t~^,t,sH,t~^) i f c = d = 0, h{s~^, s~^, s, s) X h{s~'^, 1, s^, 1) i / c ^ 0, d = 0, (±h{t'^,l,t~‘^,l)) X h{t,t,t~^,t~'^) i f c= 0,d ^ 0, ( ± 7 4 ) X ( ± 7 4 ) i f cd ^O (2) {B X J5)(74 X w « J ( g - XB~) = {B x B){l4 x w „ ,)(x -« ,(a ) x x -« , (c))JZ = (B X B){x-ai(a) X u>ai)R since u;«,X - a , (c)u;“j' 6 B. The stabilizer S ta b « ((5 X 5 ) (x - a , («) X w«, )) = 57 _ J h{s ^,s,)xh{s ifa^O, I h{s~^t,{st)~'^,st~^,st) X i f a = 0 (3) {B X B){U X x B~) = {B x B){U x u;„J(x-a,(a) x x-«i(c))i2 = {B X B){h X w„2X -« i(c)x -«2 (-o ))-S = { B x B ){h x Wa,x-c, 2 (-o)X-«i(c))-R = { B x B){U X u„^x-aA^))R since X-«i(«) x X-«:(o) € i2 and w»2X-«2(-G)^âJ G B. The stabilizer Stabiî((5 X B){U X w„2X-ai(c))) = [±h{{^, 1)] X if c = 0, h{s~^t,{st)~^,st~^,st) X h{s~‘^t~^,t,sH,t~^) if c = 0 (4) and (5) cad be proved in the way similar to (2) and (3), respectively. □ Lemma 2.4.17 For the R-orhits of codimension < 1, we have (a) [BxB0 (x-a, (c) X X-«,{d))R ={B X 5 )(x -a. (1) X X-a, (l))i2, (b) {BxB0(x-«,(c) X I4)R = (B X 5)(x-«,(l) X I4)R, (c) [BxB?)(/4 X X -« ,(4 )A = (^ X B)(/4 X X -m (l))A , (d) {BxB?)(x-a, («) X UJai )R=(B X B ){x-c (1) X UJa, )R, (e) {BxB0(-^4 X WcjX—«I ~ X 5)(/4 X WotjX—C»I (f) {BxB0 (^ « I X X-ai(<^))R — {R X B){ùJai X X-ori (a) {BxB?)(w„2X-a,(a) X Q R = {B X 5)(o;„jX-a,(l) X I4)R- Proof: Statements (a), (d), (e), (f), and (g) follow from the general fact: Let V be a complex connected quasi-pro j ective variety , G a complex algebraic groiq), and G 58 act on V regularly. Then there is at most one Zariski open G-orbit on V. However, statements (b) and (c) follow from easy computations of matrices: (%-«. (c) X I4) = X h{c^,c'^)](x-a: (1) X l4}[h{c^, ) X h[c^,c^)] {I4 X X-a.(d)) = [h{dk,d=T) X h{d^,à)]{l4 X X h{dk,d=T)l where h{a,b) = h{a,b,a~^,b~^). □ Corollary 2.4.18 The Bruhat cells of codimension < 1 can be decomposed into R- orhits as follows: (a) {B X B){B- X B-) = {B X B)R U ( S x jB )(x -« ,(1) x h)R U (H x g ) ( I , x X-., (1))A U (B X B)(x_m (1) X x_m (1))A. (b) {B X B)(/4 X ){B~ X B~) = {B x J3)(/4 x ) B U ( B x B)(x-«i(l) x w«, )R. (c) {B X B){l4Xuja2){B~ X B~) = (BxB)(/4Xo;„j)BU(BxB)(/4Xa;„2X-ai(l))B. (d) {B X B)(w«, X /4)(B- X B-) = {B x B)(w «, x h)RU {B x B)(w «, x x - « , (e) {BxB){tjJa^xl4){B~xB-) = {BxB){uj„^xl4)R\J{BxB){u>„iX-aAl-)xh)R- Theorem 2.4.19 Let — £2 be the short root of Sp{2) and «2 = 2£2 the long root of Sp{2). Let fo = X-ai(l) x X-m (!)• Define following two embeddings of SL{2) into Sp{2) corresponding to the roots «i and a 2, respectively: /ah 0 0 \ / 1 0 0 0 \ c rf 0 0 0 e 0 / and t, 0 0 rf -c «2 0 0 10 \ 0 0 —6 a ) \Q g Q h ) 59 and set = im{iai) = *^(* 02)- Then the open R-orbit and its codimension one boundaries can be described as follows: (a) {B X B)^oR is a Zariski open subset in Kg = Sp{2, C) x Sp{2, C). ( B X B)(x_.,(l) X f4)A , (B X B )(/4 X x_.,(l))A, (B x B)(x_«,(l) x w ^,)B , (BxB)(w„i xx-a,(l))i2, (BxB)(/4Xo;ajX-ai(l))-R; «««f (BxB)(W(,;X-m(l)x l 4)R are all codimension-one boundaries of {B x B)^qR K$. (c) Those codimension-one boundaries can be distinguished in the following way: the disjoint union of the boundaries (B x B)(x-ai(l) X Ia)R, (B x B)(x-«i(1) x a;„, )R, and (B X B)(o;a 2X-ai (1) X I4)R is contained in (B x B)(G„2 x )^oR. and the disjoint union of the boundaries (B x B )(/4 x x-«, (1))B, (B x B)(u;„, x X-ai(l))B, and {BxB){l4XUa2X-ai{l-))R is included in (BxB)(G«, xG„2)^oB. Proof: The proof follows directly from Lemma 2.4.16 and 2.4.17 and the general fact about the orbital decomposition on a group variety, see [9]. □ Corollary 2.4.20 Let X = {B x B)[(G„, x Ga^) U (G„2 x )](foB. Then the codimension of [(5p(2, C) x Sp{2, C)) — Â"] is greater than 2. Finally we describe the intersections of the open dense double coset (B x B)^oR with (B X B){Gai X Ga^)^oR and (B x B)(G «2 x )^oB, respectively, and choose six special paths that live in (B x B)^qR and go to each of those six codimension-one double cosets living in the boundary of (B x B)^qR. 60 In {B X B){Gai X Gc2)^qR, we set Zi,2 := [((?«! x Gag)^o%] fl [{B x S)(fo-R]- Then [{B X B)^oR] n [(5 X B){Gai X Gag)^o-R] = {B X B)^oZi, 2 SKq, which is a Zariski open subset in {B x B)(G«, x Ga2)^o-R- Any g in Zj,2 can be written as: ^ a b 0 0 \ ( 1 0 0 0 \ c d 0 0 0 e 0 X / 9 = ( 0 0 d —c 0 0 1 0 )^o(l4 X X-ai (2.47) \ o 0 -b a I 0 9 0 h ^ * — l 0 0 0 _ (I " a " V I 0 ('“) 0 tf X I 0 0 t o I )^o(l4 X X-ai-a2(x )) 0 0 0 ht / -2 where (ts)'^ = 1, w = {t — t~^) — t~^cd~^ = = gh~^ + cd~^ + 1^0, X* = + gh~^), and k{w,t,g,h) is in Kg. Let I be the parameter that will goes to zero. Then three special paths are defined by 7i(0 — %-«] (Z — 1) X 1-4- x_a; (—1) X 7^. / - 1 1 0 0 72(0 = 0 0 -I \ -° I Xl4 H^ ( -’0 0n 0n ?1 I XI4. (2.48) 0 0 - 1-1 0 0 — 1 —1 ( 1 0 0 0 \ 0 0 1 0 I X-cvi( — l) X Waj. 0 —1 0 —/ j It is clear that those three paths are in the unique open subset Z\^2 and go to the three codimension-one boundaries: Y\ := {B x 5)(x-«i(—1) x l4)^oR, V2 := {B x -S)(w«iX-«i(-l) X Q^oR, and Y3 := {B x B)(x-a,(-l) x Wa,)^oA. Similarly, in {B x B)(Gag x G„, )^oR, set .^2,1 := [(G„g x G„, )^oÂ] n [{B x B)^oR]- Then [{B x B)^qR] H [{B x B)(G«g x G«, )^o72] = {Bx B)(oZ 2,\SKg, which is a Zariski 61 open subset in {B x B){Ga2 x Any g in ^ 2,1 is in form: /1 0 0 0 \ f a b 0 0 \ 0 e 0 / c d 0 0 X )^o(^ X ^_Q,j_a 2(x)) (2 .4 9 ) = ( 0 0 1 0 0 0 d —c I 0 a 0 h I 0 0 - b a / s“l 0 0 0 (s2rf)-‘ 6 0 0 0 («/«)-' 0 fs 0 d o 0 — (1 n ' n n I X 0 0 da^ 0 )^o(l4 X X—a, —«2 (x )) 0 0 - 6*2 d“* • (/l2(s,S,5~\.S"^) X2 (3^,l,a"^,l))t(zi,Z2), where f = 1, zi = gh~^, Z2 = {x + zi){s^ + Zi) — {cd~^ + \)x — Z\s^, = gh~^ + cd~^ + 1 7^ 0, X* = s “^(a; + gh~^), and k{z\,Z2) is in Kg. Three special paths 7 4 , 7 5 , and 7 6 from the open dense double coset to those three boundaries: I 4 := {B X B )(/-4 X x-a,(-l))^oi2, ^ := {B x B){h x w«.x-m(-l))&^, % := {B x B){ijJc2 X x-cvi ( —l))^o-R, respectively, are defined as follows: 74(0 = h X X-»I — 1 ) 1-^ /4 X X -ai( —!)• / - 1 10 0 0 \ - 1 1 0 0 0 0 -1 0 0 0 (2 .5 0 ) 7 s( 0 = /4 X ( -I I - I I ^I4X 0 0 0 1 0 0 — 1 —1 0 0 —1 —1 1 0 0 0 0 0 0 1 7 o( 0 = 0 0 1 0 X X-cvi (1 — 1 ) ^0:2 X X-«i (“ !)• 0-10 -/ 2.4.6 Homogeneity rfj((mi,m 2 ;ni,n2)) By Kostant-Rallis algorithm, the integers dj((mi,m 2;«1, 712)) are the eigenvalues of Xo in the subspace (mi,m 2;« i,«2)(e), where Xo €. t such that [;Co,e] = e and is diagonalizable in the representation space (m i,m 2; «1, «2)- Our proof is organized as follows: 62 (a) We realize (m i,m 2;n i,n 2)^» as the space of right A'l-invariant holoinorphic sections of the line bundle attached to the highest weight (mi, m 2) 0 (rii, «2) by Borel-Weil-Bott Theorem. In other words, (m i,m 2; n\,ni)^e can be realized as a space consisting of all regular functions f : Ke C satisfying the following property of quasi-invariance: for ( 61, 62) € {B X B), k € Kq, and g E Ke-, = (mi,m2)(6,~^)(ni,n2)(&J^)/(5)- (b) The restriction to the unique open dense A-orbit {B x B)^qR gives a canonical embedding: z* : (m i,m 2;» i,«2)^3 ^ C[{B x (2.51) where C[(S x B)^oR]^^ is the ring of all right /lT|-invariant regular functions on {B X B)^oR. Since Kg is a normal subgroup of R by Proposition 2.4.15 in subsection 2.4.5, the embedding t* is actually a homomorphism of i?-modules. Then the image of i* consists of all right ATJ-invariant regular functions / on {B X B)^oR, enjoying following two properties: (1) For ( 61, 62) € (B X B), k e K$, x E X, and sES, f{{b\,b 2)(oxsk) = (m i,m 2)(6r ’)(n i,n 2)(&2 ^)/(^o^s), (2) / can be extended to be a regular function on the whole group Ke- (c) The implement of these extensions. Since any afl&ne algebraic group is normal as an algebraic variety, see [27], it follows from [39] that any regular function / on the open dense double coset (B x B)^qR is regular on the whole group Ke or 63 the set of points where / is not defined has codimension one in K0. By Theorem 2.4.19 (c) and Corollary 2.4.20, the subset X includes the unique open dense double coset and all six codimension-one double cosets with respect to the pair of subgroups {B x B,R), and the subset Ke — X has codimension two. Hence if a regular function / on the open double coset {B x B)(oR can be extended to be regular on X, then / will be extended automatically to be regular on the whole group Ke- One can find a similar argument in Bernstein-Gelfand- Gelfand [4]. First we consider the extensions along those special paths { 7, } of regular functions from the unique open dense double coset to its codimension- one boundaries. In other words we consider a subspace (mi,m 2;ni,U 2)|'.^®j of C[(BxB)^o-R]^® consisting of all functions / satisfying following two conditions: (1) For ( 61, 62) E {B X B), k G x e X , and s G S', fi{bi,b2)^oxsk) = {rn^,m2){b^^){nun2){b2^)f{^oxs), (2) lim j^o/( 7i(I)) exist for f = 1,2,• • • , 6. (Note that this condition is weaker than the condition that / can be extended to be a regular function on X and then to be regular on the whole group Ke). It is evident that (mi,m 2;ni,U 2)^''é C (m i,m 2;n i,n 2)j!^®.j. Further we have a filtration of subspaces: (mi,7ri2;ni,U2)(e) C (m i,m2; Ui,712)^^ C (mi,m2;7Zi,772)|^^p (2.52) 64 After computing the dimension of the subspace (mi, m 2; , U2){^.}, we will show that dim (m i,m 2;ni,U 2)(e) = dim(mi, m 2; «1, 712)^®.}. (2.53) In other words, we will obtain that (mi,m 2;n i,n 2)(e) = (mi, m 2; 771, 7*2)^^- (d) We will determine explicitly the eigenvalues and their multiplicities of the dif ferential operator Xo in the subspace (mi,m2;7%i,7%2)(e). Before going to compute the dimension of the subspace {m\,77 i2’,ni,n 2)[^.y, we give first some necessary conditions about the existence of the Ag-invariant func tions (77ii,m2;ri],7i2)^» and the occurance of eigenvalues Vp of the subgroup S in (7ni,7n2;7li,7l2)*fl. Lemma 2.4.21 Let (m%,m2 ;ui,7%2)^'^K'^p) the set of all S-eigenfunctions in the space (m i,m 2;n i,7î2)^"» with the eigenvalue Vp {f{gs) = Vp{s)f{g) = s‘'f{g)). Then (a) (m],m2;7îi,7i2)^'» ^ 0 implies that (m% m 2 + 7*i 712 ) = 0 {mod 2). (b) The space { 7n■^,m2^,n\,7l2)^^^'p) 7^ 0 (or (/ui, 77*2;,»2){^}(z/p) f 0) implies that (mi 4- m 2 + 7ii -1- 7*2 ) = 0 [mod 2) and 77%i 4- m 2 = p (mod 2). Proof: {&) Since (5 X j5)n^oA^|^o * = {±(I4 X A)}, for any / 6 (7711,7712; 7ii, 712)^'», we have that f{^ox) = /(^o^(—(J4 x I4))) = (—l)"u+'"2+"i Thus / 7^ 0 implies that 7771 4- 7712 4- 7ii 4" 772 = 0 (mod 2). 65 (b) Since {B x B)n(oSxiô^ = x Zj}, for any / € ((m i,m 2; ni, U2))^"é(//p), we have that f{^on{-h)xh)) = ( - 1^ 0®) = f {{{-!,) xQ^ox) = (-l)-'+ ’"=/(6 ^). Therefore / ^ 0 implies that mi + m 2 = p {mod 2). □ Theorem 2.4.22 Let C[X] be the affine algebra of the one-dimensional section X as defined in proposition 11. Let ip{x) = Yli=i ««a;', (a) any S-eigenfunction f with eigenvalue .s^ in C[{B X B)^qR]'^» can be expressed as f{{buh)^o^f>:) = i'nii,m 2){bf^){jii,n 2)ib^^).s’’f{^ox) = i m , m 2){bf^){ni , ri2)(62 ’ )-sV(^’) for (61, 62) E. {B X B), k 6 Kg, x £ X, and s £ S. We call ^{x) the polynomial off. (b) the S-eigenfunction f with eigenvalue .s'* belongs to (m i, m2; M], if and only if the polynomial g>{x) (or(p{x)) of f satisfies following four conditions: (cl) 2 q < mi + m 2 + 2ni — p, and a,- = 0 if 2 i (c2) 2q < Ml + ri2 + 2mi — p, and a,- = 0 if 2i < ni + «2 + 2m 2 — p, (c3) if2j < mi — m 2 + 2ui — p, bj = 0, and (c4 ) if 2 j < Ml + » 2 + 2mi - p, bj = 0 . 66 Proof: Paxt (a) follows straight from the quasi-invariant properties on both sides and the regularity of the function / on the unique open dense double coset. Part (b) will be more complicated and its proof will be separated into cases (bl) and (b 2). (bl) We consider the conditions on the existence of the extensions of the functions f to {B X B){Gon X Ga2)îoR along those three special paths 71, 72, and 73 as chosen in the previous subsection. In other words, we consider the conditions under which those three limits lim/_yo/(7,(0) exist for i = 1,2,3. When g in Zi,2 := x n [{B X it can be written as in (2.48). Thus for any 5-eigenfuiiction / with eigenvalue in C[{B X B)(oR]^^ , one has f{g) = (hf)”==rv(r''(.-c 4- # - ') ) . (2.54) We are going to consider the existence of the limit lim/(g) when g goes to the codimension-one boundaries along those three paths 71 (/), 72(1), and 7a(/) as in (2.49), where / is the parameter going to 0. Along the path 7i(/), t'^ = I goes to 0 and /(7i(/)) = <2'">+">+”2-V (r'^x). When t goes to 0, the limit of / ( 7i (/)) exists if and only if 2ç ^ 2?7î] -b H2 — p. (2.55) Along the path 72(0) goes to 0 0 and /( 72(f)) = 67 If t goes to oo, then the limit of /(72(0) exists if and only if fli = 0 when 2i < 2ni 2 + + «2 — P- (2.56) Finally, along the path 73(1), = I — goes to 00 (as I goes to 0) and /(7 3 (0 ) = As t goes to 00, the limit of /( 73(/)) exists if and only if bj = 0 when 2j < 2rni + + n-z — p. (2.57) In other words, the conditions on the existence of the extension of f to [Bx B)(G„, x Ga2)^oR along those three special paths 71, 72, and 73 are (c2) and (c4). (b2) We consider the conditions on the existence of the extensions of the function f to {B x S)(Gc,j X Gai)^oR along those three special paths 74, 75, and 73. Again the element g in the open dense subset Z-2,\ = [(G«z X )^oA’] D [{B x B)^qR] can be written as (2.50). Hence any 5-eigenfunction / with eigenvalue in C[{B x B)^oR.y'"0 can be expressed as f{g) = a"»: (hs)""(da^)'" d-":s-P(p(s-^(z f pA -')). (2.58) By the same argument as that in case (bl), we obtain that /(74(/)) = .s’”'+’"^+2«--V(.s-'^x),(.s2 = /-^ 0 ) (2.59) /( 75(f)) = (^2 = J ^ Oo) (2.60) /( 76(f)) = .s-(’" '- ’”=+"’“ -'') The following lemma gives a way to count the dimension of (m i, m^; n i, for each possible integer p. Lemma 2.4.23 Let aij = (—1)' ^ (j) = (—1)* ^ « / , o • «m,0 • • • « n,0 ^ «1,1 • «m,l ••• «n,l «(,( • «m,I • • • «n,i , 0 < 1 < m; 0 < 1 < n. V 0 «n,m / / «1,0 « n,0 «1,1 ••• «n,l , 0 < 1 < n; 0 < m < 1. \ «l,m ■ ■ ■ «n,m / Then rankM{l^ m, n) = min{m + 1, n — f + 1} and rankN{î, m, n) = min{m + l,?i I + 1}. Proof: Since ai+xj + axj = a,j_i, we have, by elementary transforms of columns, ^ aifi 0 • • • 0 \ M{l,m,n) =>- «1,1 M(f, m —l,n —1) V 0 By induction, we obtain vankM{l, m, n) = mim{m + 1, n — f + 1}. Similarly, we can prove that rankN{l, m, 71) = mim{rn + 1, n — f + 1}. □ 69 Theorem 2.4.24 Let Xo is such an element in t, as chosen above, that [xo,e] = e and {mi,m 2 ]n\,n 2 )[^.y{p) the eigenspace of Xo in (m i,m 2 ;n i,n 2 )|!y®.}. Then we have (a) //( m i + m 2) > {n-i + M2) > (mi — m 2) > (mi — M2 ), then the eigenvalues of Xo are p = 2imi — Ml + M2) 2/Ml — Ml + M2 + 2, • • •, mi + m 2 + 2mi and (al) ifn i+ M2 — mi + m 2 < 2mi — 2 m2 , then d im ( m i, m 2 ; Ml, M2 ){!|,®}(iJ) (2.62) 1 + |(mi — M2 — 2mi + p) if 2 mi ■ Ml + M2 < p < m i + m2 4 - 2 m2 , 1 + |(m i + M2 - mi + m 2 ) i / m i + m 2 + 2m2 < p < 2mi + Mi - M2 , 1 + |(m i + m 2 + 2 m i- p ) i f ‘.if 2mi + Ml — M2 < p < mi + m 2 + 2mi . (a2 ) if ui + M2 — mi + m 2 > 2mi — 2 m2 , then i 1 + |(m i — M2 — 2 m i + p) i f 2 m i — mi + M2 < p < 2 m i + M] — M2 , 1 + (Ml - M2) i f 2mi + Ml — M2 < P < Mil + 7M2 + 2m2, 1 + |(mii + m 2 + 2mi - p) if +1712 + 2m2 < p < mi + niz + 2mi ; (b) //( m i + m 2 ) > (mi + M2 ) > (mi — M2) > (mi — niz), then the eigenvalues of Xo are p = 2m2 + M] — M2, 2mi2 -|- Ml — M2 + 2, • • •, 2mi + mi + M2 and (bl) if U2 < mi — 1 1 1 2 , then d im ( m i, m 2 ; mi , M2 ) ^ ^ ( p ) (2.64) 1 + ^(m2 — Ml — 2m2 + p) if 2mi2 4 Mi — M2 ^ p ^ 2mi2 4 mi 4 M2 , 1 4 M2 i f 2m 2 4 Ml 4 M2 < p < 2 m i 4 Mi - M2 , 1 4 |(2m.i 4 Ml 4 M2 - p) i f 2mi 4 Mi - M2 < p < 2mi 4 Mi 4 M2 , 70 (b 2 ) ifu 2 > m-[ — 1712, then dim(mi,m2;ni,n2)|^®.}(p) (2.65) 1 + 2 ( ^ 2 — n-i — 2m 2 + p) if 2 rri2 + w j — fi2 ^ p ^ 2mi + — ii2 , 1 + (mi — m2) if 2mi +n-i —n 2 < p < 2m 2 + »i + »2, 1 + | ( 2mi + »i +n -2 — p) i f 2m 2 + »i + »2 < p < 2mi + U] + m2; (c) If {n\ + m2 ) > (mi + m 2 ) > (mi — m 2 ) > (»i — 7*2 ), then the eigenvalues of Xo are p = mi — m 2 + 2ri2, mi — mi + 2u2 + 2, • • •, mi + m2 + 2ni and (cl) if m 2 < 1 1 1 — U2 , then d im ( m i, m 2 ; , «2 )^} (p) (2.66) 1 + l(m 2 — mi — 2 h 2 + p) if mi — m 2 + 2n2 < p < mi + m 2 + 2 ti2 , 1 + m 2 7/ m i + m 2 + 2»2 < P < — "'2 + 2rii, 1 + |(m i + m 2 + 2»i — p) i f mi — m 2 + 2ni < p < nii + mg + 2rii, (c2 ) if m 2 > Til — 772, then dim(777i, m 2 ; 7ii, »2){!^}(p) (2.67) 1 + l ( m 2 — 77li — 2712 + p) i f m i — 7712 + 2772 < p < 777] — 777g + 2 7 7 i , 1 + (771 - 772) 7/ m i — 7772 + 77i < P < 7771 + mg + 277g, 1 + |(m i + mg + 2771 — p) 7 / mi + mi + 277g < p < mi + m 2 + 277] ; (d) If {u\ + 77g) > (m i + m 2 ) > (771 — 77g) > (m i — m g), then the eigenvalues of Xo are p = 277g + nil — mg, 277g + mi - mg + 2, • • •, 2777 1 + 77i + 77g and 71 (il) ifm\ + m 2 — ni + U2 < 2m\ — 2mi, then dim(mi,m2;ni,n2)|^.}(p) (2.68) { 1 + |(m i — m2 - 2ni + p) i f 2ni - mi + m 2 < p < ni + + 2m2, 1 + l(mi + m2 - ni + n2) i f ni + ng + 2m2 < p < 2ni + mi - m 2, 1 + |(ni + n2 + 2mi - p) i f 2ni + mi - m2 < p < ni + n 2 + 2m i, (d2 ) ifjïii + m 2 — ni + U2 > 2mi — 2 m 2 , then dim(mi, m2; ni, n2)^®}(p) (2.69) 1 + |(m i — m2 - 2ni + p) i f 2ni - mi + m2 < p < 2ni + mi - m 2, = < 1 + (mi — m2) î / 2ni + mi — m 2 < p < ni + U2 + 2ni2, 1 + i(ni + ri2 + 2mi — p) if ni + 712 + 2m2 < p < ni + 712 + 27ni ; fej If (mi - m2) > (ni + 712) or (ni — U2) > (mi + m2), then dim(mi,m2;ni,n2){.^®}(p) = 0. Proof: Assume that y(z) = Si=i n,z', (p(z) = ]Ej=i bjx^ 6 C[X] such that ip{x +1) = (p(x), i.e. hj = E L i(-l)‘"^ 0) (a) (mi + m 2) > (ni + n2) > (mi - m2) > (ni - n2). In this case, we have 2m2+ni + n 2 < mi +m2+2n2, mi +7n2+27ii < 2m 1 +7ii 4-7*2, and 7ni — m 2 4- 2ni < 2mi 4- ni — U2, and further we have p < 7ni -f- 77*2 4- 27*1 , 7ni 4- m 2 4- 2n2 — p < 2i < m-i + jn 2 + 2ui — p, and bj = 0 if 2j < 2m 1 4- ni — 7*2 — p by theorem 14. (al) If ni 4- ri2 — m-i + m 2 < 2ni — 2u 2, then 2mi — n.i 4- ri2 < mi 4- rrii + 27*2 < 2mi 4- ni — U2. we shall compute dim(mi, m 2; n i, n2)j^.^(p) case by case for p. (ali) Case 2mi 4- ni — n2 < p < mi 4- n *2 4- 2ni. Since 2mi 4- ni — n2 — p < 0, there are no conditions for bj and because 77*1 4- n*2 4- 2n2 — p < 0 we know that for such a p, the degree of the polynomial (p{x)could be 72 any integer between 0 and |(m i + m 2 + 2ni — p). Thus dim (m i,m 2;n i,n 2)|^®.}(l>) = 1 + ^(mi + m2 + 2ni -p). (alii) Case mi + m2 + 2^2 < < 2m 1 + ui — n 2. In this case, the degree of (fi{x) = J2l=i o«a:' will takes values of all integers between 0 and I (mi + m 2 + 2n-i — p) and there is the restriction for bj, that is, bj = 0 if 0 < j < |(2mi + Ui — ri2 —p). This means that dim(mi,m 2; ni,U 2){^.}(p) is equal to the dimension of the solutions of the system of homogeneous linear equations bj = 0 for 0 < j < ^(2mi + rii — n-2 — p). Since the system of the equations bj = ^(-l)*"’’ 0 ^ ai = 0, j = 0,1, ■ • •, ^(2m i + ni - »2 ~ p) - 1 has 1 + l(mi + m2 + 2ui — p) variables and l( 2mi + ni — «2 — p) equations and its rank is v |( 2mi + rii — 712 — p), we know that the dimension of the solutions is 1 + I (mi + m2 + 2ni — p) — ^(2mi + Ui — ri2 — p) = 1 + ^(ui + ri2 — mi + /U2)- (aliii) Case p < mi + m 2 + 2^2 In this case, we have a,- = 0 if 2; < mi + m2+ 2^2 —p. The system of homogeneous linear equations bj = ^( — ^ a,- = 0, j = 0, 1, • • •, - ( 2mi + rii — 712 — p) — I has 1 4- (ui — 7 1 2 ) variables and |(2m i + ^i — « 2 — p) equations and the corresponding matrix of the coefficients is M (|(m i + m 2 4- 2 ^ 2 —p), |(2m i +ni — « 2 —p) — 1, |(m i + 77i2 + 2ni -p )). By Lemma2.4.23, its rank is rmm{i(2mi +rai — ri 2 —p), l + (rii —7(2 )}. Since [i(2m i+7ii —7Î2—p)] —[l + (u i--« 2 )] = |(2m i—711+ 7x 2 —p) — l, it is evident that 73 if p < 2m\ — ni + ng, then the number of the variables equals the rank of the system of equations and is less than the number of the equations, and this means that the system of equations has no solutions, in other words, dim(mi,m 2;n i,n 2){^®.}(p) = 0; and if 2mi — n% + < p < mi + m 2 + 2n2, then the rank of the system of equations is | ( 2mi + ni -n-i—p) and the dimension of the solutions of the system of equations is 1 + |(ni — n2 — 2mi + p). (al) is proved. (a2) If ni + n-z — mi + m 2 > 2ni — 2u2, then mi + m 2 + 2n2 > 2mi + Ui — n -2 > 2mi — ni + U2 . we shall compute dim(mi,m 2; Mi,»2 ){^}(p) case by case for p. (a2i) Case mi + m2 + 2n 2 < p < mi + m 2 + 2ni. Since 2mi + ni — n2 — p < 0, there are no conditions for bj and because mi + m 2 + 2«2 — p < 0 we know that for such a p, the degree of the polynomial 1 dim (m i,m 2;rei,n2){^®.}(p) = 1 + -(mi + m2 + 2ni - p ) . (a2ii) Case 2mi + ni — n 2 < p < mi + m 2 + 2^ 2- In this case, the degree of 2ri2 — p, and there are no restrictions for bj for 2mi -b ni — ri2 — p < 0. Therefore dim (m i,m 2;n i,n 2)^.y®.}(p) = 1 -b (% - Mj). (a2iii) Case p < 2mi 4- »i — 112- 74 In this case, we have a,- = 0 if 2i < mi + m 2 + 2n2 —p- The system of homogeneous linear equations hj = 0 ^ a,- = 0, i = 0, 1, • • •, ^ (2mi + ni - «2 - p) - 1 has 1 + (« 1 — «2) variables and |(2m i + «i — » 2 — p) equations and the corresponding matrix of the coefficients is 7V(|(mi + m 2 + 2 ^ 2 — p), |(2m i + ni — « 2 — p) — 1, ^(/ui + ?n2 + 2ni —p)). By Lemma2.4.23, its rank is mim{^(2mi + ui — « 2 —p), H -(ui — «2 )}. Since [|(2mi +«] —« 2 —p)] —[l + («i —^2 )] = |(2 m i — rii + «2 —p) —L it is evident that if p < 2mi — n\ + M2 , then the number of the variables equals the rank of the system of equations and is less than the number of the equations, and this means that the system of equations has no solutions, in other words, dim(mi,m 2 ; Mi,7%2){!,^}(p) = 0; and if 2mi — rii + ri2 < p < 2mi + n\ — M2 , then the rank of the system of equations is \{2m\ + Ml — M2 — p) and the dimension of the solutions of the system of equations is 1 + |(mi — M2 — 2mi + p). Case (a) is proved. (b) (mi + m 2) > (Ml + M2) > (Ml - M2) > (mi - m2). In this case, we have 2m2+Ml + M2 > m i+ m 2+ 2M2, m i+ m 2+ 2ni > 27Mi+M]+M2, and mi — m2 + 2mi < 2mi + M] — iiz, and further we have p < 2mi + mi + M2, 2/7(2 + Ml + 7(2 — p < 2( < 2mi + Ml + M2 — p, and hj = 0 if 2j < 2m\ + 7(1 — 7(2 — p. (bl) If M2 < 2/mi — 277(2, then 2/7(2 + /(i — M2 < 2/M2 + mi + M2 < 2mi + /(] — 7(2. we shall compute dim(/Mi, m 2; Mi,M2){^^(p) case by case for p. (bli) Case 2mi + Mi — m,2 < p < 2mi + Mi + M2. 75 Since 2mi + — Mg — j) < 0, there are no conditions for bj and because 2mg + Ml + « 2 — P < 0 we know that for such a p, the degree of the polynomial ip{x) could be any integer between 0 and \{2m\ + M] + Mg — p). Thus /c® 1 dim(mi,M2g;Mi,Mg){^.^(p) = 1 + -(2rMi + mi + Mg - p). (blii) Case 2mg + Mi + Mg < p < 2rri\ + Mi — Mg. In this case, the degree of ip(x) = Yli=i will takes values of all integers between 0 and ^(2mii + Mi + Mg — p) and there is the restriction for bj, that is, bj = 0 if 0 < j < \{2mi + Ml — Mg — p). This means that dim(mi,/Mg; Mi,Mg)j^^(p) is equal to the dimension of the solutions of the system of homogeneous linear equations bj = 0 for 0 < j < \{2mi + mi — Mg — p). Since the system of the equations bj = 0 ^ a,- = 0, ; = 0,1, • • •, ^(2/Mi + Mi - Mg - p) - 1 has 1 + |(2/Mi + Ml + Mg — p ) variables and \{2m\ + «i — Mg — p) equations and its rank is \{2m\ + n\ — Mg — p), we know that the dimension of the solutions is 1 + ^(2/Mi + Ml + Mg — p) — ^(2/Mi + Ml — Mg — p) = 1 + Mg. (bliii) Case p < 2/Mg + Mi + Mg In this case, we have a,- = 0 if 2/ < 2/Mg + //] + //g — p. The system of homogeneous linear equations bj = 0 ^ ai = 0, j = 0, !,•• • ,^(2/Mi + Mi - M g - p ) - 1 has 1 + (/M] —//ig) variables and |(2 /Mi +M] — Mg — p) equations and the corresponding matrix of the coefficients is M(i(2/Mg+Mi+Mg—p), |(2 /M]+Mi —/ig—p) —1, |( 2 /M]+Mi + 76 îi2 —p)). By Lemma2.4.23, its rank is mim{j{2m\+ni —ng—p), l + (mi—m2)}. Since [1 + (mi — m 2 )] — [|(2m i + rii — «2 —p)] = 1 + 1(—2 m 2 — ui + U2 +p), it is evident that if p < 2 ni 2 + «1 — rii, then the number of the variables equals the rank of the system of equations and is less than the number of the equations, and this means that the system of equations has no solutions, in other words, dim(mi, m 2 ; " i, M2 ){..|^.}(p) = 0 ; and if 2 m 2 + n\ — U2 < p < 2 m 2 + ni + ri2 , then the rank of the system of equations is \{2 m\ + Til — « 2 — p) and the dimension of the solutions of the system of equations is 1 + \{ri2 — n-i — 2 m 2 + p). (b2) If ii2 > 2m\ — 2m2, then 2m2 + + «2 > 2mi + ni — n2 > 2ni2 + iii — 112. we shall compute dim(mi, 7U2 |» i, (p) case by case for p. (b2i) Case 2m2 + n-i + ri2 < p < 2mi + »i + «2- Since 2m.] + — nz — p < 0, there are no conditions for bj and because 2iU2 + 7Ji + «2 — p <0 we know that for such a p, the degree of the polynomial ip{x) could be any integer between 0 and ^(2mt + Ui + 712 — p). Thus /c* 1 dim(mi,m,2;rai,n2){..^j(p) = 1 + - ( 2mi + rii + «2 - p). (b2ii) Case 2mi + «i — ri2 < p < 2m2 + n-i + »2- In this case, the degree of (p{x) = a,æ* will takes values of all integers be tween ^(2m2 -I- ni -f ri2 — p) and |(2m i -f »] + «2 — p), = 0 if 2z < mi 4- 7712 -f- 2712 — P, and there are no restrictions for bj for 2mi + 7i\ — 712 — p < 0. Therefore dim(mi, 7712; 7ii, M'2){^.}(p) = 1 + (mi - m2). (b2iii) Case p < 2mi 4- Ui — 7*2. 77 In this case, we have a,- = 0 if < 2m2 + ni + nj — p. The system of homogeneous linear equations bj = 0 ^ tti = 0, y = 0,1, • • •, i(2mi + ui - ri2 -p ) - 1 has 1 + (mi — m2) variables and | ( 2mi + wi — ^2 —p) equations and the corresponding matrix of the coefficients is 7V (|(2m2 + ni +U2 — p), | ( 2mi + u i — n2 —p) — 1, | ( 2mi + Ml +«2— p)). By Lemma 2.4.23, its rank is m im {|(2mi + % — n2 —p), l + (mi — m 2)}. Since [l + (m i—m2)] —[|(2m i+ u i—«2—p)] = l + |( p —2m2—«1+^2), it is evident that if p < 2m2 + «1 — ri2, then the number of the variables equals the rank of the system of equations and is less than the number of the equations, and this means that the system of equations has no solutions, in other words, dim(mi,m2;ni,n2){.^j(p) = 0; and if 2m2 + Ui — 712 < p < 2m% + ni — ri2, then the rank of the system of equations is |( 2mi +rii — U2 — p) and the dimension of the solutions of the system of equations is 1 + ^(u2 — Tl\ — 277*2 d" p)' (c) (7*1 + 7*2) > (77*1 + 77*2) > (m i - m 2) > (7*1 - 7*2). In this case, we have 2m2+7*i + 7*2 < mi + m2+27*2, mi + 7*7,2+ 27*1 < 27**i + 7*, + 7*2, and 77*1 — m2 + 27*1 > 2mi + 7*1 — 7*2, and further we have 0 < p < 77*1 + 77*2 + 2** 1, 27*2 + 77*1 + 77*2 — p < 2* < 27*1 + mi + m 2 — p, and hj = 0 if 2j < 2**i + 77*1 — 77*2 — p. (cl) If 777.2 < 2**1 — 27*2, then 27*2 + mi — m2 < 2**2 + n*i + m2 < 2**i + mi — 77*2. If we make a change of variables: vn\ 44- 7*1, m2 44- 7*2, then case (cl) will go to case (bl) when (mi — m2) > (7*1 — 7*2), and to case (al) when (77*1 — m2) = (7*1 — 7*2). 78 Hence if (mi — m 2) > (»i — na), dim (m i,m 2;ui,ra2)|!y®.}(p) = 1 + |( m 2 — mi — 2 ri2 + p) i f m i — m2 + 2 n -2 < p < mi + m2 + 2n 2, 1 + m2 i / mi + m2 + 2n2 < p < mi — m2 + 2« i, 1 + |(m i + m 2 + 2ni — p) i f m i — m 2 + 2ni < p < m i + m 2 + 2 u i , and if (mi — m2) = (ni — na), dim (m i,m 2;n i,rî2){^®}(p) = 1 + |(m i — ma — 2ni + p) i f 2n^ — m i + m 2 < p < ni + « 2 + 2 m a , { 1 -|- 2 (m i -t" ma — ni + na) ni + Ua + 2ma ^ p ^ 2ni + ?7ii — 77* 2 , 1 + 2 ( 7*1 + 7*2 + 2m i — p) '*/ 27*1 + m i — ma < p < 7*i + 7*2 + 277* 1 . However, both expressions are actually same since 77*2 — mi — 27*a = 77*1 — 77*2 — 27*1, 77*1 — m2 + 27*2 = 27*1 — mi + ma, and mi + m2 + 2t*2 = 7*i + 7*2 + 2ma; 277*2 = 77*1 + ma — 7*1 + 7*2; and 77*1 + ma + 27*i = 7*1 + 7*2 + 2mi. We are done in this case. (c2) If ma > 7*1 — 7*2, then 77*1 + 77*2 + 27*2 > 2**i + mi — 77*2 > 77*1 — m2 + 27*a. Similarly, if the variables are changed by mi ■H- 7*1 and ma < 4- 7*2, then case (c 2) goes to case (b 2) if (77*1 — 77*2) > (7*1 — 7*2), and to case (a 2) if (mi — m2) = (7*1 — 7*2). Just as case (cl), we will get our results for case (c 2) as stated in the theorem. (d) ( 7*1 + 7* 2 ) > (mi + ma) > ( 7*1 - 7* 2 ) > ( 77*1 - m a ) . In this case, we will make the same change of variables: 77*1 < 4 7*1 and 77*2 < 4 7* 2 , and case (d) is switched to case (a). The results for case (d) follow from the same argument as in case (c). (e) ( m i — m a ) > ( 7*1 + 7*2) or ( 7*1 — 7* 2 ) > (mi + m 2 ). 79 If (rrii — m2) > (ni + U2), then (mj + m2 + 2ri\) < (2mi + ni — «2) and mi + m2 + 2ri2—p <2i< mi+m 2+2ni —p. We have bj = 0 for 0 < j < |(mi+m2+2ni —p) = q. Therefore this system of equations has no solutions since the rank of the system is equal to the number of the variables and less than the number of the equations, the result follows. If (ni — TI2) > (mi + m2), just changing the variables by mi <->• rii and m 2 44- 112, we will obtain the result. We are done. □ From the proof above, the eigenvalue p of Xg must be nonnegative. Corollary 2.4.25 The dimension of the space (m i,m2; Ui,U2))j^). is determined as follows: (a) If {m\ 4- m2) > (ui + «2) > (mi — m2) > (ui — «2), then dim(mi, m2; «1, 712))^» = ^(ui + U2 - mi + m 2 + 2)(ni - 7*2 + 1); (b) //(m i 4- 77*2) > (ui 4- 7*2) > (771 — 772) > (mi — m2), then dim (m i,m2; 771, 772))^^» = (772 4-l)(mi — 7772 4-1); (c) If {n-i + 772) > (7771 4- 777,2) > (7771 - m2) > (771 - 772), then dim (m i,7772; 771, 772))^^® = (m2 4- l)(ni - 772 4-1); (d) If [n\ 4- 772) > (mi 4- m2) > (771 — 772) > (7771 — 7772), then dim (m i,7772; 771, 772))^'» = l(m i 4- m2 - 77 i 4- 772 4- 2)(mi - niz 4-1); 80 (e) If {mi — m2) > (ni + n-2) or (ni — n2) > (mi + m 2), then dim(mi,m2;ni,n2))^'« = 0. Proof: According to the Theorem, we have dim(mi,m2;ni,n2)|^} = ^dim(mi,m2;ni,n2)|'^®.}(p), (2.70) where the summation takes over the sets of integers of different types subject to the conditions on m i,m 2, ni, and ri2, which are (a) If (mi + m2) > (ni + ng) > (mi — m 2) > (ni — U2), then p 6 {2mi — ni + U2,2mi — ni + U2 + 2, • • •, ?ni + m2 + 2ni}. (b) If (mi + m 2) > (ni + U2) > (ni — Hi) > (mi — m2), then p E {2m2 4" ni — n2,2m2 ni — n2 -1- 2, • • •, 2zni ni + U2}. (c) If (ni + U2) > (mi + m2) > (mi — m2) > (ni — U2), then p E {mi — m2 + 2n2, mi — m2 + 2nj + 2, • • •, t71 i + m2 + 2ni}. (d) If (ni + n,2) > (mi + m 2) > (n-i — 712) > (mi — m 2), then p E {2n 2 + mi — m2,2 /i2 -t- mi — m2 + 2 , • • •, 2 mi 4- ni + 712}- (e) If (mi — m 2) > (ni 4-n2) or (ni — 712) > (m-i 4- m2), then the summation is over an empty set. Thus the dimension of the space is zero. 81 By means of the conditions of congruence on those integers m i,m 2,n i,n 2, and p: mi + m2 + ni + «2 = 0(mod2) and m% + m 2 = p{mod 2), it is not difficult to calculate the dimension of the space, which is exactly as same as stated in the Corollary. □ In the subsection 2.4.4, we used the Branching Rule to compute the multiplicity of the representation (mi, m 2; n i,«2) occurring in the space H of all harmonic polyno mial functions on p, which is Z((mi,m2; Ui,n2)) and is equal to dim(mi, m 2; ni,n 2)(z) for some regular semisimple element 2 in p. Following Kostant-Rallis [40], the multi plicity /((mi, m 2; Ml, 712)) is also equal to dim(777i, 7712; iH, 772)(e) for any regular iiilpo- tent element e G p. Following Corollary 2.4.25, we obtain Corollary 2.4.26 Notations are as above. (a) We have following equalities /((mi, m2; 771, 772)) = dim(mi, 7772 ;77i, 772)(e) = dim(mi, m2; 771, 772)^.^^. (h) We have (mi, 7772 ;77i, 772)(e) = (m i,m2; 77i, 772)^^«. Proof: Part (a) follows from Corollary 2.4.12 in subsection 2.4.4 and Corollary 2.4.25 in this subsection. Part (b) holds because we already know the following filtration of subspaces: (7771, 7772; 77i, 772)(e) C ( m i , m 2 ; 771, 7 7 2 )^ 9 C (7771, 7772; 77i,772)!^ } . 82 □ Finally we can write down explicitly the eigenvalues and the dimensions of the corresponding eigensubspaces of Xo in (m ],m 2;n i,n 2)(e). Theorem 2.4.27 Let (mi, m2 ; ni, «2 ) the finitely dimensional complex represen tation of Ke with highest weight (rni,m 2) ® (%,M2). Let e be the regular nilpotent element in p as in proposition 1.1 and Xo in t such that [xo, e] = e. Then the eigenval ues p and the dimensions of the corresponding eigensubspaces (m i,m 2;ni,U 2)(e)(p) of Xo in the subspace {mi,m 2',ni,n 2){e) (which is defined in [ 40]) can be determined as follows: (a) / / ( m i + m 2 ) > («1 + 112) > (m i — m 2 ) > (ui — ^ 2), then the eigenvalues of Xo are p = 2mi — n \ + U2,2mi — rii -f- M2 + 2, • • •, mi + m 2 + 2ui and (al) if n\ + U2 — m,i + m2 < 2n\ — 2ii2, then dim(mi, m2; ni, n2)(e)(p) (2.71) 1 + ^(ui — n,2 — 2ni\ + p) i f 2m,i — n\ n 2 ^ p mi + m 2 + 2îi2, 1 + %(ui + U2 — mi + m2)i f m\ + m 2 4" 2u2 ^ p ^ 2mi + ni — ri2, 1 + |(mi + m2 + 2ui - p) i f 2mi + Ui — U2 < p < mi + m 2 + 2ni, (a2) if u\ + U2 — mi + m 2 > 2ni — 2u2, then dim(mi,m2;ni,n2)(e)(p) (2.72) 1 + \{ni — 712 — 2mi + p) i f 2mt — ni + U2 < p < 2mi + »i — «2 , 1 + ("'1 — «2) i f 2mi + ni — U2 < p < m.i + m 2 + 2u2, 1 + i(mi + m 2 + 2ni — p) i f m \ + m 2 + 2^2 < p < mi + m 2 + 2?ii ; 83 (b) If {mi + m2) > (ni + n2) > (ni — n2) > (mi — m2), then the eigenvalues of Xo are p = 2m2 + ni — n2,2m2 + ni — n2 + 2, • • •, 2mi + ni + n2 and (hi) ifu 2 < mi — m2, then dini(mi,m2;ni,n2)(e)(p) (2.73) 1 + |(n2 — ni — 2m2 + p) i f 2m2 + n i —n 2 < p < 2m2 + ni + U2, = < 1 + n2 i f 2m2 + ni + m2 < y < 2mi + ni — 112, 1 4- ^(2mi + ni + n2 - p) i f 2mi + ni - m2 < p < 2mi + ni + 112, (b 2) ifuz > nil — m2, then dim(mi,m2;ni,n2)(e)(p) (2.74) 1 4- |(n2 — n] — 2m2 4- p) i f 2m2 4" ni — n2 ^ p ^ 2nii 4- ni — n2, = < 1 4 - (mi — m 2) i f 2mi 4- ni — n2 < p < 2m2 4- ni 4- m2, 1 4- |(2m i 4- ni 4- ri2 — p) i f 2m2 4- ni 4- n2 < p < 2mi 4- ni 4- m2; (c) If {ni 4- m2) > (mi 4- m2) > (mi — m2) > (ni — n2), then the eigenvalues of Xo are p=z nil —m 2 + 2n2, mi — mi 4- 2n2 4- 2, , mi 4- m 2 4- 2ui and (cl) if m 2 < ni — m2, then dim (m i, m 2 ; n i, m2)(e)(p) (2.75) 1 4- I (m2 — m i — 2 u 2 4- p) i f nil — m 2 + 2 ii2 < p < nii 4- m2 4- 2?i2, 1 4- m2 i f mi 4- m2 4- 2ri2 < p < mi — m 2 4- 2ni, 1 4- |(m i 4- m 2 4 - 2ni — p) i / mi — m 2 4- 2ni < p < mi 4- m 2 4- 2 n i, 84 (c2) if m2 > n\ — 712, then dim(mi,m2; 7ii,n2)(e)(p) (2.76) 1 + 2 (^^2 — m\ — 2u2 p) i f mi — m 2 4- 2îi2 ^ ^ nii — 7712 + 2^ii, 1 + («1 — TI2) i f mi — m 2 + 7li < p < 77li + 7712 + 2712, 1 + |(m i + m 2 + 2ni — p) i f mi + mi + 2»2 < P < + m 2 + 27ii ; (d) If {ni + «2) > (m,i + m2) > {ni — ^2) > (mi — m2), then the eigenvalues of Xo are p = 2n2 + m i — m 2 ,2»2 + mi — m2 + 2, • • •, 2m i + n i + 712 and (dl) if mi + m2 — ui + U2 < 2mi — 2m2, then dim(mi,m2;ni,n2)(e)(p) (2.77) 1 + I (m i — m 2 — 2ni + p) i f 2ni — mi + m 2 < p < rii + ” 2 + 2m2, 1 + I (mi + 7712 - rii + ri'i) i f «i + »2 +2 ni 2 < p < 2ni + rri] - m2, 1 + i(rii + »2 + 2mi — p) i f 2ri] + mi — m 2 < p < rii + ri2 + 2m i, (d2) if nil + m 2 — rii + ri2 > 2mi — 2m2, then dim(mi,m2;rii,ri2)(e)(p) (2.78) 1 + |(m i — m 2 — 2rii + p) if 2»i — mi + m2 < p < 2ri] + mi — 7712 , 1 + (mi — m 2 ) if 2»i + mi — m2 < p < rii + ri2 + 2r?i2, 1 + ^(rii + Ü2 + 2mi — p) if rii 4- ri2 4- 2m2 ^ p ^ rii 4~ ri2 4- 2m 1 ; (e) If {mi — m 2 ) > (rii 4- ri2 ) or (rii — ri2 ) > (mi 4- m 2 ), then dim(mi,m2;rii,ri2)(e)(p) = 0. Proof: By the filtration of those three subspaces: (mi,m2;rii,ri2)(e) C (m i,m 2 ; n i , 122)^» C (m i,m 2 ; r i i , «2 )^!^®.}, 85 we have dim(mi,m2;ni,n2)(e) < dim(mi,m2;ni,n2)^® < dim(mi,m2; «i, By corollary 11, we have ^dim(mi,m2;ni,n2)(e)(p) = dim(mi,m2;ni,ri2)(e) p = dim(mi,m2;ni,n2){^®.j = ^ dim(mi, m2; »i, (p). p Therefore, for each eigenvalue p, we have dim(mi,m2;7M,fi2)(e)(p) = dim(m,, m 2; , M2)|^}(p). The theorem follows from Theorem 2.2.24. □ 2.4.7 Application of Kostant-Rallis’ Algorithm We are able to apply Kostant-Rallis’ algorithm in this subsection to compute the polynomials P(m],m 2,n i,n 2;X), which is the main part of our computation of the local Langlands factor 7Ti„ (g) 7T2„, (pi 0 Pi)«)- Theorem 2.4.28 Let and n -2 be any fournonnegative integers. Then the polynomial P{m -\, m2, n \ , Ui', X ) can he expressed as follows: (a) If (m-i -t- m2) > (ui -f «2) > (m] — m2) > (nj — U2), then we have n _ Y«i+«2-fni+m2+2Wi _ V2(»11-n2+l)\ = 86 (b) If {mi + m 2) > (ni + M2) > (ni — n2) > (mi — m2), then (c) If {ni + U'z) > (mi + m2) > (mi - m2) > (ni - n^), then P{m,, n„n,;X) = X"" (d) If {ni + M2) > (nil + ni2) > (ni — n2) > (mi — m2), then (1 _ Y'ni+m2-ni+ti2+2)\ Il _ y2(mi-,»2+])'\ P (m ,, m2, » „ »2iX) = X""------(1 ^ x 5 ) ------ (e) If {nil—m 2) > (n i+ n 2) or (n i—112) > (mi+ma), then P{îni,m 2,ni^n 2',X) = 0. Proof: By Proposition 2.4.5, one lias P (m i,rn 2,n i,n 2;X ) = eacli term of which can be interpreted as follows: the term piX‘ indicates that the irreducible representation (mi, m 2) © (ni, n2) occurs, with multiplicity p/, in the PTg-submodule of "H consisting of all homogeneous functions of degree I. In other words, I is an eigenvalue of Xo in (m i,m 2;n i,n 2)(e) and pi = dim(mi,m 2; n i, M2)(e)(Z). Applying Theorem 2.4.27, we will achieve the expression of the polynomial P(m i, m 2, n i, n.2; X) by means of m i,m 2, ni, and n.2. Note that mi + m 2 + ni + n2 = 0 (mod 2) and / = mi + ni 2 (mod 2). (a) If (mi + m 2) > (ni + M2) > (mi — m2) > (mi — M2) and if mi + 712 — nii + m 2 < 2mi — 2m2, then the multiplicity p; can be expressed as 1 -|- ^(mi — M2 — 2m 1 + /) i f 2mi — Mi -t- M2 ^ ^ mi + m2 4- 2m2, pi = 1 + g(Mi + M2 — mi + m2) i f mi + m 2 + 2m2 ^ I ^ 2mi + mi — M2, 1 -|- %(mi m2 ~\' 2mi — I) i f 2mi -|- mi — M2 ^ Z ^ mi 4- m2 4- 2m.i. 87 Thus the polynomial P{m i,m 2,ni,n 2',X) can be deduced as = +"2 + 2%^"" -"‘+”2+2 + ... + i(m + n-2 - mi + m2)X"“+”*2+2"2 -2 + [1 + + »2 - mi + m2)](%""+'"2+""2 + • • ■ + + &»i + U2 - mi + m2)A’2'">+">-"2+2 + 2X’"‘+”*2+2"‘-2 + x'">+'"=+^"'. It is not difficult to see that P (m i,m 2,n i,n 2;X ) = X^"*‘-" ‘+"2 • P'{X) where the polynomial factor P'{X) has form P \X ) = 1 + + ... + + U2 - mi + m2)%"' + [1 + \{ni + ri2 - mi + m2 + • • • + + g(Mi + 712 — mi + m2)X^"' 2«2+2 _|_ 2 X ^ " ' +”*2—2 _j_ x ^ ”l +’"2 Factoring the polynomial P'{X)^ we obtain that P(mi,m2,Tii,772; X) = x'^”*'""'+"2 . P'(X) y 2 , n , (1 - X».+»2-»».+>n2+2)(i _ X'^(m-n.+l)) (1 - %2)2 Similarly, we can show that if 7%i + U2 — mi + m2 > 2wi — 2ti2, the polynomial P{ni],m 2^n-i,n 2\X) is also equal to _ ^2m, +n. (1-% ":+ "^-""+ '”^+") (l_% 2(»,-»3+D) (l-%2) (1-X2) This proves (a). By using the same argument, we will obtain the following identities: 88 (b) If (mi + m 2) > (»i + «2) > («1 — «2) > {'mi — m 2), then (I — y2("2+1)'| (I _ Y2(tni-ni2+l)') P ( m ,,m^,n„nr,X)^ ^ (c) If {tI] + 712) > (mi + m 2) > (mi - m2} > («1 - ÎI2), then P (m .,m2, n,,m-,X)= X ' " ' [iT x I)'-' - (d) If(ni + 722) > (mi + m2) > (ni — «2) > (mi — m2), then (1 _ ymi+m2-ni+n2+2)\ _ y2(mi-m2+l)\ P (m i, m 2, n „ „ 2i X) = X "" i------( Y T i n ------( l _ % .) ■ (e) If (/Ml—m 2) > (721+ 722) or (n i—712) > (mi +m 2), then f (m i, 7722, 721, 722; %) = 0. The theorem is proved. □ Combining Theorem 2.4.28 and Proposition 2.4.5, we obtain Corollary 2.4.29 Let (mi, m 2), (221, 722) he the finite-dimensional complex represen tations of Sp{2) with highest weight m\£\ + m2£2, m^ > m 2 > 0, 72161 + 72262, 221 > 722 ^ 0, respectively. Let (mi, m2|72i, 722) he the trace of the tensor product of repre sentations (m i,7222) and (221, 222) evaluated at the matrix {p\ 0/)i)„(ti„ x T2„). Then 00 '^trace{Sym}-fi{{pi 0pi)„(ti„ x t2„)))X^ 1=0 (1 _ yni+n 2-rni+m2+2Wi _ y-2(»»i-«2+I) 1 (m,, m,|n,, n,)X"" U— ------m,+m2>n,+n2> (l-X^)^ mj —m2 >nj —ng mi+m2>ni+n2 V / ni —7%2 >m\ —m2 ni+n2>”*l+"*2 \ / m j —m 2 > n j —TI2 / 1 y m i 4 -f)i2 —" 1 4 - " 2 4 -2 \ / 1 T ^ ‘2{Trii —Tri2 4 - 1 ) ^ + E (.ni,m.|ni,nJJI^"'-'">+"-l------_ " 1 ni+n2>mj+m2> \ / ”1 — ~mi2 89 2.4.8 The Proof of Theorem 2.3.14 Recall from (2.36) and Proposition 2.4.4 that (1 and r /•« + ! _ _ r \ \ E/=o*^’«ce(5ym^((pi (8 ) /?t)„( Theorem 2.3.14 will be proved if we prove the following identity: ■PW = Yltrace{Symy^{{pi ® X )))%'. (2.79) 1=0 In Theorem 2.3.13, we use (mj, m 2) to indicate the irreducible complex representation of Sp(2) with highest weight rniSi + m 2(£i + £2}, mi, m2 > 0. In order to make the notations compatible with these used in this section, we have to set some substitutions and rewrite the rational function P(X). Let A,’i = mi + m2, A=2 = m 2; Z] = 2tii — mi + «2, I2 = U2. Then (0) A-’i > A.'2 ^0 , Zi ^ Zg ^ 0 and kj + k2 "h Zi + I2 = 2(m2 + ni -f 7^2) = 0 (mod 2); (a) mi + m2 > ui + 7*2 > mi and 27ii > m \> n i are equivalent to A;i > ^(Zi + I2 + fci — k^) ^ k\ — &2 and Zi — Z2 ki — ^2 ^ k\ — k'l ^ ^(Zi — I2 A,'i — ^’2)) that is, k] d" Zu2 ^ Zi -j- Z2 ^ k\ — &2 ^ Zi — Z2, and also 3mi -|- 2m2 — 27ii = 2A:i — Zi -t- Z2, 2 (77,1 d" ^2 — 777i d" 1) = Zi d" Z2 — k\ d" ^’2 d" 2, and 2(277i — mi d" 1) — 2(Zi — I2 d" 1); 90 (b) mi + m 2 > ni + U2 and ni > mi are equivalent to k\ + k-2 > h + h and h — h > k\ — k-2, and also 2m2 — mi + 2ni = 2&2 + h — h, 2(u2 + 1) = 2(12 + 1), and 2(mi + 1) = 2{ki — &2 + 1); (c) rii +U2 > mi + m2 and 2ni > mi > ni aie equivalent to ni + ri2 > mi + m2 and mi — m 2 > Ui — U2, and also mi + 2n2 = ki — ^2 + 2Î2, 2(m 2 + 1) = 2(&2 + 1), and 2(2ni — mi + 1) = 2(l\ — I2 + 1); and (d) ni + 112 > mi + m2 > ni and ni > mi are equivalent to ni + 112 > m\ + m2 > ni — U2 > m, —m2, and also 4ni+2n2 —3mi = 21i — fci + A’2, 2(mi + m 2 —ni + 1) = k\ + &2 — Zi -f" I'i 4" 2, and 2(mi -|-1) = ‘2(k\ — k2 + 1). After substituting those data into f (%), identity (2.80) follows easily. Therefore Theorem 2.3.14 is finally proved. 2.5 The Fundamental Identity According to the results in previous sections, our degree 16 L-function L^(s\ tti ® 7T2,/)i eg) p\) can be expressed by the normalized global zeta integral Z*(»^(j)\,(j)2^ f) with a product of finite ramified local zeta integrals. More precisely, let tti = ®„7ri,„ and 7T2 = ®u7T2,u be irreducible automorphic cuspidal representations of G5p(2, A). The (global) degenerate principal series representation I^(s) is defined as in section 2.2.2. ?/> is a generic additive unitary character of the adelic group N'^(A) of the standard maximal unipotent subgroup N'^ of GSp{2). We have the following global result. 91 Theorem 2.5.1 (Fundamental Identity) Assume that the data ore factorizable, i.e., F that S contains all archimedean places of F and at any finite place v which is not in S, the local representations 7Ti,v and 7T2,u are unramified (or spherical) and the local character ipy is unramified and generic. Then the global zeta integral can be factorized as Z*(s, (pl, This fundamental identity is our starting point to study the degree 16 standard L-function L^(s', tti ® 7T2,pi <8> pi). Next we have to study the Eisenstein series, the analytic propertities of which will determine those of the global integral, and the ramified local integrals. fZ -d IZ 'd XV'^).z Og-rf ‘(‘/‘?.^‘i^‘s-)2' ^X'rf '^2 ig -d ‘09-rf g9' Og-cf ‘( 7 ‘‘--‘% ^ 3 ' 6 Vd X 7 ‘^ ‘^)^^ 2,g-(f ‘Hÿ ‘( 7 f p gT'rf gx d ‘ ®c/ÿ = ÿ Tg-d ‘(X 08'<^ '(A)X'... ‘ ( m ) X g g - d '•[a'"n'‘x^fi'-Cfi)X gg d <(%)*% 2S'(f ‘ff TT'(^ 6T-d ‘/ y ‘V II m pasfi saoi;%;of^ 9'g 36 CHAPTER III The Local Theory of Rankin-Selberg Convolution The analytic properties of the global integral Z{s, <^i, <^2) fs) is in general determined by those of the normalized Eisenstein series and the representation-theoretic, proper ties of 7Ti and 7T2. Following the fundamental identity (Theorem 2.5.1), the analytic properties of the i-function tti Z^(s,W uW 2, fs) = I fs{'yo{9\,g 2))Wi{gï)W2{g2)dgidg2. (3.1) JCN^’^{Z2Xh)\H More precisely, we will prove that those local integrals absolutely converge for Re(f>) large, and have, in nonarchimedean case, a meromorphic continuation to the whole complex plane, and also for an appropriate choice of the integrand data, those local integrals can be made to be a constant independent of s (in archimedean case, we assume that the section /, is smooth). To this end, we have to estimate Whittaker functions on the splitting torus, similar to [33], [35], and [68], and estimate an integral of mixed type in the sense that the special case of it will be either an intertwining 93 94 integral or certain partial ‘Fourier Coefficient’. Since we only consider local situations, we will drop v from the subscripts of all notations. Therefore, the underline field F is the real archimedean field or any nonarchimedean one with characteristics zero. 3.1 Some Estimates The estimates to be established in this section will be used to study those ramified local integrals later. The estimates of Whittaker functions are made in the standard way, while the estimates of the integral of mixed type will be more complicated. 3.1.1 Estimates of Whittaker Functions: Let 7T be any irreducible admissible representation of GSp{2,F) and W(7T, (/>) the Whittaker model of tt with respect to the generic additive unitary character of N'^, the standard maximal unipotent subgroup of GSp{2) in form: / 1 X z w \ 1 w' y iV^ = {u{x,y,w,z) = 1 }• —X 1 y As usual, the generic additive unitary characters ?/> can be assumed in following form: ?/)(u(x, y, w, z)) = ?/)o(x + y), where i/>o is any additive character of F. Recall from [36] that a finite function on a locally compact abelian group is a continuous function whose translates span a vector space of finite dimension. When the locally compact abelian group is any finite function on is a finite 95 linear combination of functions of the following types; X(«, = Xi(o)x2(&)|a|“‘ |6|“*(log|a|"‘ )(log|a|”‘ ), (3.2) where and X'i are characters module 1, u\ and u-2 are real, and and are nonnegative integers. We consider first the case that the underlying field F is a nonarchimedean local field. We shall use the methods from [33], [35], and [68] to estimate the Whittaker functions W in W(7t,0) on the splitting torus. Since the representations are as sumed to have trivial central characters, one has an identity W{h{abc, be, a~^c, c)) = W{h[ab,b,a~^,1)) for W 6 W(7r,V’) and h{abc,bc,a~^c,c) G Tg, the maximal split torus of GSp{2). So we can assume that any element in the maximal torus is of form h{ab,b,a~^,l) when deal with Whittaker functions. We denote Tj = {h{ab,b,a~^,l) G Î 2}. As in [68], we have Lemma 3.1.1 Let F be a nonarchimedean local field. Then we have (1) The function (a,b) H- W{h{ab,b,a~^,l)) vanishes when |a| or |6| is sufficiently large, and (2) the function (a,c,d) H- W{h{cd,c,c~^d~^a,c~^a)) vanishes when ]d| or is sufficiently large. Proof: The proof will be similar to that of Lemma 2.1 in [68]. Let u{x, y, w, z) G N'^. By the admissibility of W(7r, ?/>), we can pick up an element u = u{x, y, w, z) so close 96 to (but not equal to) I 4 that W{gu) = W{g) for g G GSp{2) and W G W(7ri,V-’)- Now for any t G Ag, that is, t = h{ab, a, 6~ \ 1), we have W{t) = W{tu) = W{tui~^t) = il){bx + ay)W{t). For a given u = u{x,y,w,z), if |a| and |6| are sufficiently large, rj;{bx + ay) will not be one and then 1F(<) must vanish. This proves part (1). part (2) can be proved in exactly same way and will be omitted here. □ Actually, Whittaker functions can be estimated more precisely by a formula ana logue to that in [33]. The argument to obtain such a formula is also from [33]. In the case under consideration, the reductive group is GSp(2). We have two maximal parabolic subgroup and P |> to conjugation. These are = M fN f with the Levi part = {GL{1) x GSp{l))° and = M |A^Siegel parabolic). Let Hi be the center of Mj, i = 1,2, respectively. Then by the same reason, we can assume that ifi = (A(a, 1, 1)} and H2 = (A(6,6,1,1)}. It is clear that A'2 = H1H2. Let W(7t,i/’)(A,) = {TTi{ni)W — W : rii G Ni, W G It is easy to check as in [68] that for W G W(7t, ?/))(AT,) and G Hi, W'(aj) = 0 when [a,I is sufficiently small or sufficiently large. The .Jacquet module W(7t, ?/’)jv,- = W(7r,j/))/W(7r, i/i)(iVj) is a finitely generated, admissible M,-module for % = 1,2. We denote by r, the rep resentation of Hi on W(7r,î/));v,-. Then the algebra spanned by {ri(a,) : G Hi} is of finite dimension. Now let V be the Ag-module obtained by restricting W(7r,i/>) to A'2 as in [33], and V, the subspace of vectors v E V which vanish for |c^| small enough and the representation of Hi on V/Vi. It is not difficult to see as in [33] that W{ir,ij;){Ni)\ni C V,- for z = 1,2. Therefore the representation ai is a quotient of 97 the representation r,- ajad the algebra spanned by { Proposition 3.1.2 There exists a finite set X of finite functions on so that for any Whittaker function W in there is for every % G a function G (3 3) xex where t = h{ab, 6, 1) in ^4^. Next we are going to estimate the Whittaker functions when the underlying field F is the real archimedean field. Since the results and their proofs are exactly same as those in [69], which are the analogues of the corresponding results in [35]. For convenience, we shall introduce the relevant notions and state the results. The proofs of those results will be omitted. Let g be the Lie algebra of the reductive group G = G5p(2,R). According to Casselman [15], any irreducible admissible (g, A' 2)-module V° can be realized as the (g, A72)-module of a continuous irreducible representation t t of G on a Frechet space which is smooth and of moderate growth in the sense of [15]. We assume that tt is generic. In other words, there is a continuous functional A on V„ with respect to the standard maximal unipotent subgroup and the standard generic unitary character ?/> of N^. If we set Wv{g) := ^{x{g)v) for n G 14, the space of all those Wv{g) is called the Whittaker model of t t and denoted by W(7r, t/>), which is unique after [65]. 98 One can define a gauge ^ on G as follows: Let G = be the Iwasawa decomposition of G, % a sum of positive quasicharacters of 7^ as defined in subsection 1.1, and b) a positive smooth and rapidly decreasing function in that is, given integers M, N, there is a positive constant C, so that cl>{a,b) Then the gauge ^ satisfies ^{uak) = x(a)^(a, &), (3.4) see [33]. Lemma 3.1.3 Let f 6 C^{G). There is a gauge ^ on G and a continuous semi-norm p on V„, such that \W^{f)v{g)\ < pivMg) for all g E G,v E V„. Proposition 3.1.4 There is a finite set X of finite functions on , so that for any W E W(7r,i/)), (1) there exist, for every a E X , a function (f)a E S{F^) satisfying W{h{ab,a,b~'^,l)) = ^ (j)a{a,b)a{a,b)] aex 99 (2) there exist, for every a G X , a function W{h{ab,a,b~^,l)k) = ^ (j>a{a,b-,k)a{a,b). aex 3.1.2 Estimates of Some Integrals 111 order to study the local zeta integral Z{s,Wi,W'2, fs), as in the computation of unramified local zeta integral showed in Chapter 11, it is critical to have enough knowledge about the integral of following type: I{U,h\fs,^k)= I fs{lo{uti,t 2))i>{u)du (3.5) where fs is an arbitrary smooth section in ^ is any generic unitary character of N'^, and (^1,^2) G C \ (T2 X T-2)°. To this end, we need some lemmas. Let Xo(^) denote a one-parameter subgroup of G corresponding to a root a and x{w,y,x,u,v) = - . 4 («)X-f2-=3 (i^)- (3-6) Lemma 3.1.5 Let fa G Isis). Then for u G Z2 \ and (f],t2) E C \ (Ï 2 x ^ 2)°, we have /*( 7 o(wti, t-i)) = fs ix i- ^ , -2/) a:, 1, l)hi(a, b, c, d)wo), where hi(a,b,c,d) = h(ab,a,c~^a,c~^d~^a,b~^,l,c,cd) and Wq = W 4^^4W3,4iV4 - 4. Proof: As mentioned before, for {ti, tf) 6 C \ (T2 x Tg)", we can rewrite it as, under the embedding Jf M- G, {t\,t2) = h[ab,a,cd,c, 6"^, 1, c“’a). Then its conjugation 100 7o( X—El —C4 (^)X“f2~®3 ^)X—e2~®3 ( ^)X—El —:4 ( and X—Eg—E3 ( 1)X—El—:4 ( 1)7 ° ^0" Similarly, for (tt, 1) E N'^ X I4, its conjugation 7o(tt, l)7o ' by 70 is equal to / I X —w —z — w x z w \ 1 — y — w w ' y I 1 x (-w , -y , X, 0,0) = p(w, X, y, z)x(-w , -y , x, 0,0) 1 — X 1 w ' y 1 \ —z — wx Z W I J (3.7) with p{w, x,y,z) E P3. Hence we have that /s(7o(w = fs{xi-w, -y, X, 1, l)hi{a, 6, c, d)wo). □ Lemma 3.1.6 For convenience, wesetx{w) = x(to,0,0,0,0), x{w,y) = x(t<;,;(/,0,0,0), x(z) = x(0,0,.T,0,0), • • •, x(^) = x(0i0i0)0,u), and so on. Then we have following identities: (a) h{t^,t. 2,t,3,( 4 , h , * 6, ( 7 , ts)"’x(«^iVi2;,w,v)h{ti, tg,( 3,<4,<5, te,<7 , h) = X{l3t8^w,tetÿ^y,t2ts'^x,tit^'^u,t2tj'^v); 101 (b) X- 2eAy) = A(l, 1, 1,1,1, -y , l)X2îs(2/)fcy, for |î/| > 1; (c) = /i(l,-z-\l,-æ-\l,-æ,l,-æ,l)%: 2+:X:K)&^, |æ| > 1; (d) x-ei-e^{u) = for \u\ > 1; (e) x-c2-ei{v) = h { l,- v - '^ ,- v - \l,l,- v ,- v ,l) x e 2+e3{v)K, for |u| > 1; (f) x{w,y)xe2+eAx) = X'S2-^i{-wx)xc2+e4{x)x{w,y)] (g) xMx2,a(y) = x%-eX-«':/)x3„(y)x-2,4(-"'^y)x(«;); (h) X(W,%/,Z)X,:+,X«) = Xei-S2{-Xu)Xci-e2{-Wu)Xe,+e4{u)x{w,y,x)] (■i) x{w, y, %)%,:+., (u) = P\ (w, y, X, v)x{w, ÿ,x)\ (j) x(w, y, X, u)xc2+^3 (o) = P2(w, y, X, v)x(w - xyv, y, x, u), where ky,kx,ku,kv E K4 and pi{w,y,x,v),p 2{w,y,x,v) 6 PsiF) and also h2(P\{w,y,x,v)) = 6p^{p2{w,y,x,v)) = 1. The proof of the lemma is straightforward. The integral I{ti,t2', fa, t/») as in (3.5) can be reduced as follows. Proposition 3.1.7 Let 62 = 5fj(h{ab,a,b~^,1) x h{cd,c,c~^d~^a,c~^a)) = Then the integral I{t\,t 2‘, fa, equals 52(|a|^|6||c|“^)*'^^ j ^ fa{x{w,y,x,abc~^d~^,ac~^)wo)tl2o{a~^cdx + a~^c^y)dwdxdy. 102 Proof: By Lemma 3.1.5, the conjugate h~^{a , , b, c, d)x{w, y,x,l, l)hi{a, h, c, d) is equal to x{f>'(^~^d~^w,ac~^y,ac~^d~^x,ahc~^d~^,ac~^) and fsiloiut^^h)) = (3.8) • fs{x{('‘d~^d~^w, ac~^y,ac~^d~^x, ahc~^d~^ ,ac~^)wo). Thus we can deduce our integral J(ti,< 2;/s ,’/’) as follows: I fs{lo{uU,t 2))il}{u)du J Z2\N^ = |n=6c-'|'+=|a|-#('+=) • I fs(x{ac~^d~^w,ac~^y,ac~^d~^x,ahc~^d~^,ac~^)wo)ij:{x+ y)dwdxdy Jpi = (^&(|n|#|6||c|-:)'+: • / ,ac~^)wo)il:{a~^cdx + a~^c^y)dwdxdy, (3.9) Jpi here we change the variables by ac~^d~^w t-4- w, ac~^y t-4- y; and ac~^d~^x t-4- x. □ It is reduced to study an integral of following type: I{u,v,a,fi]f,il:)= I f{x{w,y,x,u,v))%l:{ax + (Jy)dwdxdy. (3.10) JF^ Note that I{u, u, cv, /i; /, ^) can be reformulated as a degenerate .Jacquet integral as in [32] and [71]. It is well known from [32] and [60] that such integrals converge absolutely for i2e(.s) large. It will be interesting to know the meromorphic continuation in s and asymptotes in a, u, v of such degenerate integrals. Lemma 3.1.8 For any section /, G 7g(s) and any additive unitary character ■(/> of 103 the nonarchimedean local field F. the integral as in (3.11) equals ' y-, z)x(u, v))'tj}{ax + fiy)dwdxdy if |u|, |v| < 1; \v\~'^‘~'^SF3f{x[w,y,x)k{v)x{u))il){-vax-\-v^fiy)dwdxdy if \u\ < 1, |u| > 1; \u\~^~^Jp 3f{x{w,y,x)k{u)x{v))il^{-uax + fiy)dwdxdy if \u\ > 1, )u| < 1; !/,z) & ( w , ^ | w | > l,|u| > 1. Proof: We will evaluate the integral I{u, v, a, fi] fa, ?/>) case by case. (1) If |u| < 1 and |u| < 1, then there is nothing to do with because by definition X(w,y,x,u,v) = x{w,y,x)x{u,v). (2) If |u| < 1 and |u| > 1, then by Lemma (3.1.6) (e) and (i), fsix{w,y,x,u,v)) = fs{x{w,y,x,v)x{u)) = fs{h{l, - u " ‘, 1, 1, - V , - V , l)x{-V ~ ^W , V~'^y, -u"^z)xq+:,(u)t(u)x(u)) = |vr^*"®/s(x(-î^"’î">V^y,-u"^c)xe2+e3(u)fc(u)x(u)) = \v\~'^"~^f3{x{-v~^'^,v~'^y,-v~'^x)k{v)x{u)) Hence the integral can be deduced as I{u,v,a,fi;fa,i:) = / fa{x{w,y,x)k{v)x{u))il:o{-vax + v^fiy)dwdxdy, JF^ here the variables are changed by setting —v~^w i-)- w,v~'^y t-4- y, and —v~^x e4 x. (3) If |u| > 1 and |u| < 1, then by Lemma (3.1.6) (d) and (h), fs{x{w,y,x,u,v)) = fsix{w,y,x,u)x{v)) = fs{h{-u~'^ ,1,1, -u "’, -u, 1,1, -u)x(-u"'w , y, -u"^r)x«2+„ («)&(u)x(w)) 104 = \u\~"~^fs{x{-u~^w,y,-u-^x)x^^+,,{u)k{u)x{v)) = y, -u~'^x)k{u)xiv)). Hence the integral has form; I{u, V, a, /?; fa, V>) = / y,(x(w, y, x)k{u)xiv))i>o{-uax + (iy)dwdxdy, here the variables are changed by setting —u~^w ^ w,y y, and —u~^x t-4 x (4) If |u| > 1 and |u| > 1, then by Lemma (3.1.6) (d), (e), (h), (i), and (j), we deal with the variable v first and then with u and obtain that fs{x{w,y,x,u,v)) = fs{h{l, -t;~\ 1,1, -V, -V, l)x{-v~'^w, v~'^y, -v~'^x,u)xe2+^ai^)Hv)) = \v\~'^‘‘~^fa{x{-v~'^w,v~‘^y,-v~^x,u)xe 2+6a{v)k{v)) = \v\~'^^~^fa{x{-v~^w + v~'^yx, v~'^y,-v~'^x, u)k{v)) = m —s—3 \vL. I —a—6 ^Vi{vv) ^x)k{u,v)). Hence the integral is equal to I{u,v,a,/i-fa,il)) = fa{x{w,y,x)k{u,v))tl)o{uvax + v'^fSy)dwdxdy, here the variables are changed by setting (uv)~^w i->- w,v~^y t->. y, and (uv)~hv t-4 x. □ We consider two special cases of the integral I{u,v,a, /i] fa,xl}): (1) o = = 0 and (2) u = u = 0. It is clear that the integral in case (1) will be related to an intertwining integral as in following lemma, while the integral in case (2) is a partial ‘Fourier transform’ in some sense, which will be estimated in the next subsection. 105 Lemma 3.1.9 Let / , € Then the following integral 1 X 1 1 w y = / /-K g)dwdydx (3.11) ■IJpi /'( 1 1 — X 1 1 / defines an intertwining operator from I^„{s) to and takes the normalized spherical section f° in 7g(.s) to the normalized spherical section in where ! \ 0 \ 0 1 0 0 0 0 1 10 0 0 1 0 1 - 1 0 0 \ 1 0 / 3.1.3 Estimate of the Partial ‘Fourier’ Transform 111 this subsection, we assume the underline field F is nonarchimedean. We will estimate the ‘Fourier’ transform / fs{x{w,y-,x)fil){ax-\-(iy)dwdydx, (3.12) J where a and fi are in and x{w,y,x) is a unipotent subgroup of GSp{4:,F) of following form: x(w, y, x) := X-S3-Ï4 [y)x-e2-ei (*)• (3.13) 106 To do this, we need some Lemmas. Lemma 3.1.10 Let ip be any additive unitary character of the field F with conduct 5^, dx a Haar measure on F , and e any integer. For any a E F ^, one has following formulas: (i) / i,(ax)dx = [ \ J\x \< q -^ I (2) { I-q ~ ^ if ordy{a) > ord„(5^,), / ip{ax)dx = < —g"’ if ordv(oc) = ordv{S,p) — 1, [ 0 if ordv{a) < ordv{S,i,) — 1. (3) If ord„{a) > e + ord^{5^) — the integral /|a;|>,< \x\~^~'^ip{ax)dx equals a linear combination of 1 and with coefficients in C(g~*), the fi,eld of rational functions in g~® with complex coefficients; otherwise the integral vanishes. Proof: Formulas (1) and (2) can be verified easily. We omit the proof here. We are going to check formula (3). Since / \x\-^-^iP{ax)dx = It is easy to see that if ordv{a) — e < oi'd„{5,i,) — 1, then the last integral vanishes for any i > e. So we may assume that e < ordv{a) — ordv{S^,) + 1. In this case, the integral can be computed as follows: f la:I ^~^ip(ax)dx 107 ordv (a)-ordu E ' t=e ord„(o:)-ord„(i^,)+l-ordv{S^,)+1 E :=e« ./O -/O ordv(a)-ordv(S^) g-(»+l)*(l _ q - ^ 'j _ ç-(«+l)(ord„(a)-ord„((5^)+l)-l^ It is not difficult to conclude that the last expression equals a linear combination of 1 and |a|*"''^ with coefficients in C (g"'), the field of rational functions in with complex coefficients. □ Lemma 3.1.11 For any section fs 6 lUs) and %(u;) = (w), we have J^fix{w))dw = [^Q~^\wiMix{wiJ) + J2\wiJ^+'^f{k{wiJ)— ^ — ], (3.14) where the summation is over a finite set of indexes {%«,}. In other words, the integral is equal to a rational function of q~^. Proof: Since the degenerate principal series representation is admissible, there is a positive integer e so that /(%(u;)) = /(I) for |w| < q~^. Let us denote by the set of representatives of the cosets of ir^O in O. Then the integral can be reduced as follows: j^f{x{w))d%^ ' " iu) = [ , fix{w))dw + / f{x{w))dw = J^Q~^\wiJf{x{wiJ)+ I |u;r('+^V(t(u;"'))d*u;. V W >1 108 By changing the variable w t-4- ia“^,the last integral is equal to J\w\>l Jo<\w\ = Ç ^ |u;|®+Mu; □ Recall that i/’ is the additive character of F that defines the standard generic additive character ^ of the standard maximal unipotent subgroup N'^ of GSp{2). We set ^ki{a) := il){xiO.) and •= '*Kyj^)i where z, and yj are elements in F. The main result of this section is Theorem 3.1.12 For any fs 6 /|(s ) and any additive unitary character ?/> of the p-adic field F, the following integral l{oi,0-,fs,'il^)= I f{x{w,y,x))ip{ax +/3y)dwdydx (3.15) J f 3 is, if ordv{ci) > e and ordv{l3) > e for a positive integer e, a linear combination of and lal*"*"'where i,j varies in a finite set of indexes, with coefficients in C{q~^), the field of rational functions in q~^ with complex coefficients; otherwise the integral I{a, fs,ik) will be zero. Proof: Basically, we deduce each variable once by means of the admissibility of the representation pg, the degenerate principal series representation First of all, let 109 u)e2+£4 be a Weyl group element of following form: /I 0 \ 0 0 1 1 0 0 1 0 0 1 0-10 0 1 \ -1 0 0 / Then there exists a positive integer e* so that f{gX-e2-sA^)) = fis) and psiw^ 2+e,)if){gx-e2-eii^)) = Psiwe2+u)if)is)^ for any |x| < by the admissibility of pg. The integral /(o;, /3, ; /, i/i) can be deduced as follows. = I [ fixi^,y)x~c2~eiix))i^[ax)dxdwdy JF^ J\x\ = 7( Then we consider the integral , fix i‘w,y)x-e2-^A^))‘‘Ko‘^)dx. (3.17) J\x\>q^x By Bruhat decomposition, one has X-£2-£4(.ï) (3.18) h[l, X ,1,'C ) 1? ^)Xe2+^4 (•^)X—e2~®4 ( ^ )*^«2+e4' Plugging in the integral 7(w,y;/,), we have I{w,y;fs) = / , y)X - , : - = 4 (-a:"^)«;.2 + . 4 )^(aa;)da: ./ T >a«z = / , kl ' ^fsixi^ ^w,y))wg^+g,)jl^{ax)dx. (3.19) J\x\> q ^x 110 So, the integral /(> g®*) becomes / , )tl}{ax)dxdwdy JF^ V|r|>g«z = JF^L ' ^ i M J|x|>g** [ , y)M)e:+.4 = / , \x\~^'^tKo‘x)dx f tKl3y)f{x{w,y)w^2+ei)dwdy. (3.20) J\x\>q'^x Jp2 Denote by {xj} the set of the representatives of where V denotes the maximal ideal in the ring O of integers in the p-adic field F. Then we obtain that /( < = X) / f f{x{w,y,xi + x))7l){a{xi + x))dxdwdy (3.21) “JF^ J\x\ = / tl^{ax)dx I {ps{x{xi))f){x{w,y))xl^{/3y)dwdy. i JF^ In other words, we obtain that 7(a,^,;/,V) (3.22) = / I rk{^y)fix{w,y)w^^+et)dwdy J\x\>q^x Jp2 + J2il^{axi) f il){ax)dx [ p^{x-ei-c,{‘->^i)){f){x{w,y)yKfty)dwdy. Next we have to consider the integrals of following type: /(/); fs, %A) := fsixiw, y))‘il^{/3y)dwdy, (3.23) for any section /, in I^(s). Applying the same procedure to the integrating variable y, we obtain + Y^'‘K0yj) j ' Note here that the positive integer Cy depends on more sections / ’s than does and {yj} is the set of all representatives of 'P~^v j'P^v, Finally we obtain that = / \x\~^~'^il){ax)dx f \y\~^~^^{^y)dy f f'{xM )dw j\x\>q‘^ J\y\>q'‘V Jf + IZ jp,fAx{w))dw + 53V’(«a:.) I i){ax)dx f \y\~"~^^KMdy [ fi{x{w))dw (3.25) + ^^(aa:.)/ ^{ax)dxtl^{^yj) f il}{/3y)dy I fij{x{w))dw 7 X f := P.(W2:,U,.;+:J(/,) fj ■= Ps{X-2e,{yj)w,^+,,){fs) fi •= Ps{uJ2c3X-e2—sti^i))ifs) fhj /^s(X-2e3 (ÿi)X-e2-ï4 (^0)(/»)‘ According to Lemma 3.1.10 and 3.1.11 above, there is a positive integer e such that when 07'dt,{a) < e or ordv(^) < e, the integral 7(a, /), ; /, i/») will vanish. On the other hand, if oi'd„{a) > e and ord„{^) > e, the integrals jjx| \x\~^~'^rl7{ax)dx and /|y|>,= \y\~^~^tl^{o‘y)dy are equal to linear combinations of 1 and with coefficients in C (ç“*), the field of rational functions in q~^ with complex coefficients. Therefore, if ordu{a) > e and ord„{l5) > e and if we set î/)i(cv) := 7j;{xia) and 0j(/3) := 7j;{yj0), the integral J(o!, /), ; /,?/)) is a linear com- 112 bination of tlJi{a)ipj{/3), and with coefficients in C(ç“*), the field of rational functions in q~‘ with complex coefficients. □ 3.2 Nonvanishing of Local Zeta Integrals The local zeta integral we are going to study in this section is Z{s,Wi,W2,f)= f f{'yo{g\,g2),s)Wi{gi)W2{g2)dgidg2, (3.26) JV(F) where V := CN'^’^{Z2 x I4) \ H and / , is any smooth section in /^(.s) and W^, W2 are smooth Whittaker functions of GSp{2, F) in the Whittaker models W(7Ti, i/>), W(7T2, î/>), respectively. We shall prove that one can choose appropriate data (/«, W), W 2) so that the integral Z(.s, Wi, W 2,/) is a constant independent of s. 3.2.1 Nonarchimedean Cases: We begin with proving the nonvanishing property for a certain type of integral similar to the integrals studied extensively by .Jacquet and Shalika in [35] and [36]. We first recall some notations. Let be the maximal parabolic subgroup of GSp{2) with Levi decomposition Pf = M^Nf. The Levi factor Mf and the unipotent radical N'-^ are in forms: a \ / 1 X z w \ ( " ^ 1 } and Nf = {n(x,w,z) = f ^ ° i } We choose such an embedding of GL(2) into the Levi part of the parabolic 113 subgroup Pf of GSp{2) as ^ detg ^ at ~I3 ( 7 â ) I—»'m(detg,g) = 1 -7 ^ J Let B be the standard Borel subgroup of GL{2) and B = N T in the usual sense. We are going to study the following integral I{s,W ,,W 2,^) = f _ W,{m{detg,g))W2{7n{detg,g))mO,l)g)\detg\^dg, (3.27) JU2\GL{2) the properties of which will be used to prove the nonvanishing property of the integral yV{s,Wi,W2,fs) for an appropriately chosen data {Wi,W2,fs)- Lemma 3.2.1 (a) For any Whittaker functions Wi G W(7ri,V’) and W 2 G W(7r2,?/>), and any Schwartz-Bruhat function $ G S{F'^), the integral 7(s, Wi, Wg,$) converges absolutely for Re{s) » 0. (b) For a suitable choice of Whittaker functions W% G W(7ri, i/i) and W2 G W(7r2, i/>), and for a suitable choice of a Schwartz-Bruhat function <5 G S(F^), the integral 7(.s, Wi, W2, $) is a nonzero constant independent of s. Proof: The proof of the absolute convergency of the integral will follow as a special case from that of absolute convergency of the local zeta integral, which will be given in the next section. Here we only give the proof for (b). In terms of Iwasawa decomposition of GL(2), U2\GL{2) = [U2\P]F^K', where K' = GL{2,0), F^ is the center of GL{2), 114 and U2\P = ( 0 i )• Then the integral /(a, Wg,as in (3.27) equals l,rsr for any Whittaker functions Wi and W2, and any Schwartz-Bruhat function . We denote the inner integral by Wi, W 2), that is, J{g,-^,Wi,W 2)= I \detp\^W^{pg)W 2{pg)dp. (3.28) JÜ2\P Then this integral enjoys following two properties: (1) J(p£T, s, W ,, W2) = J(g, s, Wi, W2) for p G P n K'. (2) We can choose Whittaker functions W\ and W2 such that the integral J(l,.S,^l,W2) = l. Property (1) is straightforward, while property (2) is not so direct to see. We are going to verify property (2). Note that there are W hittaker functions W, G W(7Ti,'0) and W2G W(7T2, (/>) so that W,(l) = 1 for z = 1,2. Then we can pick up a Schwartz-Bruhat function G |o|" #i(/,,2(a, a, 1, l ) ) #2(A2(a, a, 1, l))$ '(u ),P a = 1 (3.29) where h,2{a, b, c, d) denote a diagonal matrix element in GSp{2). 115 We then consider the Fourier transform 4' of with respect to the additive character V’o of F, i.e., ^'(y) = ^'{x)ipo{yx)dx, (3.30) which is also a Schwartz-Bruhat function on F. We define a convolution of the Whittaker function W2 with the Schwartz-Bruhat function by W2{g) := W2{gu{Q, y, 0,0))$'(y)dy ( 3.31) for g e GSp{2). It is easy to check that Wz(^) is again a Whittaker function in W(7T2,t/’). With those choice of Whittaker functions Wi and W2, the integral J{g, s, Wi, W2) at y = 1 will equal one since J(l,.Si,Wi,W2) |a|*'Wi(A2(a, a, 1, l))W2{h2{a, a, 1, l))d^ct = \a\"^Wi{h2{a,a,l,l)) J^W2{h2{a,a,l,l)u{0,y,0,0))é'{y)dy(Fa = \a\^^W-i{h2{a,a,l,l))W2{h2(a,a,l,l)) J^é'{y)‘ip{ay)dyd^a = \a\^^W,{h2{a,a,l,l))W2{h2{a,a,l,l))^'ia)d^a = 1 . This prove that the integral J(y,.s, Wi, W 2) enjoys property (2). Note that with the chosen Whittaker functions Wi and W2, the integral J{g, s, Wi, W2) defines a locally constant complex-valued function on GSp{2,F), which we denote by J{g). Then there is a small open compact subgroup /f' contained in K' so that the restriction of J{g) to K[ is a nonzero constant independent of s and (0,1)/^' C We choose the 116 Schwartz-Bruhat function $ = ch^o,i)K'„ the characteristic function of (0, l)/iT'. Then the desired property of the integral /(s, Wi, W 2, #) for the chosen data (Wi, W2, can be deduced as follows: I{s,WuW2,^) = J^J^^J{ak')m O ,l)ak')\a\“^d^adk' = J^ J{k')mO,l)k')dk' = m It is sure that the last integral is a nonzero constant independent of .s. This proves the lemma. □ Now we turn to deal with the local zeta integral J(.s, W\,W 2, fs)- We need more notations. The center Z2 of Nf is the one-parameter subgroup X 2ei (2). By means of the isomorphism Z2\N f = as algebraic varieties, it is well known that there is a surjection from the space S{N^) of Schwartz-Bruhat functions on N f onto the sijace S{F'^) of Schwartz-Bruhat functions on F^. The surjection is defined by the integral: p. : y I— > ^{x,w) = j^ip{n{x,w,z))dz. (3.32) We also define the Fourier transform $ of $ with respect to the additive unitary character ?/>o(— utw — vx) of F^ by $(z, u), ) = J ^{u,v)ij){—uw — vx)dudv. (3.33) Note that we can choose a Schwartz-Bruhat functions Theorem 3.2.2 (Non-archimedean Cases) We can choose an appropriate sec tion fa G Isis), and Whittaker functions W\ G W(7Ti,V’) o-nd Ws G W(7r2,?/’), so that the (unnormalized) local zeta integral Z{s,W \,W 2, fa) equals a nonzero constant independent of s. Proof: Let Wi, W2 be the WhittaJser functions and $ the Schwartz Bruhat function on as chosen in the proof of the previous lemma. We define now another Whittaker function Wi in W(7ri,V-’) as follows: for g G GSp(2) and (p G S{Nf), W,{g) := lw ,igu)p{u)du. (3.34) JNf Since the representations tv\ and are admissible, there is a small open and compact subgroup Ko contained in the maximal open and compact subgroup K2 of GSp{2) so that Wi and W2 are right Jfo-invariant. Next, we are going to make the choice of the section /,. Since Pf'joH is the unique open H-orbit in Pf\GSp{4:), it is easy to check that the following isomorijhism of if-modules holds { / e Isi-^) : -^uppif) C PsJoH} = filoigug-i)) = /*( Here P('^{Z2 x I4) is the stabilizer of jo in H and f* has compact support modulo X 74 ). Applying the Iwasawa decomposition of 77 = {GSp{2) x GSp{2))° to our situation, we can decompose the group H into H = P('^[Z2 x / 4)([Z2\P i] x 74)°(A2 x K 2Y. 118 We claim that P ^ ^ (Z 2 X h ) n [([Z2\Pi] X h y { I < 2 X K -iT ] = X h ) n (j^2 X l u y . (3 .35) This will be proved at the end of this proof. We denote Q = P^{Z2 x I4) D {K2 x K2)° and Qo = Pi'^{Z2 x I4) fl (Ko x Ko)°- Then the quotient Q/Qo is a finite set, which is denoted by { 51,521 ••• Let fio = U|_., 5j(/<’o X Ko)°- Then fio is an open compact subset in (.^ 2\Pi x l 4)°{K2 x K2)°. It is evident that there exists a section /* € (|a|^|a'|"^) so that the •*1 ("2Xi4) restriction of f* to {Z2 \ Pi x I4)°{K2 X ^"2)° is equal to the characteristic function of do and its support supp{f*) is P ^ { Z 2 X h^K o x Ko)°- In this way, we obtain the section / in /^(s) via the correspondence indicated above. With the data (Wi, W2, /) as chosen above, the local zeta integral Z[s, W|, Wi, /) equals oy., M /(7o(5i,<72),s)W,(<7i)W2(<72)d(7id(72 JCN^<^(Z2XU)\HI = f \a\ï^W-[{pki)W2{pk2)dpdkidk2 J[C2N^\Pi](KoXKo)'> = p{{KoXKo)°)f \detgyWi{m{detg,g))W2{Tn{detg,g))dg JU2\GL{2) where p{{KoXKo)°) denote the volume of {KoXKo)° with respect to the Haar measure on the group GSp{2). Replacing Wi by the corresponding integral, we obtain that I \detg\^'W4 {m{detg, g))W 2{m{detg, g))dg JV2\C1L(2) = / \detgY' / Wi{m{detg,g)u)ip(u)duW 2{m{detg,g))dg JU2\GL{2) Jn^ = / , , \detg\^'Wx{m{detg,g))W 2{m{detg,g)) *'V2\OL{y2) 119 • / ip{u(x,w,z))i{}{'yw + 5x)dzdxdwdg JN^ = / _ \detgY'W]{rn{detg,g))W 2{m{detg,g)) I ^{x,w)i}){'yw + Sx)dxdwdg JU2\GL{2) Jp^ = f \detg\^'Wi{m{detg,g))W2{m{detg,g))^({0,l)g)dg, (3.36) JU2\0L{2) where we let g = ^ ^ ^. This implies that $(( 7 , Lemma 3.2.1 (b), with the chosen data (VFi, WSj,/*), the last integral (3.37) equals a nonzero constant independent of s and so does the local zeta integral 2{s^ , W2, /*). Finally we are going to prove the claim: ‘P f’^(Z 2 X I4) D [{Z2 \ f i x l4)°{K2 x Ag)"] = P^’^{Z2 X I4) n (7^2 X In fact, for any x E P^'^{Z2 x I4) n [{Z2 \ P i x I4)°{I<2 x A 2 ) ° ] , X can be written as X = {pz,p) = {piki,k2). Then we have p = k2 and pz = p% &i. This implies that ki G Pi D IC2 and the Levi part ni] of pi belongs to Mf D K2. Hence we obtain that x = (pz,p) = (niiu,p) with u E Z2 \ U\ and p E P4 C\ Kz. This implies that u E U\ C\ K2 and z E ZzH Kz- Therefore x E P f’^(Z 2 X I4) (~1 {K2 X KzY- Our claim is true. □ C o r o l l a r y 3 . 2.3 With an appropriate choice of the data {Wi,W2,fa), we can make the local zeta integral Z(s,Wi,W2, fa) = 1. 120 3.2.2 Archimedean Cases: We assume in this subsection that the local field F is the real archimedean field. We denote by Ki the standard maximal compact subgroup of GSp{2, F). In other words, K2 = GU{2). Let tT] and tt2 be irreducible admissible representations of GSp{2,F) and 7Ti,o and 7T2,o the /^ 2-finite vectors of and 7T2, respectively. We denote by Wo(7ri,t/)) and Wo(7r2,V’) the /iT 2-finite W hittaker functions in W(7ri,?/’) and respectively. However, we assume that the degenerate principal series representation 7j(.s) consists of all smooth sections (not necessarily fiT^-finite). Our local zeta integral is .2(.s, W ,,W 2,/) = / f{li9i,92),s)Wi{gi)W2{g2)dgidg2. (3.37) J[CN^’^(Z2Xl4)\H] It is the goal of this subsection to prove an archimedean version of the nonvanishing property of local zeta integrals. The statement will be Theorem 3.2.4 (Archimedean Cases) For a given value .sq of s , there are Ki- finite Whittaker functions Wi G Wo(7Ti,^) and W 2 G Wo(7T2,V’); o,nd then there is a sm.ooth function / , G /^(-s), so that the zeta integral 2 ( sq, W], W2, / ,) dose not vanish. The idea to prove this theorem is similar to that used in our proof of non- archimedean cases. In other words, we have to prove the theorem for an integral of the following type 7(.s,Wi,W 2 ,$ ) = / \detg\W^{m{detg,g))W2{Tn{detg,g))^{0,l)g)dg. (3.38) JU2\GL{2) 121 Again such types of archimedean integrals were studied extensively by Jacquet and Shalika in [35] and [36]. Lemma 3.2.5 (a) For K 2-finite Whittaker functions Wi and Wz and Schwartz function #(;c,y) G the integral I{s,W i,W 2,^) converges absolutely for Re{s) » 0. (b) For a given value s, we can choose Kz-finite Whittaker functions W\ and Wz, and a function $ in C^{F'^) so that /(.s, W], W2, $) does not vanish. Proof: Statement (a) will follow from the estimates of W hittaker functions. The jjroof for more general cases will be given in the next section. We are going to prove (b) here. Let K' be the standard maximal compact subgroup of GL{2,F), i.e., K' = 0(2). Applying the Iwasawa decomposition, the quotient Uz \ GL{2) can be written as [Uz\P]F^K' where F^ is the center of GL{2) and U z\P = Thus the integral /(s, Wi, W 2, $) can be deduced as follows: I{s,Wx,Wz,^) = f _ \detg\^Wi{m{detg,g)Wz{m{detg,g))^{{0,l)g)dg JV2\0L(2) = f I f \detp\‘’'W,{pak')Wz{pak'))dpmO,l)ak')\a\^^d^adk'. (3.39) J*'U2\P As in the non-archimedean cases, for a fixed Si, we denote by J(flr,.Si, Wi, W 2) the inner integral of (3.40), that is, J{g,si,Wi,Wz) = f \detp\‘'^Wi{pg)Wzipg))dp. (3.40) 122 Note that the function J{g, 5i, Wi, W2) is /C'-finite. We are going to pick up some Whittaker functions Wi and W2 so that the integral Wi, W2) ^ 0. We choose /<’ 2-finite Whittaker functions W\{g) in W(7Ti, •) and W2{g) in W(7T2,ï/’) so that Wi(e) = Wile) = 1- We choose a function 6 C^{F^) whose support is such a small neighborhood of 1 that the following integral (3.41) does not vanish. For the sake of convenience, we can make it equal to 1. We define the Fourier transform of with respect to the additive character iJjo as é'{ij) = j^^'{x)'ipo{yx)dx, (3.42) which is again in C^{F^). Then a convolution of the Whittaker function W 2 and the Schwartz function is defined as Wz(^) := W2(^u(0, y, 0,0))$'(y)4, (3.43) which is a Whittaker function in W(7T2,V’)- For such chosen Whittaker functions W\{g) and Wzlg), the integral J( .7 (l,.si,W,W2) = \detpY^W\{p)W2{p)dp W^{h2{t,t, 1, l))W2{h2{t,t, 1, l))dt^ Wi {h 2{t, t, 1,1)) W2{h 2{t, t, 1, l)u(0, y, 0,0))é'{y)dydf^ Wi{h 2{t,t, 1, l))W2{h 2{t, t, 1,1)) i>'{y)^l;{ty)dydt^ 123 = Wi(Az(<, 1, l))T^(A2(t, 1,1))$'(<) = 1. (3.44) With the chosen Whittaker functions Wi and W2 and the fixed Si, we set J{g) := Wi, W2), which is a /f'-finite function on GL{2) and J(l) = 1. With the preparation above, we can prove the statement (b) as follows: (3.40) = f [ I \detp\^^Wi{pak')W2{pak'))dpmO,l)ak')\a\^^d^adk' JJJU2\P = J(afc')<5((0, l)ak')\aY^d^adk' = j^ J{k'a)mQ,l)k'a)dk'd^a. (3.45) We define $((0, !)&'«) := (3.45) = |a|*^ J{k'a)J{k')dk'(j){a)d^a. (3.46) Note that if we set (p(a) = j)^-, J{k'a)J(k')dk\ then the function tp[a) is continuous in a and y(l) = 1. We can assume that the support of the function ^(a) is in such a small neighborhood of 1 that the restriction of y(u) to the support of (j) is always positive and the integral (3.47) does not vanish. This finishes the proof. □ Proof: (The Proof of Theorem 3.2.4) For a given complex number .s, we choose W] and W2 as in Lemma 3.2.5. As in the p-adic case, we have the decomposition of the group H, H={PrX P ,y { K 2 X I<2)° = P ^ ' \ Z 2 X h){[Z 2\Px] x h Y { K 2 X K 2T, (3.47) 124 and the following isomorphism of ^-modules { / 6 ind%{5^) : supp{f) C P3J0H} ^ filo{gu92)) = /*(<71,<72)- Let °Mi = Pi n K-2- Let kg and °mi be the Lie algebras of °Mi and 1^2, respectively. Then we know that k 2 = °mi ® ko (orthogonal direct sum with respect to the Killing form on k 2) and ko is locally homeomorphic to the quotient °Mi \ K2. Choose an open neighborhood do of 0 in ko so that the exponential image Do of do is an open neighborhood of the distinguished element e in the quotient °Mi \ K-z, (the size of which will be determined later), and P ^ { Z z X I^){[Z2 \ Pi] x I^)°{Do x Do)° is an open subset in H. Therefore we can choose a section /* in the following form: /s((Pî-^4)(di.d2)) = (t)o{p)^o{d\,d2), (3.48) where d>o{p) is smooth and compactly supported over Zz \ P\ and yo(di, dz) is smooth and compactly supported over {Do x Do)° (whose size of the supports will be de termined later). Since Pi = Mi Pi, we may require that (f)o{miUi) = ^o(mi) • f \detp\^'Wi{ppidi)W2{pd2)dpdpid{di,d2) JCN^\Pi • f \detp\^'W\(puiTnidi)W2{pd2)dpdpid{di,d2). (3.49) J C N^\P\ and the inner integral equals I <^o(mi)<)io(ui) I \detpfWx{pu^midi)W2{pd2)dpdpi J{[Z2\Pi]'xh)'’ JCN^\Pi = / |deimi|“^o(mi) / \detgY'W\{m{dttg^g)m\dx)W2{m{detg^g)d2) J(M \y.î\)^ JU2\Oh('Z) • j (j)Q{u])xj){Ad{m{detg,g)u\)du\dgdmxd. JZ2\Ui We set #((0,l)g) = (j>o{ux)ip{Ad{m{detg,g)ui))dux. Then the last integral becomes / |deimi|"(;6o(mi) ■ I \deig\‘'Wx{m{deig,g)mxdx)W 2{m{detg,g)d2)^{{0, l)g)dgdnix. JU2\GL(2) Let denote the inner integral in the above integral by (mid], dg); Wi, W 2, $), that is, ^ ( / , (m idi, dg); Wx, Wg, $) = / |det<7|'’'Wi(m(det5r,<7)midi)Wg(m(det(7,<7)dg)§((0, l)<7)d5f. JU2\GL{2) Then it is easy to figure out that X(.s% (1,1); Wj, Wg, $) = /(s', Wi, Wg, $). According to Lemma 3.2.5, if we pick up such a function By means of this nonvanishing property, we may choose the function ^o{di,d'2) in C°°{{Do X Do)° and the function in C^(M i) with their compact support in such small neighborhoods of the identities in the group Mi and the group {Do x Do)°, respectively, that the integral / ^o{di,d2) I \detmi\°‘^o{mi)A{s\{midi,d2)]Wi,W2,^)dm-id{di,d2) J(Do'X-Do)'*' J(Mi'kU)'‘ does not vanish. The theorem is finally proved. □ In our set-up, we made an assumption that at a real archimedean jjlace, the sections /, be -finite in the degenerate principal series representation when we established our Rankin-Selberg global integral in the first chapter. In other words, we have to refine Theorem 3.2.4 so that the same result holds for any Ka,oo- finite section fa in before we use the theorem to study the analytic properties of the degree 16 L-function L^{s\7t (g3 7t,/9i Cx) p\). We will return to this later. 3.3 Absolutely Convergence of Local Zeta Integrals We shall prove in this section that the local zeta integral I{s,W \,W 2, f) converges absolutely for Re{s) large- for both archimedean and nonarchimedean cases. We first prove some lemmas that hold for both cases. Lemma 3.3.1 Let fg,f° € o,nd f° the normalized Kn-spherical section. Then one has following estimate: \Ms)\ < Ckj-JM 127 where Ck j is a constant independent of s. Lemma 3.3.2 Let F be a nonarchimedean local field. Let be a right K^-spherical section in as in Lemma S. 1.9. Then we have following estimates: $°(x(w,u)) = max{l, |«|}“*“^ max{l, where x{u,v) = X_e,+,z(u)x_„+,4(^) Lemma 3.3.3 Let F be a real archimedean field. Let be a right K 4-spherical section in as in Lemma 3.1.9. Then we have following estimates: #°(x(w,v)) = (1 + 3.3.1 Nonarchimedean Cases: We are going to prove that the local zeta integral Z(.s, Wi, W2, fs) converges absolutely for the real part of 5 large, that is, to prove the following theorem. Theorem 3.3.4 (Non-archimedean Cases) Letip be any generic character of . For any fs G and any Wi G W(7Ti , j/i) and W 2 G W(7T2, tp), the local zeta integral Z(.s, Wi, Wg,/) = / f{'yo{gi,g2),s)Wi{gi)W2{g2)dg4dg2 (3.50) JCN^’^(Z2XU)\H absolutely converges for Re{s) >>0. 128 Proof: In terms of the Iwasawa decomposition H = (Pi x Pf)°{K2 x K 2Y 1 we can write the quotient CN'^'^{Z2 x l4)\H as CN^'^{Z2 X h ) \H = {Z2\N^)[C\{T2 X T2Y]{K2 X jTz)". (3.51) Then the local zeta integral reduces as follows: = , ^^^fs{lo{gug2),s)Wi{gt)W2{g2)dgidg2 JCN^'^{Z2Xh)\H / fs{'io{uti^t2){kx,k2)Yl>{u)dudadk. (3.52) By the admissibility of the representations involved, we can pick up a small open compact subgroup in K2 so that the Whittaker functions Wi and W2 are right Æg-invariant and / , is right {K'2 x A"2)°-invariant. We denoted by {(^•,-,^f) : i = 1,2, the finite quotient {K2 x K 2)°I{K'2 X PTg)". Then integral (3.52) can be reduced further and becomes a sum of integrals of nice shape. (3.52) = f^X*i,(2)Wi((itj)W2((2A:n ,=1 ^ JCMTiXTz)^ X I fs{'yc{uU,t2){k'i,k'/)Yp{u)duda X / fY\jo{uti,t2))‘il:{u)duda (3.53) ./Z2VV2 where is the volume of the open compact subgroup {K!^ x Kl^Y with respect to the Haar measure on the group H and we denote Wi’^ = Tr]{k'{)W\ and = 7r2(A,f)W2, 129 and = Pa{m, It thus reduces to prove the absolute convergence of following type: I(s,Wi,W2,fs) := f Wi{ti)W2{t2) f /,(7o(M Taking absolute value and applying Lemma 3.3.1, we obtain that \I{s.W^,W2js)\ < f \W,{U)W2{t2)\ I \M-/o{utut2))\dudhdt2 JC\(T2XT2)“ < Ck f \Wi{U)W 2 {t2 )\ f f°{'yo{utuh))dudUdt2 •/C\(T2XT2)® = I{s,\W^l\W2\J:). (3.55) Basically, we reduce the problem of absolute convergence to the case where the section f° is /^4-spherical. We set I{h,h\fs)-= I ^ Ja{lo{uti,t2))du. (3.56) ./Z2VV2 Applying Proposition 3.1.7 to the case of V’o = 1, Lemma 3.1.9, and Lemma 3.3.2, we have = ^2(|at6c"*|)*+' / f°{x{w,y,x,u,v))dwdydx Jp3 - S2\ahc ' 1^+^ ^ max{l, |u|} " ^nax{l,|u|} \ C(-s + 2)C(-‘* + 3) where u = abc~^d~^ and v = ac~^. Recall that in the coordinates chosen on C \ (Î 2 x T^)", U = h{ab,a,h~^,l) and ti = h{cd,c,c~^d~^a,c~^a). By Proposition 3.1.2, we have I^i(ti) = I]9!>xi(«>^)Xi(«,&) and Wz(t2) = ^<^xa(c^a"\d)X 2(c^a"\(() (3.58) Xl X2 130 where the summation is tahen over a finite set of finite functions %'s and ^%'s are Schwartz-Bruhat functions over It reduces to prove the absolute convergence for integrals of following type: / S'2\ahc-^ 1"+^ l^x. (g, b)x\{a, b)^^^{c‘^a-\d)x2{c'^a~^ , d)\ , J max{l,|u|}'+:max{l,|u|}2'+2 d{a,b,c,d) (3.o9j where the integration is taken over the domain: 0 < |a|, |6|, |c|, |d| < oo. Note that, in general, the absolute value of a finite function of two variables can be written as %(«,&) = lal'lfej^'llog |a|"*log |6|"|, and when t goes to oo, |t'lo g |t|’"| is bounded by for any e > 0, and when t goes to 0, log |i|"*| is bounded by for any e > 0. Without loss of generality, we may assume that X,(«,6) = |a|“-|6|“= and X2{c^a~\d) = a~^ \d\^^ (3.60) for our purpose to prove the absolute convergence of the integral in (3.56). Hence we have to show the convergence of following integral: r (a, 6 )^ (c ^ a -', d) . J ---- m=a{l,|„|).+.max{l,|„|P*« ‘ ’ ’ ’ ' ' ’ where the integration is taken over the domain: 0 < |a|, |6|, |c|, |d| < oo and the Schwartz-Bruhat functions are positive and .s is a large real number. Changing the variables (a, 6, c, d) to (a, b, u, v), the integral becomes J ------max{l,|„|)-fmax{l,|HP«------where the integration is taken over the domain: 0 < |a|, |6|, |u|, |u| < oo. We write the integral as a sum of following two integrals: 131 ■ f |u|*'“ ^(j>2{av ^,bvu ^)dudvdadb (3.63) /|u|>l and • f \u\'°'"(j)2{av~^,hvu~^)dudvdadb. (3.64) •/|u|>0 When 1 < |u| < oo, there is a constant Ci so that |u|'~-'^(on-", bvu-^) < Cl |u|^“-". (3.65) Thus we have /*oo , ^ , too / |u| “~®^2(ov~ ,bvu~ )du < Cl |u| '‘~^du, (3.66) V|u|>l which converges for s large. Because of this estimate for the integration of the variable u, integral Ii is bounded by The integral |a| 2«+*“|6|^+*'<>^,(a, 6)dad6 converges since is compactly sup ported and .s can be chosen suitably large. While the integral X Z o = L I"!-"'-'" + L which is convergent for s positive and large. Hence integral Ii is convergent. On the other hand, when 0 < |u| < 1, there is a positive integer e so that / \u\^''(j)2iav~^,bvu~^)du < C2 f (3.68) J\u\>0i | | > 0 ' ' " - 7,-e|fcü|<|u| = C2 1 — g-(tu+l) 132 So integral I2 is bounded by a two-term linear combination with complex coefficients of integrals of following type: which is convergent following the same argument as used in the case |u| > 1. □ 3.3.2 Archimedean Cases: In this case, the proof of the absolute convergence of the local zeta integral will be very close to that of nonarchimedean case. The theorem we are going to prove is Theorem 3.3.5 Let F be a real archimedean field. Let tj) he any generic character of N^. For any smooth fs G and any K 2-finite Whittaker functions G W(xi,t/>) and W 2 G W(7T2,?/>), the local zeta integral JCN^’^(Z2Xh)\H absolutely converges for Re{s) » 0. Proof: By Iwasawa decomposition, the local zeta integral Z(s,W ^,W 2, fs) can be written as / / ( < i (^2^2) / fs{')o{ut\h,t2k2)fil:{u)dud(idk. JiKzxKiy JcMTixnr JZ2W Since VFi and W2 are assumed to be ATg-hnite, there are a finite number of -finite functions Oi{kx) and 9j{k2), and a finite number of Whittaker functions and 133 so that T^\{h)W]{g-i) = X)^i(A:i)W/'^(ÿi) and 7r2(&2)M^(g2) = Y^&3{hW^^\g2). « 3 We thus have % W l,W 2 ,/.) • f / 6i{ki)0j{k2)fs{'yo{uti,t2){h,k2))dkidk2ip{u)dudtidt2 / fl,''^\'yo{uU,t2))tl3{u)dudUdt2, (3.71) where we set := ^i{k\)&j{k2)fa{g{h,k 2))dkxdk2. Note that is {K2 X ^C2)°-finite. It suffices to prove the absolute convergence for the integral of following type. I(^,W:,W2,A) = / ^'h {tuhW\{t\)W2{t2) f fa{jo{uti,t2))^/3{u)dudUdt2- ./C\(T2XT2)» JZ2VV2 By the same argument as that used in the proof of nonarchimedean case, it is enough to prove the convergence of the special case where /, = / “, the spherical section in /^(.s), that is, the integral X\(T2xT2)« y;'(7o(wfi,<2))(f«Ai(f<2. (3.72) 134 The proof here will go similarly to that of nonarchimedean case. By Proposition 3.1.7, Lemma 3.1.9, and Lemma 3.3.3, we have = d^(|a^5c"^|X+^ f°{x{w,y,x,u,v))dwdydx where u = abc~^d~^ and v = ac~^. For Whittaker functions, we use proposition 2 and obtain and W2(i2)’= \d ) x 2{c^a \d ) (3.74) X l X2 where ti = h{ab,a,b~^ ,1) and <2 = h{cd,c,c~^d~^a,c~^a), the summation is taken over a finite set of finite functions y’s and (^^’s are Schwartz functions over F^. For our purpose of proving the convergence, we may assume as in the nonarchimedean case that the finite functions are in form: Xi(a, 6) = |a|“‘|&|“2 and X2{c^a~\d) = |c^a"^|“=>|d|“", (3.75) and the Schwartz functions are positive and s is real. Thus it reduces to prove the convergence for following integral (when s is large) r|a|M'«|5|^+'''|c|-"+''|d|'''(^i(a,6)^2(c'^a-\d) J (l + „.)i(-M)(l+ „.).«------4 where the integration is taken over the domain: 0 < |a|, |6|, |c|, |d| < oo. Similarly, by changing the variables (a,6, c, d) to (a,b,u,v), the integral becomes J ------(i + ..)K .«)(i+ „.)•«------<3.77) 135 where the integration is taken over the domain: 0 < |a|, |6|, |u|, |u| < oo. We write the integral as a sum of following two integrals: fOO , fOO [îd ® " * ’ * ” • / ------LI--..——(l)2{av~^,bvu~^)dudvdadb (3.78) and poo , poo L.ls+fci) • / ------11 hvu~^)dudvdadb. (3.79) It is clear that the integral I\ is bounded by poo , , , poo l u l ^ “ (3.80) which is convergent by obvious reasons. When 0 < |u| < 1, for any large integer T, there is a constant C t s o that (j>2{av~'^,bvu~^) < Ct|1)u|~^|u|^. (3.81) We can pick up T so large that Therefore the integral I2 is bounded by which is again convergent. This proves the theorem. □ 136 3.4 Meromorphic Continuation of Local Zeta integrals We will apply the estimates obtained in the first section of this Chapter to prove, in nonarchimedean case, that the local zeta integral Z{s^ Wi, W2 , fs) has a meromorphic continuation to the whole complex plane as a function of s. The archimedean version of such a result has not been proved as yet. 3.4.1 Nonarchimedean Case: Our proof of the meromorphic continuation of the nonarchimedean local zeta integral is based on the direct computation of the integral. In principal, such computation will be reduced to the integrals of one-variable integration as follows. I{as + f3-,n-,x,p; (3.82) where A is a character of the group of the units in the integral ring O with its conduct f\ (A’er(A) = 1 -t- is a additive unitary character of F with its conduct 6,j, {ker(r[>) = and ch[x+vp}(i) is the characteristic function of the subset {x-\rV^}. Lemma 3.4.1 Let a, be two real numbers and k,l,n,p integers such that k > I and n is nonnegative. Let be the characteristic function of {x + 'P’^} with X 0 V^, Ip any additive unitary character of F , and A(t) the ramified character of 137 O'^. Then we have, forp > ord{S,j,), that I{as^•,n\x,p\\\'ij)-,l,k) (3.83) 0 ifl> ord{x) or ord{x) > k; = < 0 i f l ^ ord{x) < k and p — ord{x) < f\] log |æ|" if I < 07'd{x) < k and p — ord{x) > f\. Proof: Since the positive integer p is assumed to be larger than ord{5,j,), the integral /(a.s + /?,ri,p; /, k) is equal to ^(a;)7(o!S + ^, 7i,p; A; I, k). It is obvious that /(a.s + (), n,p; A; /, k) will be zero if I > ord{x) or ord{x) > k. When I < ord{x) < k, we have / ( a s + ^ ;n ;x ,p ; A; ?/);/,&} = [ |f|"'’+^A(f)log JjfOrd{x)0X = \{t)ch[,,+-p„y{t)d^t = \xr+f^iog\x\^ f \{t)d^t. J x + V P It is easy to see that fx+-pj> X{t)d^t is equal to zero if p — ord{x) < f\. When p — ord{x) > f\, one has / X{t)d^t = X{x)q~’^. J x + V P This proves the lemma. □ Lemma 3.4.2 Let a, /i be two real numbers and k,l,n,p integers such that k > I andn is nonnegative. Let ch-pp{t) be the characteristic function ofV^, amj additive unitary character of F, and X{t) the ramified character of . Then we have, for p > ord{S,j,), that /(as + /3; n; p\ A; ?/>; I, k) (3.84) f 0 i/A ^ 1; 138 Proof: Similarly, we may forget the additive character tl> upon the positive integer p being chosen large enough. Then the integral can be computed as follows: I(as + /3,n,p-,X,tj}-,l,k) = J<|“*+'^A(t)log|i|"c/ipp(t)d^t i= m a x {l,p ] Hence, if A is not trivial, the last integral is zero; otherwise, it is equal to 1 — q~^. The lemma is proved. □ Note that one has the following obvious formula: k (3.86) i=:f lq-l(as+0) _ ( ^ _j_ -|^^ç-(fc+l)(as+^) _ (^ _ ■j^^ç-(/+l){«s+/3) _j_ ^^-(fc+'2)(ors+/3) (1 — ç-(“s+/3))2 The theorem we are going to prove is as follows: Theorem 3.4.3 Let ?/> be any generic character of N^. There is a positive real num ber So, which is independent of the finite place v, so that for any / , € 7g(.s), and any Wi E W(7Ti,i/’) and W -2 E W(7r2,V’)) ihe local zeta integral Z{s,Wi,W2,f)= f f{l/o{gu92),-^)Wi{gi)W2{g2)dgidg2 (3.87) JCN^>^{Z2XU)\I^ is of form P{q~^), P{T) is a rational function in T with complex coefficients. In other words, the integral Z(.s, W\, W 2, f) has a meromorphic continuation to the whole complex s-plane. 139 Proof: As in the proof of the absolute convergence of the local zeta integral, it suffices to prove the rationality of the following integral: I(s.Wi,W2, fs) := f Wi(ti)W2{t2)IiU,t2-,fa,4>)dtidt2. (3.88) JC\{T2XT2) o By Proposition 3.1.7, we have lihih] fs,'^) = w here/' = Ps{wo){f). By meansof Lemma3.1.8, the local zetaintegralZ(.s, W^i, IP 2,/s) can be rewritten as a linear combination of integrals of the following four types with complex coefficients: / |ac-i|< ,-e SH^it\,t2)i\a\^b\\c\-^)^+^Wi{ti)W2{t2)I{a-^cd,a-^c^-J,rl:)dUdt2, (3.89) I «^;'( |o 6c-'d -' |>,: / Sfi^ {ti,t2){\a\~^c\^\d\)‘+^Wx{tx)W2{t2)I{abc-\a, f,il:)dUdf.2. (3.92) |ol Let €i = 0, or 1 for z = 1,2 and (^ 1,^2) = (h(ab,a,b~^,l),k(cd,c,c~^d~^a,c~^a)). Again by the estimates in Theorem 3.1.12, we have to prove the rationality for the following four types of integrals: (1) The integrals |zA}->(<^)’A]-‘^(^)[|a^6c-'||«n/3r=]»+’l^i(t,)^2(f2)dtidt2 (3.93) 140 where the integration is taken over the domain: (a, b, c, d) G satisfying the following conditions: |a 6| < £i|cd|, |o;| = |a“’cd|,|/3| = |a“^c^| < £2, |a| < £i|c|, and |a|, |&|, |d| < £3; (2) The integrals I Mr' H^Y'^^W,{U)W2ih)dtxdt2 (3.94) where the integration is taken over the domain: (a, h, c, d) G satisfying the following conditions: \ab\ < £i|cd|,|c| < £i|a|, |c''^| < £a|a|, and |a|, |fe|,|d | < £3 ; (3) The integrals / V»!"" (a-^c^)[M^d| |6|"|a-(/i ) W2((z) A, dfg (3.95) where the integration is taken over the domain: (a, b, c, d) G satisfying the following conditions: |cd| < £i|a 6|, |a| < £i|c|, |c'^| < eaMI? Ml) Ml) k4 < ^3; (4) The integrals j V'j-"(o&c-^)TA]-"(o)[M"^c' where the integration is taken over the domain: (a, b, c, d) G F^'‘ satisfying the following conditions: |cd| < £i|a 6| < £i£2|c|, |c| < £i|a|, |c^| < £2|a|) Ml) Ml) Ml < S3. Note that the integration domains of the above four integrals are determined by the conditions of vanishingness of Whittaker functions given in Lemma 3.1.1 and that of the integrals of type /(a,/?;/s,?/>), and the integration domains of four integrals 141 from (3.90) to (3.93). In each of those four cases, the data ( 61, 62) has four different choices: (0,0), (1,0), (0,1) and (1,1). We actually have to deal with sixteen different types of integrals. By asymptotes of Whittaker function on the torus, we have that, for = h{ab, a, 1), W(ti) = ]^<^x(a,%(o, 6). X where the summation is taken over a finite set of finite functions % and (f)^ are Schwartz-Bruhat functions over F^. For the proof of the absolute convergency and the rationality of those integrals, we may assume that W{t\) = (j)^{a,b)x{a,b). In other words, Wi(fi) = (a , 6)xi(a,i) and W-zit-z) = (c^a"^,d)%2(c^a''^, d). Therefore we have to show the rationality for the integrals of following sixteen types: (1) The integrals J cl)i{a,b)x\{a,b)(l>2{f3,a^oi^~'^)X2{(^,a^CifS~'^)dUdt2 (3.97) where the integration is taken over the domain: (a, b, a, /3) G satisfying the follow ing conditions: |o;|, \/S\ < £2, |a|, |6| < £ 3 , |6a| < £1, |a^^“ ?| < £1, and < £3. (2) The integrals J |*+V i(a, 6)Xi(a,&)^2(c'^a"\d)x 2(c'^a"\(i)dtidt2 (3.98) where the integration is taken over the domain: {a,b,c,d) G satisfying the fol lowing conditions: |a 6| < £i|cd|,|c| < £i|a|, \c^\ < £3|a|, and |a|, |Z»|, |d| < £3. 142 (3) The integrals (3.99) where the integration is taken over the domain: {a,b,c,d) € satisfying the fol lowing conditions: |cd| < £i|a6|, |a| < £i|c|, |c^| < £ 3 |a |, and |a|, |6|, |d| < £3. (4) The integrals J /3a^a~^)Xi{a, a~^)^2io(,d)x2{a, d)dUdt2 (3.100) where the integration is taken over the domain: {a,/S,a,d) 6 satisfying the following conditions: | a | , |^| < £2, |o|, |d| < £ 3 , \d^~^\ < £1, jaza": | < £1, and |/ia 2a“ 2| < 63. Note that, in general, the finite function of two variables can written as x(o,6) = Ai(o)A2(6)|or'|5r'(log|ar')(log|6r) where A,-, i = 1,2, are ramified characters of F^, Ui are real numbers, and ni are nonnegative integers. From now on, .s can be any complex number with Re{s) large so that the local zeta integral converges absolutely. The proof will again break into many cases to discuss as in the proof of the absolute convergency of the local integral. We will pick up case (2) to compute and the other cases will follow in the same way. We set, for simplification of our notations, that ft{a; A; ijr, x) := |f|"*+*At(t) log |t|*cAa,+yp(t)^(t). (3.101) 143 Then the integral of case (2), which is denoted by can be written as fa{^2 ; A a; V’a; Xa) /c(^; Ac; Xc) /d(ei; Aj; tpd; Xd) fb(l; A&; Xb)d{b, d, c, a). (3.102) The domain of the integration is 0 < |6| < 0 < |d| < 0 < |c| < |a|}, and 0 < |o| < We first consider the integration with respect to the variable b, that is, B2 := fb{l; A&; Xb)db (3.103) and have to separate the case Xb G 'P'’*’ and the case Xb 0 P'’*. When Xb G P'*'’, by lemma 3.4.2, if the character A(, is not trivial, B-i = 0; and if \b is trivial, one has Pi(g“®, 1) if |a“•Ic 2C2d| i < By = Qi(ç~*) if |a“ 2cld| > g®'-'*'’, (3.104) Æi(g“^, 1) if la“ 2cld| > , where P\{q~^^u) is a linear combination with coefficients in C(g“®) of |a“ lc 2d|'^®"*'* and [u'icldl"®"*"* log [a'Icld j. When Xb^ V^‘‘, by Lemma 3.4.1, we have Jo if min{g-^=,g " |n 2 cU|} < I Gi(g-) ifmin{g-':,g-"|n-lcld|} > |zb|. ^ ^ Note that min{g“®®, g“®* la“lcld|} > |æb| implies that < |a“lcld|. Without loss of generality, we can reformulate the results on the integral B2 as follows: Plugging B2 into the integral I2, we obtain that I2 = ^/a(f2 - ^;---)^/c(l;---) J^/d(ei + l;Aj;i/)d;a;d)d(d,c,a) + I I f fd{^vAd\^h\ Xd)d{d,c,a) Ja Jc Jd = ^21 + I22 (3.107) 144 where the domain of the integration with respect to the variable d in /21 ( or I22) is |d| < min{g“®®,(jr®*~'’'’|a 2c“?|} ( or resp. < |d| < The computation of the integral i? 2i will go in the same way as that of the integral B2 did. So we obtain that ,ei + 1) if |c“ 2a?| < ) if > gPh-G'-Pd, (3.108) - { w l- ': where P2{q~^, ei+1) is a linear combination with coefficients in C{q~^) of |c“ 2a 2 and |c“ 2a 2 log |c“ 2a 21. The integral D22 can be computed as follows. We need to distinguish two cases according to xj G Vj and xj ^ Vj. When xj G Vj, D22 will vanish if Aj is not trivial. If Ad is trivial, one has, from Lemma 3.4.2, that where ^ 2(9 """,Ci) is a linear combination with coefficients in C(ç~®) of |c~ 2a 2|‘i*+* and |c“ 2a 2 log |c"&U 21. When xj ^ V^, we have, by the same reason, that n - I ^ 2(5 ") if "Me < kl < ? /n Hence we can put the results on the integral D22 in form: where ^ 22(9 "^, Ci) is a linear combination with coefficients in C( According to those computations, the integral /21 is a linear combination with coefficients in C(ç“^) of integrals of following types: and the integral I22 is a linear combination with coefficients in C(g“®) of integrals of following types: ■“ L 2 ’ ' ' * 1|.|) ^‘^~2 ’ ' ' h-n := I /.(■ ■ •) U f ■ ■ ')4 « , «) Since the positive integer pj can be chosen so large that ^^(pd+ei-pb) ^ g-^i (choose Pd after p*, was chosen), there are no c’s which satisfy the condition < |c| < mm{ç~®“, 5“®'|a|}. In other words, the integrals / 211, / 221, and 7222 are equal to zero. On the other hand, the integral /212 is, following similar argument, a linear combination with coefficients in C(ç“®) of integrals of following types: T f r /.262 262 + 1 \ , /2122 := / fa{e2]---)da. 7q«lV l -J'c<|a| By Lemma 3.4.1 and 3.4.2 again, we conclude that the integrals J 2121 and /2122 are rational functions in q~^. This prove the rationality of the integral I2. The rationality of other three cases can be proved by following the same argument. □ 146 3.5 Notations Used in Chapter III X> Xi, P - 9 3 Xa{t), p . 9 8 x{w,tj,x,u,v), p.98 x(w ,y,z), p. 104 5^,, p. 104 Hi,H2, p.95 p.lOl P-98 1) 104 Z(a,/3; p. 104 J( p ,5,W i ,T^2), P-113 p.103 P i P i p.95 V, p.109 p.96 ^(o, 6), p. 96 5 (F ), p.96 TTio, 7T2 o , p.118 p.97 $=, p. 104 p., p-107 Q, Qo, p-116 Wo(7T,?/i), p.118 %, p.95 & p.97 C H A PT E R IV The Poles of Eîsensteîn Series Attached to Sp(n) The theory of Eisenstein series was initiated by Selberg and developed for general reductive groups by Langlands, which has been playing very important roles in the modern theory of automorphic functions. One of the most important problems in the theory of automorphic representations is to understand the spectral decomposition of the Ir'^-space L^(G(F)\G(A)) for any reductive group G defined over a global field F. According to the general theory of Langlands, one has the following decomposition L^(G(F) \ G(A)) = Lcusp ® -^rea ® ^cont- How to describe the residue spectrum, i.e. Lrea, become an interesting topic in the recent study of automorphic representations. In order to understand the residue spectrum of a reductive group G, we need explicit knowledge about analytic properties of Eisenstein series of G. In other words, we have to know the location and the order of poles of Eisenstein series, and the specific description of the residue representations of Eisenstein series at the poles. Those analytic properties of Eisenstein series can also be used to study other topics in the theory of automorphic representations. Eisenstein series associated to some degenerate principal series representations 147 148 of symplectic group Sp{n) induced from a maximal parabolic subgroup have been studied extensively by Piatetski-Shapiro and Rallis, Kudla and Rallis. In the case that the maximal parabolic subgroup is of Siegel type, Kudla and Rallis have determined the precise location of the poles of the (normalized) Eisenstein series. Those precise information about Eisenstein series has been used in their paper to study automorphic L-functions and theta correspondences. In this Chapter, we will apply the methods, developed by Piatetski-Shapiro and Rallis in [54] and [55], and Kudla and Rallis [43] and [44], to study a family of non- Siegel type Eisenstein series of Sp{n). More precisely, we assume that F is a totally real number field. Let G„ = Sp{n). As usual, G„{A) denotes the adelic group of Gn and 2» modular character 5 ^ ^ (more precise definition will be given in section 4.2). We define, for any section fg = /(-,s) € /,"_i(s), an Eisenstein series as follows: K-\is,s;fs)= X) f(l9',s)- (4.1) (F)\G(F) For the given section / , 6 there is in general a finite set of places of the number field F including all archimedean places, so that /, = ((8«Gs/..«) €) (4.2) where /°„ is the normalized spherical section in the local degenerate principal series representation /,"_i,„(-s) such that /°_„(1) = 1. we define as follows, see [57], the 149 normalizing factor for the fa) attached to the section = II + %)(«('^ + % — 1) H C«(2-s+i)]- (4.3) viS ^ Z Z j=l,j=„(2) We will modify the Eisenstein series s; /,) in the following way We will call /a) the normalized Eisenstein series of G„ associated to the section fa. According to general theory of Eisenstein series, one knows that the Eisenstein series s;/*) converges absolutely for Re(s) large and has a mero- morphic continuation to whole complex plane with finitely many poles and satisfies certain functional equation relating s to —.s. The main result we are going to prove in this Chapter is Theorem 4.0.1 (Main) Assume that F is a totally real number field. For any holo- morphic section /(-,.s) € the normalized Eisenstein series enjoys the follow ing properties: (a) The set of possible poles (of order at most two) of the normalized Eisenstein series K ’fi{g,s; fa) is , n + 2 n n —2 « — 2nn + 2. (b) The (normalized) Eisenstein series achieves a simple pole at s = | and a double pole at s = and has at most a double pole at s = • • •, where e{n) = 1 if n is odd and e(n) = 2 if n is even. 150 We describe the contents of this Chapter. In section 4.1, we study certain orbit decomposition on the flag variety \ G„, which will be used in section 4.2 to obtain an inductive formula that relates certain constant term of < 7; /«) to some Eisenstein series of Sp{n — 1). In section 4.3, we will study some intertwining operators, the analytic properties of which are very important for us to determine the location and the order of the poles of our normalized Eisenstein series. We will prove our main theorem in the last section. One of the most technical parts is to cancel the higher order poles of Eisenstein series, which is treated in section 4.4. 4.1 Orbital Decomposition on P”_i \ Sp{n) We will start with study the f "-orbit decomposition on the grassmanian variety P"_i \ Sp{n). Such a decomposition will be used in the next section to give a nice inductive formula for a constant term of our Eisenstein series. Let (Vi ( , )) be a 2n-dimensional non-degenerate symplectic vector space over a number field F and Sp{n) be the symplectic group of (V, ( , )), i.e., Gn = Sp{n) = {<76 GL{2n) : {gu,gv) = (u,u) for u,v E V}. Let {ei, 62, • • •, e„; e^, 62, • • •, e(,} be a symplectic basis of V so that the symplectic form corresponds the matrix ^ 0* )’ vector space V is identified with (row vectors) and G„ with the usual group of 2n x 2n matrices acting on to the right. Let F" = Stabsp(n){Fe[ © Fcg © • • • © Fe'). Then F" = M"7V" is 151 a parabolic subgroup with its Levi factor M" and its unipotent radical TV", that is, / a 0 0 0 \ 0 0 X2 M" = {m = e GL(r) X G(ti - r)} (4.4) 0 0 0 0 Z3 0 «4 / and / Ir X W y \ N" = {n(x, w, y) = 0 I n - r ‘y 0 0 0 /r 0 }• \ 0 0 - ‘æ In-r Lei C be the flag variety of (n — l)-diniensional isotropic subspaces in V. Then £ is 5p(n)-equivariant to P"_j \ Sp{n) as 5p(n)-homogeneous algebraic varieties. Proposition 4.1.1 The Sp{n)-flag variety £ is decomposed, under the restriction to P" of the right translation of Sp{n), into Pf-orbits as: £ = £q U £ ] U £2, where Cq = {L e C : Fe-i C L), £1 = {£ 6 £ : Ft\ (f. L, Fe\ C L^), and £2 = {£ € £ : Fe.gLL^}. Proof: It is evident that £,■ ( i = 0,1,2) are stable under the action of P". We are now going to show that P" acts transitively on £,• for z = 0,1,2, respectively. Case of £0 : If £], £2 E £0, then Fe[ C £1, £2- Extending the vector e\ to a basis of £ 1, £2, respectively, we obtain an element g E GL{n — 1) so that L\g = £2 and e\g = e\. Since L\ and £2 are isotropic (n — l)-dimensional subspaces of V, we can extend g to an element g E Sp{n) by W itt’s Theorem. Moreover, g is in P" because e\g = e\. Case of £ 1 : If £1 , £2 E £1, then L f and £.f are (n-f-l)-dimensional subspaces of V and e[ E £ f , £2 • Since £1 and £2 are the radical subspace of Lf, L2 , respectively. 152 we know that Fe\ ® Li C Lj-, i = 1,2 are maximal isotropic subspaces in V. Again by W itt’s Theorem, there is g 6 Sp{n) such that {Fe[^Li)g = (Fe\(BL-2), L\g = L2, and Fe\g = Fe\. Such a g is required. Case of £2 : If I 'i,£ 2 € C\, then Fe\ 0 uY g TY, z = 1,2, and Zi © Fe^ and £2 © Fe\ are not isotropic. Let Ui be a dual vector in £, of uY [i = 1,2). Set Li = L'^®Fui. Then we have that LiO)Fe\ = L'i ± {Fui®Fe\), i = 1,2. Since Fui®Fe\ (i = 1,2) are non-degenerate symplectic planes with form ^ ^ ^ under the basis {ui,e\}, respectively. We can construct an isometry g' from L\ © Fe\ to £2 © Fe\ so that L[g' = £3, u\g' = u^, and e\g' = e\. Once again W itt’s Theorem says that g' can be extended to an isometry g from V to V, i.e. g 6 Sp{n) so that £ 1*7 = £2 and Fe\g = Fe\. Therefore in all three cases, we can find an element g 6 P " so that L\g = L-i-O Corollary 4.1.2 The symplectic group Sp{n) has a decomposition of double cosets as 5p(n) = P,"_iPi* U P;,_^WiPf U P”_^W2Pl' where I “ 0 1 \ 0 I n - 2 0 0 / 0 0 1 0 \ 1 0 0 0 I n — 1 0 0 Wl = -, W2 = (4.5) 0 0 1 -1 0 0 0 0 0 In —2 0 \ 0 0 0 I n - l / V 1 0 0 Proof: Choose 153 L\ = F e'2 ® F 0 • • • 0 F e'^ L2 = Fei 0 Fe'2 0 • • • 0 Fe;,_v Then L q G jC q,L \ G >Ci, and Tg G £ 2 - Since the map, g 1— > L o g , gives the S'p(ri)- equivariant bijection from P"_j \ Sp{n) to £, there are three Weyl group elements /gn, w-i, and W2 so that Lohn = To, LqWi = L\, and LqWi — Tg. It is easy to check that {/g,„ toi,ujg} satisfies the requirement. □ Lemma 4.1.3 Let P" = where M" is the Levi factor o /P " and is the unipotent radical of P{\ ThenforwÇ. {l 2niW\,W2), the N'f-orbits on are parametrized by {w~^P^_^w fl M") \ M". In other words, there is a bijection between the set of double cosets Pn^ipN’f in P,"_i \ the set of cosets (iü “ 'P,"_]tx^ n M")m in (to“ ^P,"_,tü D M”) \ M{'. Proof: Since P,"_iiüP" = U,neM;;‘Pn--[WmNl\ we define a fibration map p : M r (4 .6) m I— y P"_iiomiVr. It is clear that p is a surjective map and {w~^P|^_■^w fl M'f)m C p”^(p(w)). We will next prove that n M")m = p~^{pim)). If P^_-iWmN!j‘ = P,'^_^w7U\N’f, then wm = pwm^n, p G P " _ i, M G ^,nd further we have ) = w~^pw. In other words, mmî^{min~^mf^) G P]" D w~^Pll_^w. According to the Levi decomposition P f = and P ”_ , = we have E since w G {Tg„, wi, wg}. This implies that wm = p{wm]nm^^w~^)wm\ belongs to 154 P 'l_ ^w m \. Therefore we have mmj"^ G {w~^P^_-yW fl M"). In other words, the fiber at p{m) is {w~^P^_-^w fl M{')m. □ Lemma 4.1.4 ^4s Sp{n — l)-homogeneous varieties, we have Proof: Let /q = Fe[ and /g = Fe^. If w = Lg,., then /g C Lg and Lq fl (/o ® Zg is an (n — 2)-dimensional isotropic subspace of (/g © Zg )"*■, which is an 2(n — l)-dimensional non-degenerate symplectic space with respect to the restriction of the form on V. Hence Sp{n - 1) = Sp{{lo © Zg)-^). Since P"_j = Stabgp(„)(Lo) and M ” = Stabgp(„)(Zo) n Stab5p(„)(Zg) D Stabsp(„)((Zg © Zg)-"") = (Stab 5p((/„0,v))(/g) n Stab 5p((/„®/v))(/^)) x Sp{{lo © Zg It is easy to see that = Stabgp(n)(Zg) n Stabsp(n)(Zg ) fl H (Zg © Zg ) ) = (Stab,sp({/o(g/v)j(Zg) n Stab.çp((f(, 0fV))(/^)) x Stabgp((,^ 0(V)j.)(Z,o fl (Zg © Zg )•*•). Therefore we obtain {Pn-\ H M") \ M" = \ Sp{n — 1). li w = W2, then wM['w~^ = M". It is easy to construct a Sp{n — 1 )-equivariant bijection between (P,"_i D M") \ M" and (u;^’P"_jZü2 fl M") \ M". If w = u)i, then wf^P^_-iWi = Stabgp(„)(Li). Since = Xi © (Zg © Zg) and Li = Li n (Zg © Zg )-*- is a maximal isotropic subspace in (Zo © Zg we deduce that n M" 155 = Stab5p(n)(^o) n Stab5p(„)(Zo) H = {Stahsp({io®q)){lo) n Stabap((foe;v))(Z^)) x St&hsp{{to®i^)-^)(Li). Thus we obtain that = PnI^\Sp{n — l) as algebraic varieties. □ 4.2 The Pf-constant term of Eisenstein Series We assume that the number field F is totally real in the rest of this paper. As usual, G„(A) is the adelic group of Gn and Gn{F) is the group of P-rational points of G„. Let P " = M^Np be the standard maximal parabolic subgroup of G„ with its Levi factor M" and its unipotent radical iV", as defined in section 4.1, that is. / a 0 0 0 \ 0 xi 0 X2 M" = {m = G GL(r) X G„_r} 0 0 ‘a -' 0 \ 0 23 0 24 / and ( I r X W 2/ \ N" = {?i(x, w, y) = 0 I n - r ‘ y 0 0 0 /r 0 }• \ 0 0 - ‘ 2 I n - r ) We will use the notation that a{p) = a(m) = det(a). Then the modulus character Spn = |a(m)P"“’'+\ For each place v of the number field P , we choose a maximal compact subgroup K„^„ of Gn{Fp) as follows: If u is a non-archimedean place of P , we let Kn,v = Sp{n^Ov)i and if real archimedean place u, we let Kn,v — U(n) via the canonical embedding from Z7(n) into Gn{Fv). We let K,i = Ilw , which is a maximal compact 156 subgroup of G„(A). By Iwasawa decomposition, we have = P’'{Fv)K,i,v for each place v aiid G„{A) = P"(A)/iT„ globally. We denote by /"(.s) = the degenerate principal series represen tation of G„(A), which consists of smooth functions f{-,s) : G„{A) C satisfying the following condition; f{pg,s) = |a(p )|^ + ^ '^ /((7 ,5 ) (4.7) for p 6 P"(A ), <7 E G„(A). The group action is the right translation. By smoothness of the function / we mean that the function / is locally constant as a function over the non-archimedean local variables and smooth in the usual sense as a function over the archimedean local variables. A section f(p, s) is called holomorphic or entire if the section /(p, s), as a function of one complex variable s, is holomorphic or entire. We also assume that sections f{g,s) is right -finite. This implies that /"(.s) is, in fact, a representation of (g„,oo, K,i,oo) x G (A /), where g„,oo is the Lie algebra of G„,oo and A f indicates the group of finite adeles. We define, for any section /« = /(•, s) E /"('^); Eisenstein series as follows: Er{g,s'Js)= /(7<7;-s)- (4.8) yePp(F)\G„(F) According to the general theory of Eisenstein series [47] and [2], Such Eisenstein series is absolutely convergent for Re{s) large and has meromorphic continuation to the whole complex plane as function of s and satisfies certain functional equation. In practise, the knowledge about the precise location and the exact order of the poles of Eisenstein series is important and interesting. 157 In the rest of this Chapter, we concentrate our study on the family of Eisenstein series ( Eisenstein series. From the general theory of Eisenstein series [47] and [2], Eisenstein series and its certain constant terms share the same analytic properties. We start with computing the constant term of fa) along the maximal parabolic subgroup P". We can do this because the Eisenstein series s;/,) is concentrated on the Borel subgroup of Gn in the terminology of Langlands [47] and .Jacquet [31]. The unipotent radical of P ” is / 1 z y \ 0 In-1 y 0 N î = { 0 0 10 }• y 0 0 -'^Xn-l 1 y As usual, the P ”-constant term of the Eisenstein series fa) is defined as p , (4.9) In order to study the P"-constant term EH_i pn{g,s; fa), we have to introduce some intertwining operators i*_j, o U"^{s), o and which are defined in the following ways: By means of the canonical embedding of Sp{n — 1) into the Levi factor M[* of P", that is, 5p(n—1) = 5p(Vi,„_i ), where = (PeiSPci)-*- in V, we define, for any section fa G 7,"_|(.s), to be simply the restriction of fa to the subgroup Sp{n — 1), and (■«)(/«)(5) = / fs{wing)dn, (4.10) (A) ^,"j(-5)(/*)(<7) = / fa{w2ng)dn, (4.11) ./yvSj{A) 158 CC.(a)(/,)W) = / fs{wong)dn. (4.12) where wi and W2 are the Weyl group elements defined in section 4.1, which represents two different -orbits on \ Sp(n), and Wo is the longest Weyl group element; A^"., '< = 1,2 are such subgroups of iV", the standard maximal unipotent subgroup of Gn, that iV” = N'“' = {w'[^P"_-iWi)r\N’^. By [60] and [32], those intertwining operators converge absolutely for Re{s) large and have meromorphic continuation to the whole s-plane. Those intertwining operators enjoy following properties. In the next section, we shall locate the poles of those operators. Lemma 4.2.1 For mi{ti,g') E M" (M" = GL{1) x Sp{n — 1)) and / , E we have (a) fs{rn^{U,g')) = fsim ihg')), (b) U’^,is){fs){m{h,9')) = \tA’'~^U’^,{s){fs)imi{l,g')), and ((^) U’^h){fs){mi{U,g')) = (s)(A)(mi(l,y)). Proposition 4.2.2 For generic complex values of s, we have the following intertwin ing operators: Ifn>3, then C l : C i W — (b) 0 17”, (.s) : — > C i(.s ), and 159 C M : C i W — If n = 2, then Zj o C/^,( 5) : I]{s) — > I\{s)^ and Zj, i\ o U^^{s) map I'f{s) to the trivial representation of Sp{l). By the generic complex values of s we mean that for such values of .s those in tertwining operators are holomorphic. We will prove, in the next section, that there exists a normalizing factor to each intertwining operator so that the normalized in tertwining operator will be holomorphic for Re(s) positive. Our main result in this section is the following theorem, which suggests an inductive way to study the analytic properties of Eisenstein series. Theorem 4.2.3 For generic complex values of s, the constant term E’,l_i pn{g,s;fs) can be written, ifn>3, as a sum of Eisenstein series of G„-\, that is, 1 where m^{ti,g') E M ". Ifn = 2, then the constant term E \p 2{m\{t],g'),s-,fs) can be expressed as follows: Elp2{nii{tug'),s;fs) (4.14) 160 Proof: The Haax measure on iV"(F) \ iV"(A) is so normalized that the volume of iV"’"'(F) \ iVf’"'(A) is 1. We first consider the case of n > 3. Unfolding the Eisenstein series, we have = / f8{ing)dn = f fsijng)dn. (4.15) Te-Pn-I (F)\Sp(n,F)!N”(F) JN^'’’(F)\N^(A) According to Corollary 4.1.2, the double cosets can be described as follows: C ,? -i\S p (» )W (4.16) = \ C,f,"/ivr] U K_, \ C-i*»iC;'Wl u [C_, \ C_,%f,"/ATI. So, one obtain that (4.15) = Yh fn-, fs{ing)dn + y ] f fs{ing)dn 76(P”_, \p"_, «1, p,"/Af")(F) •^^r'''(^)\^r(A) + y2 f fs{ing)dn = E] -f Eu,, + E ui2 . Applying Lemma 4.1.3 and 4.1.4, we can compute each term as follows: The first term E\ is easy to calculate. El = Y f fs{ing)dn 161 H /s("^l (1, 7 )5 ), 76tP"r2\M”-i)](n since /* is left iV"(A)-invariant. The computation of the term Ewi will be a little longer. = E) / /s(u;inmi(l,7)5r)rfn, since iV” = and fs is left AT"’'"' (A)- invariant and the volume of A^"’’"' {F)\ AT"’'"* (A) is one. It is easy to check that AF"^,^ = Nw^. Hence we can write the term Eu>i in terms of the intertwining operator (s), that is, = E C/:,(/;)(/.)(mi(l,7)^). 76[p;^r,'\sp(,.-i)](F) By the similar reason, we can deduce the last term E^,, as follows: = 76[P,?_, \P"_,^ W2PC/Np]{F)...... 162 E L„ fs{w2nrm{l,'^)g)d: 76[P”rj‘\Sp{„-l)](F) 7GK\gp("-l)](F) Therefore the constant term E'^_-i pn{g, s; /,) can be expressed in the following way: Eu-i,pn{g,s-Js) = E fs{mi{ln)g) 7eK\gp(»-i)](P) + E %W(/.)(mi(l,'y)(7). 7e[P„"rASp(n-l)](F) After restricting pn( 1 + - |;Ci ° %(«)(/.)), for generic complex values of s. In the case of n = 2, one has that p2\p2p2/AT2 = p2\P>2Pi^/AT2 = 1. Following the same calculation, we will obtain the expression for the constant term Elp2{jni{tug'),s-,fs). □ 4.3 Local Analyses of Intertwining Operators In this section, we assume that P„ is the local completion of the number field F at the prime v, finite or real. As usual, G„,„ = G„(P„) is the P„-rational point of the 163 group Gn- We are going to determine the holomorphy of those intertwining operators (.s) and C/^2 (•®) involved in our inductive formula for the constant term of the Eisenstein series, and that of the intertwining operator (.s) attached to the longest Weyl group element Wo, which will be used to determine the poles of Eisenstein series on the neg ative half-plan. Since those intertwining operators are eulerian, we can deal with the problem for each corresponding local intertwining operator CC, and The methods used in this section were developed by Piatetski-Shapiro and Rallis in [54] and [55], and by Kudla and Rallis in [43] and [44]. Our process consists of two steps: (1) work with general holomorphic sections in and (2) work with the spherical sections in Since in step (1) we need to decom pose our intertwining operators, we may gain extra poles for our original intertwining operators. This is the reason we need step (2) to confirm the poles of our original intertwining operators. 4.3.1 Intertwining Operator general sections We recall some notations from previous sections. Let / 0 0 1 \ ( 1 X n - 2 z 0 In -2 0 0 0 In -2 0 0 1 0 0 0 0 1 Wl , Nw. = { 0 0 1 1 0 0 0 0 In -2 0 0 —*Xn - 2 In -2 0 1 0 0 / -z 0 1/ 164 and for any section /* G the local component „(s) of the intertwining operator (s) attached to w\ is defined as ZC.,«(a)(A)W = / fs{wing)dn, (4.17) Nwi (rt,) which is a G»,«-intertwining operator from to | det ), the unnormalized induced representation of G„,„ = Sp(n,Fv) from a parabolic (not maximal) subgroup P ”„_i, which is in the following form; f t * \ 0 m * : m E GL(n — 1, Fy)}. (4 . 18 ) A%-i — { <-* 0 0 \ Since the analytic properties in .s of the family : fs varies as holomorphic sections in are independent of the evaluation of £/,” „(s)(/,) at g, we know that the analytic properties in s of the family {U ’^^^vifs)} is determined by that of the restriction of the family {17^,,„(/«)} to GL(n,Fv), which is canonically embedded into Sp(n,F„) via 9 (^Q tg-\ Then (7^^ ^(s) defines a GT(n, F„)-intertwining operator from mdp_^^"j^’p^j(|detm|®‘'‘^ ) to I (i^t ), where and are maximal parabolic subgroups of GL(n) in the following form: Pn-^^ = {( 0 / )^ i 0 m More precisely, for any section /, G detml*"''^), the restriction of the intertwining operator U'^^ „(s) is Pwuvi-^)ifs)i9 ) = / fs(w\n(Xn-2,z)g)dXn-2dz JFy 165 / 1 \ /I »n-2 Z \ with wi = I /„_2 and n(x„_2, z ) = 0 /„_2 0 . \ \ I V 0 0 1 / The following lemma is a Gi(n)-version of the key Lemma in [55]. Since the proof is similar, we omit it here. Lemma 4.3.1 Let Wo he the longest element of the Weyl group of GL{n) i.e. Wo = Jn Jn-2 j and S :={(/) e : supp{f) C }• Then the analytic properties in s of the family • fa varies as holomorphic sections in coincide with that of the family {tC,,„(-s)(<^)(îïïo)} with (j) E S. According to this Lemma, it is sufficient to determine the holomorphy of the family {CC,^ E 5}. By Bruhat decomposition, / - 2 “ * - a ; „ _ 2 2 “ * 1 \ / I Xn-2Jn-2Z~' \ WiJl(Xn-2, z)Wo = I 0 In -2 0 1 Wq I 0 In -2 0 I \ 0 0 2 / \ 0 0 1 / The analytic property of ,u(-^)(^)(^o) can be determined in the following standard way: (j>{w-in{Xn- 2,z)Wo)dXn- 2dz •'F'v = \z~^\^'^^H ‘ai^n-2Jn-2Z~^,2~^))dx„-2dz. JF„ By changing the variable Xn- 2Jn- 2Z~^ x„-2, and then z~^ t-4- z, and using the rule of separation variables for smooth functions i.e., deduce that CCi,«(a)(^)(ü;o) = j^^_Jz\^-^+^ = H \z\"~^^'^(l>'l{z)dz'^ • f Since the integration over the variable æ „_2 is independent of s and the integration over the variable z equals, by standard computation, to a product of (^„(.s — ^ + 2) times a polynomial of s. Therefore we can express „(.s)((^)(üJo) in the following way: ?7”|_„(s)(^)(ü;o) = Cu('S — — + 2) X an entire function in .s. This proves the following Proposition. Proposition 4.3.2 The modified intertwining operator ,«(-s) is holomor- phic. In other words, for any holomorphic section fa in ,u(’^){/s) is a holomorphic section in mdp^^"’^|’^^j(|t|”“^|detm |*‘*' 2'). 4.3.2 Intertwining Operator general sections In this case, we will deal with the following Weyl group element W2 and the corre sponding unipotent subgroup /1 Zn-2 X z 0 y \ / 0 0 1 0 \ 0 In-2 0 0 0 0 0 In-1 0 0 0 0 1 0 0 W2 and N" = { y -10 0 0 0 0 0 1 0 0 VO 0 Q In-1 f 0 0 0 —*®n-2 In-2 0 I 0 0 0 —X 0 1 167 The local component ^(g) of the intertwining operator (g) is defined as follows: for any section G and generic value of g, (4.19) which takes, if n > 3, sections in ///-i.uC-s) to sections in the induced representation ^ I det ), which is the unnormalized induced representation of Gn,v from the standard parabolic subgroup P ]"„_2 of form: * * \ * * * * * * For general section /,, the analytic property of the local intertwining operator 17^2 „(.s) is in some sense hard to deal with. Our idea is to decompose this intertwining operator into two intertwining operators, the analytic properties of which are easy to determine, and then to verify that our method does not create extra poles for the original intertwining operator which will be done in the last subsection of the present section. Any element n(x„_ 2, .x', j/, z) G can be written in the following form: / I 0 X z 0 y \ ( 1 X „_2 0 0 In-2 0 0 0 0 0 In-2 0 0 0 1 2/ 0 0 0 0 1 7i(x„_2,x,y,z) = 0 0 0 1 0 0 1 0 0 0 0 0 0 /„_2 0 0 —‘x „ _ 2 In-2 0 \ 0 0 0 —X 0 1 y 0 0 I = n(x,y,z)n(x„_ 2 ). The later factor is a subgroup in the center of Then for any /« € 7,"_| „(g), our 168 intertwining operator ^(g) can be decomposed in the following way: = / fs[w2ng)dn t/yVwg {rv) = fs{w2n{x,y,z)n{xn-2)g)dxdydzdxn~2 = f_^Ms{fs){n{Xn-2)g)dXn-2 Note that if n = 2, = Ms. Those two operators M s and Ms will be implicitly described in the following Lemma. Lemma 4.3.3 Assume that n > 3 . Let Q"„_2 be a •parabolic subgroups of G„ m the following form: /to* * m * * 0 0 ft b Q l,n -2 — { } = (GL(1) X GL(n - 2) x Sp(l))N'’;„_2 . * 0 0 0 ‘m -‘ 0 \ c * * d ) Then the Gn,v~i'ntertwining operators M s and Ms can be described as follows: For generic complex values of s, (a) M s is a Gn,v-intertwining operator Ms : -4. det which is defined by the following integral Ms{fs){g)= I fs{w2n{x,y,z)g)dxdydz. JFS 169 (b) Ma is a Gn,v-intertwining operator Ma : -4- | detm|®+^) which is defined as ■/^s{fs){g) = I fa{n{Xn-2)g)dXn-2- The following lemma is an analogue of the key lemma in [55] and the proof is also similar. For completion, we will give the proof below. Lemma 4.3.4 Let B' = TN' he the Borel subgroup of Sp[n) contained in Q"„_2 nnd Xa{b) = |ti • • • t„_i 1®'*'^ a character of B'. Let Wo — ^ 0* ) ’ longest element in the Weyl group of Sp{n) and S = {(/) E I ’n-i^vi^) : ^ is smooth with compact support in P"_iU^oF,"_i}. Then we have (a) Mia is an intertwining operator from to (b) The analytic properties of the family {Mia{4'){wo) : ^ E S} coincide with the analytic properties of the family {A4a(/s) : fa varies as holomorphic sections in „(•?)}• Proof: (a) It is easy to compute, for t = diag{t\, • • •, \ • • •, ) E T and n E N', Ma{fa){tg) = / fa{w2 n{x,y,z)tg)dxdydz JFS = / fa{w2tw2^W2n{tf'^x,tf^y,tf^z)g)dxdydz JFS = |il • • • <„-l {^^"^MaifaM 170 and Ma{fa){ng) = ■M.s{fs){g)- (b) For 5o, the Laurent series of Ma{fa) at s = So is defined as /:> a By the general theory of intertwining operators, there is the smallest integer a so that the map 1-4- la(fao) is a nontrivial intertwining operator from ^(.So) to To prove (b) is equivalent to prove that there must be a function (j) E S such that lai^){'Wo) ^ 0. Now, if for any ^ E S, la{^){wo) = 0, then la{(l)){B'woPn_i) = 0 and further la{(^) = 0 as an element in indÿ‘,^p^^''\Xa^) since B'woPn-\ is dense in Sj){7i,F„). Therefore the kernel kti'{la) is a nonzero 5p(n, F’„).-subspace in Since for any .s, is dual to s) according to the non-degenerate 5p(n, F„)-invariant pair the dual subspace X-*- of X = ker{la) is a 5p(n,F„)-subspace in So)- For a /_s, E X-*- and any = / (f>{won)f-aoiwon)dn. It is not difficult to see that the space S is isomorphic to C^(N^_^{Ft,)) via Hence /_,„(îUoX”_^(F„)) = 0. This implies that /_ 5„(P,"_i't«oX”_,(F„)) = 171 0 and then /_s„ = 0 as an element in -So) by the density of P,"_]ïüo A7,*_i in Sp{n). Therefore X = ker{la) = that is la = 0. However, this contradicts the assumption that is nonzero. Thus there must be a G S' so that /a(^)(w^o) 0. □ Lemma 4.3.5 For any Proof: Considering the Bruhat decomposition of Sp{n,F„), we have 0 —xz 1 0 1 0 0 —xz / - y « \ / y^ \ 0 / n —2 0 0 0 0 0 In—2 0 0 0 0 0 0 1 - xyz y 0 -»/* 0 0 1 —xz 0 0 t02n(x,y,z ')wo = Wo 00 0-Z-'0 0 00 0 1 0 0 0 0 0 0 In-2 0 0 0 0 0 In-2 0 \ 0 0 x'^z —X 0 1 + x y z /\ 0 0 0 - y : 0 1 / Then we deduce in the way similar to that used in proof of Proposition 3 in the last subsection that for any (j> £ S, Ms{^){Wo) = / ^{w2n{x,y,z,)wo)dxdydz JFS — f \z\~^~^~^(l>{won{yz~^,—xz~^,—z~^))dxdydz JFS = f [ \z\~^~^'^'^(j){won{y,x,,—z~^))dxdyd^z JFS^ JFS J Fv : • n = C«(-s + — — 2 )x a holomorphic function in s. 172 Corollary 4.3.6 The modified intertwining operator is holomor phic, that is, for any holomorphic section fa € ^ ® holo morphic section in det For the case of n = 2, the analytic property of can be determined since = A4,. Corollary 4.3.7 The modified intertwining operator „(/,) is holomorphic, that is, for any holomorphic section fa in T{^„{s), ,«(/«) " ®holomorphic section in s). Now, we turn to consider the 5p(u)-intertwining operator jV, defined in Lemma 4.3.3. The composition of A4, and AC gives our intertwining operator „(.s). Note that the integrating. variable x „_2 in the integral which defines AC lives in the Levi subgroup M,"_i of the maximal parabolic subgroup P ".,. Since the analytic prop erties of the family {AC(/,)} do not depend on the evaluation of AC(/«) at g, we deduce that the analytic properties of the family {AC(/,)} coincide with that of the restriction of {AC(/,)} to the subgroup GL{n — 1, P„), where GL{n — 1) is canonically embedded into via g !-)■ [g, I 2) for g E GL{n — 1). As a GL{n — 1 )-intertwining operator, AC carries sections in in^'-^'^ det to sections in M,n-2V I det ), for generic complex values of s, where Pi7„-2 and P i,»_2 are maximal parabolic subgroups of GL{n — 1) in the following forms: P\~n-2 = { ( * m ) ^ 0 ^ fo rm E GL(n - 2). 173 We deduce that for holomorphic sections /« in det the analytic properties of the family {X>(/«)} coincide with that of {-Vj"(/s)}, where the GL(n — l)-intertwining operator is defined as ■^rifs){9) = I ^_^fs{üfon~{Xn-3,z)g)dXn-3dz JFv with ÜJ' = J„_i the longest element in the Weyl group of GL{n — 1) and the element / 1 0 0 \ in the unipotent subgroup is of form: n (xn- 3,z) = I 0 /„_3 0 1. It is easy to \ 2 Xn-3 1 / prove that Af~ carries sections in det m|*+^) to sections in I det ), where P,7- 2,1 is a maximal parabolic subgroup of GL{n — 1) of the following form: = {( 7 «) • "» S - 2))- Using the same argument as in the determination of the holomorphy of we need to study the analytic properties of the family )} for sections (j) in +■'^1 det ) with compact support in Pïn-ï^'oP\n- 2- The holomorphy of Af^{ ~ / n -2 (Xn-3, z)vf^)dx„^3dz J Fy = J^^_^(l>{n{Xn^3,z))dXn-3dz = 4 4 - J:;/-. , 174 = L L - , f 0 /"-3 ] ) d x „ _ 3d^z \ z X n - 3 1 / = C«(2a) X a holomorphic function in s. This yields the following Lemma. Lemma 4.3.8 The modified intertwining operator is holomorphic in the fol lowing sense: for any holomorphic section fa in det '(^'l'ïsï-^aifs) is a holomorphic section in | det Combining the results about the intertwining operators Af* and A’s, we obtain the holomorphy of the intertwining operator Proposition 4.3.9 The modified intertwining operator „(.s) is holo morphic, that is, for a holomorphic section fa in is a holomorphic section in det 4.3.3 Intertwining Operator general sections The local intertwining operator C7,"„_„(•«) is the u-component of the global intertwining operator C7^^(-s) attached to the longest Weyl group element Wo and the unipotent radical = 7V,''_i of the maximal parabolic subgroup P"_j of G„. More iJiecisely, 175 we have / 0 0 ■In — 1 0 \ ( lu-X X w Y \ 0 1 0 0 0 1 ty 0 Wo = &nA Nw„ = —In-l 0 0 0 0 0 7n—1 0 \ 0 0 0 1 I 0 0 -*x 1 / The (local) intertwining operator is defined by the following integral: for any section /, G = / fs{wong)dn, (4.21) *'Nw<,(Fv) which takes the sections in /"_i,„(5) to the sections in /"_i,„(—•?). In order to deter mine the holomorphy of this intertwining operator £7,"„,„(s), we are doing to decom pose U^^ „{s) into three intertwining operators, the analytic properties of which are easier to be determined. Let us use the notations N{X,W,Y) = A^(W,y) = N{0,W,Y), N{X) = #(%,0,0), and similarly N{W) = N{W,0) and N{Y) = N{0,Y). Then one has N{X,W,Y) = N{W,Y)N{X) = N{Y)N{W)N{X) as vari eties. The first step is to decompose into the following two two intertwining operators M w , y { s ) and M .x { s ) : = , Mw,Yis){fs)in{x)g)dn{x) JN(X )\rv) = Mxis)[Mw,Yis)ifs)]{g). (4.22) Those two operators M w,y {s) and Aix{s) are described in the following Lemma. Lemma 4.3.10 Let j and be two parabolic subgroups of Sj){îi) of the 176 following types: a * * * / 0 ^ a * \ * b * * 0 6 * * Q n -l,l — { 0 0 * } and = { 0 0 ‘a"‘ 0 } with a £ GL(n - 1). I 0 0 0 6-' I 0 0 ♦ 6-» / Then the Gn,v-intertwining operators M.wy{s) and M x{s) can be described in the following ways: (a) is a Gn,v-intertwining operator Mw.y{s) : W ^ which is defined by the following integral Mwy{s){fs){g)= ( fs{won{w,y)g)dn{w,y). (4.23) JN{W,Y)(Fv) (b) M.x{s) is a Gn,v-intertwining operator M x M : ^|p.,(W î)|-+î|ir') -> m which is defined by the integral below ^xis)ifs){g) = [ fs{n{x)g)dn{x). (4.24) JN( a )(rv ) We start with the intertwining operator M x{s) first. Since integrating variable n(.c) in the integral defining M x{s) belongs to the Levi subgroup M “_i of the max imal parabolic subgroup , the analytic properties of the family of {MxMif.) ■■ /.€in {( Q * with a 6 Gi(n - 1). With applying the argument for the intertwining operator Ms in the previous sub section to the present case, one will easily see that the analytic properties of the family of Mx{s){fs) with sections fs varying in coincides with 1,1 that of the family ofAf%(.s)((^)(l) for sections 4> varying in 2 n —1,1 with compact support inside the open Bruhat cell The possible poles of will be determined by the following standard computation: As before, we denote the longest Weyl group element of GL[n) by Wo = Jn as in Lemma 4.3.1. Then we have /^ x W (^ )(l) / I 0 Z = j <;6( I 0 In-2 Xn-2 ) )dzdx„_2 0 0 1 f / ^ 0 0 \ / 1 0 0 \ — I I 2 "Ai —2 0 J„ -Z-^Xn-2 In-2 0 j )dzdx„_ 2 . Jn -' \ 1 0 By the left homogeneity of (j) and by changing the variable —z ^a ;„_2 >->■ z»_2, we deduce that 178 1 0 0 = Iu-2 0 |)dzdx „_2 ''p'v \ 2"* 0 1 / r / I 0 0 \ = / |z| *n-2 /n-2 0 ) )d’^zdx„_2 \ 2-1 0 1 / = Cu{s — ^ ) X a holomorphic function in s. Therefore we obtain the holomorphy of the intertwining operator Afx(.s), which is stated as Lemma 4.3.11 The modified intertwining operator ^r-^^nryAdx('S) is holomorphic, that is, for any holomorphic section fs in indq^^ ^ (|a(p)|"'+ 2 x(-5)(/«) ‘is a holomorphic section mdp"’” ^ (|a(p)|~*'^^|i>l°)- We now turn to determine the holomorphy of the intertwining operator M.w,y {s). As stated in Lemma 4.3.10, M w,y{s) is a -intertwining operator taking sections in to sections in rndgn”^ ^ (|a(p)|"^+T|5|""^). The holomorphy of M w,y{s) will be determined by decomposing it into two intertwining operators M.y{s) and in the following way: For any section / , G Mw,Y{s){fs){9) = Imwm) f-M y)< ''‘MW) JN{W)JN(Y) , 0 1 / \ 0 0 0 1 179 Conjugating by the Weyl group element w-i as defined in subsection 4.3.1, one have / In-1 0 \ / 1 Y' \ ty 1 -1 0 In-l Wi _ K '^r‘Wi = 1 0 =:N'(Y') In- 1 -Y 1 0 _ty, V 0 1 / \ - ‘y' InIn-l -l J and ( In-1 0 \ ( 1 0 \ 0 1 0 Wi w. 1 _ In-l W 0 In-l 0 0 1 0 \ 0 0 0 1 J l o W 0 In — 1 If we set w* = w-iWoW-i \ then w* ^wiWo = wi and I 1 0 \ / 1 0 0 0 \ 0 0 0 w 1 In—1 w* = In—1 =: N*(W'). K)-' 0 0 1 0 0 0 1 0 \ 0 w 0 In-l ) \ 0 0 0 In — 1 / Hence we can deduce that Mw,Y{s){fa){g) ( In-1 0 \ / In-1 0 \ t y 1 0 1 = f f fsi Wog)dn(Y)dn(W) J n (W) J n (Y) In-1 - Y W 0 \ 0 1 / \ 0 0 0 I J = f [ faiw,n'{Y')w*n*iW')w,g)dn'{Y')dn*iW') Jn *{(W")(F.)w’){f „) JJN'(Y')(F,)n'(y>)(Fv) " L/w(w)(n) = Mw(s)[MYis)ifa)]iwig) = T»,, [A4vy(.s) 0 MY{s)ifs)]ig). Since the left shift operator T»,, {fs){g) = is holomorphic, the analytic prop erties of Mwyi-'i) follows from that of the composition M w(s) ° Afy(.s). On the other hand, it is easy to figure out that for any section fa in 7"_](.s), one has A4y(.s)(/s) = By Proposition 4.3.2, the modified intertwining operator is holomorphic, that is, the operator M y(s) takes holomor phic sections in (.s) to holomorphic sections in | det ), the 180 pai-abolic subgroup Pl\n-\ is described in (4.15). Therefore we only need to determine the holomorphy of the intertwining operator Mw{s). / : 0 \ Lemma 4.3.12 The Weyl group element u;* is I ^ ° ^ 1. The inter- \ - I n - l 0 / twining operator M.w{s) is defined hy the following integral ^ (4.25) JN*(W)(Fv) and takes, for generic complex values of s, sections in indpn^"'^^^j(|<|"~’|detm|*'*‘ 2) to sections m indpn^"’|^(^^j(|t|""’|detm|~*''' 2). Note that in the integral (4.24), the integrating variable n*{W) and the Weyl group element w* belong to the symplectic group Sp{n — 1) of rank n — 1, which is embedded in the Levi subgroup M" = GL{\) X Sp{n — 1) of the maximal parabolic subgroup P" via g ^ {l,g). The analytic properties of the family {M.w{s){fs)} for in | det ) coincide with that of the restriction of {M.w{s){fs)} to the subgroup 5p(n — 1). Under this restriction, Mw{s) can be viewed as an Sp{n — 1)-intertwining operator from ind^pn-~,]i.^f\ \ det m |®+2 ) to ind^n-~,]^'?\\ det m |""+2 ), where P,"ri is the standard maximal parabolic subgroup of Siegel-type. This reduces our operator Mw{s) to the case which was extensively studied by Piatetski-Shapiro and Rallis in [55]. According to the Appendix to section 4 in [55], one has the following Lemma. 181 Lemma 4.3.13 Letan-\,v{s) = C«(-s— n!^I^,j=„(2) C«(25—j+ 1 ). then the modified intertwining operator is holomorphic. Combining Lemma 4.3.13 with Lemma 4.3.11 and 4.3.12, we finally obtain the holomorphy of the intertwining operator Proposition 4.3.14 Letao,„{s) = C«(-5-t+2)Ct,(5-f+l)C«(s-f) njCij=„(2) C«(2.s- j + 1). Then the modified intertwining operator holomorphic. The method we have used to determine the holomorphy of intertwining operators 17^j(.s), 17^2(s), and U^^{s) depends on two basic techniques: (1) Decompose an intertwining operator into several intertwining operators which are in some sense easier to deal with. (2) Embeds the section into some larger spaces (by restriction) where we can apply the method developed by Piatetski-Shapiro and Rallis in [55]. In general, those two techniques may create extra poles. For instance, for n = 2, one has for all .s. We have used different way to determine the possible poles for this intertwining operator and obtained different results as shown in Corollary 4.3.7 and Proposition 4.3.14. It is clear that the method used in establishing Proposition 4.3.14 creates two extra possible poles for the operator order to make sure that our application of those two techniques to our special cases is safe, we will use the method of Gindinkin and Karpelevich, which was described globally by Piatetski-Shapiro and Rallis in [54], to prove that the poles of our intertwining operators determined by our methods are actually achieved by the spherical section. 182 By eulerian property of those intertwining operators, we can state as follows the global versions of the local results we obtained in previous three subsections. Theorem 4.3.15 (Global Versions) All of those three modified intertwining oper ators and are holomorphic, where we let ao{s) := C(-s - f + 2)C(-5 - 7 + l)C(-s “ f ) ^ ’j=hj=n{ 2) “ J + !)• 4.3.4 Intertwining Operators U^^{s), % (s), and %(a): spher ical section In this subsection, we will prove that the analytic properties of intertwining operators f7," (s), and Ul^^{s) described in previous three subsections can be realized by the spherical sections in each case. To this end, we need the formula of Gindinkin and Karpelevich. The formula of such type was first established by Gindinkin and Karpelevich for real groups and by Langlands for groups over Qp. The global version of such formula was described by Piatetski-Shapiro and Rallis in [54]. We are going to recall the formula of Gindinkin and Karpelevich (global version) from [54]. We only focus on our own case although the formula works generally. Let T„ be the standard split maximal torus of G„ = Sp{n), the elements of which are diagonal matrix of form: t = diag(ii, • • • Let B,, be the standard Borel subgroup of Gn containing T„ and its unipotent radical N" is on upijer triangular matrix. Let X*{T,i) be the group of characters of T„ and e,- the character of T„ so that £j(f) = ti for i = 1,2, Then one has X*{Tn) = n,£,}. Let 183 $Cr'„ = $(G„,r„) be the set of roots of T„ in G„, the set of positive roots of $c.'„ determined by iV„, and Ao„ the set of simple roots in Then these root data can be described as follows: $G„ = {±{£i±ej),±2ei :i ^G„ = {{ei±£j),2ei :i ^c?n = {<^« = £« — £i+i, «n = 2e„ : z = 1,2, • • •, n — 1}. (4.28) Let X»{Tn) be the set of the one-parameter subgroups of T„. Then X*(T„) pairs non-degenerately with X*{Tn) by ( , ), which is defined by a;(x''(0) = for X*{T„), and z'" G X.(T%). (4.29) Let X = SlLi -^i£i be a complex character in X*{T„) ® C and S = 2(n — i -f l)e,- the modulus character of the Borel subgroup B„. The (normalized) induced representation 7?idg"|^|(x) of G„(A) is defined to be the space consisting of smooth functions ^ : G„(A) —> C satisfying the following condition: f{bg) = x{b)S^{b)f{g). To each Weyl group element w G Wa„ we can associate a unipotent subgroup N\l of A" so that iV" = with iV“ = w~^N’*w D iV", and an intertwining operator Mw defined by the following integral Muj{f){g) = f f{wng)dn, (4.30) JNS(A) which takes sections in 7ndg|||^j(x) to sections in • %), where the Weyl group Wdn acts on X*{T,i) 0 C by w~^ • x{t) = x (^ “^^^)- It is evident that in each 7ndg"|^|(x), the subspace of A»-fixed vectors is of at most one dimension and 184 there is a unique AT„-fixed vector normalized so that ^%(1) = 1. Since the inter twining operator Mw takes -fixed vectors in to -fixed vectors in • %), there exists a function Cu,(x) so that Mxu{ The c-function can be written in terms of the following integral C(X) = / (j>x,{wn)dn. (4.31) JNSiA) Let E be any half-space in X*{T,i) ®z Q containing 0 0 » and Rw = wE for 10 e Won and = {a € 0g„ : « < 0, w~^a > 0}. According to [54], the method of Gindinkin and Karpelevich gives that the following formula L . m where iV(i?u,) = ria6-(/e„) — wNwW~^, Na is the one-parameter subgroup asso ciated to the root a, and the coroot = a ii a = Si ± Sj and j\ and = ~a if a = 2e,-. Thus by formula (4.27), the function c„,(x) is determined by Now we are going to use formula (4.28) to determine the holomorphy of our three intertwining operators 17^^(s), and U^^{s) evaluated at the spherical sections. More precisely, our degenerate principal series representation 7»_,(.s) can be naturally embedded into the normalized induced representation 7ndg”|^j(xs), where Xs is a character of T„ defined by x« = [E"T/(s — f — £«• Note that under this embedding, the image of the normalized /lf„-fixed (spherical) section /° is the 185 -fixed vector Then we have to compute the following three c-functions: (4.34) 4 ( X .) = (4.35) C ( x . ) = X . (4.36) In order to use formula (4.28), we have to check that N{Rtu) = wN^w~^ for w = Wi,W2, and Wo- Since Tn is also a maximal split torus in the Levi factor M,"_i of the maximal parabolic subgroup one can consider the set of roots = $(M,"_i, T^.) for the reductive group M,"_i with respect to T4 and the set of the positive roots is ^ n 1 * Let fi be the distinguished set of coset representatives for \ ^ 5p(n) obtained by choosing the unique element of minimal length in each coset. Then by Casselman [14]. one has a description of fl as follows: = {u ; G Wsp{n) : W C ^Sp(n)}- (4.37) It is easy to check that for w £ ft, N{Rw) = wN„^u>‘W~^ holds. So, we have to modify our Weyl group elements w\ and W2 by an element w' G so that our intertwining operators (s) and are attached to the Weyl group elements in fl, while Wo is already in 0. We choose w' to be the following element in ( 0 / „ _ 2 0 10 0 0 0 0 1 0 / „ _ 2 0 0 1 0 0 0 0 1 / Then it is easy to check that w\ = w'wi G 0 and U^.(s) = U”*{s) for i = 1,2. 186 Now, applying formula (4.29), the c-functions c” (x«} for w — w*,W 2, and Wo can be computed in the following way: = L n , A , . - . _ TT C((XiQ^^)) since wN’^{A)w~^ = N{Rw){A). Note that ~^c;„(-^«'r) ~ {s, — 6» : Î =1,2,• • • ,n — 1}; (4.38) ~ {sri-i ~ £fn£«-i+ £«■ : * = 1) 2, • • • ,n}; (4.39) -^G„(^«'o) = {£i±en,Si + 6j : i,j = 1,2, (4.40) It is a straightforward computation that = ( « 1 ) c" fs) = ~ 2 ~ 2 + ~ f) 'ff C(2.S-i-H) ^ C(a -h a + 1)((6 .{. a)((, + = _ l) ((2s + ;) ' ^ ' ^ Note that if n = 2, one has c^^(s) = c^^(s) = . Therefore we obtain that Proposition 4.3.16 For Re{s) > 0 and the normalized spherical section f° € ■f,"_i(-s), the poles of the intertwining operators U^^{s), U'f,^{s), and U'f,^{s) can be described as follows: 187 (a) (•s)(/a ) achieves the same poles as the function ({s — | + 2) does. (^) ^to 2(•*)(/») o-ahieves the same poles as the function C(2g)C(6 + | — 2) does, for n > 3 . (^) ^wo(^)ifs) a.chieves the same poles as the function ^(5 —f + 2)^(.s — | +l)C(-s— 2 ) n" :L = »(2) C(2s - j + 1) does, for n > 3. Proof: Since C7"(.s)(/°)(£r) = c^(xs) • fw-sid) for w = W], wg, and Wo, where f° is the normalized spherical section in image space under U^{s), results (a), (b), and (c) follow from the computations above and the nonvanishing of f^,^. □ Combining the results in Proposition 4.3.16 and these in Theorem 4.3.15, we have the following theorem, which is important for us to study the poles of our Eisenstein series. Theorem 4.3.17 (Global) For Re(s) > 0 and holomorphic sections fa in we have the following three statements: (a) 17," (•s)(/s) and ((.s — | + 2) share the same poles. (b) U”^{,s){fs) and C(2.s)C(.s + ^ — 2) share the same poles, for n > 3. rcj %(^)(A) - I + 2)C(^ - I 4- 1)C(6 - ((2s - ; + 1) ghore the same poles, for n > 3. For the special case n = 2, we may state our results as 188 Proposition 4.3.18 For Re{s) > 0, the intertwining operator U^^{s) and the global zeta function C(s — 1) share the same poles. 4.4 Poles of Eisenstein Series: n < 3 In this section, we are going to study, for Re{s) > 0, the poles of our (unnornialized) Eisenstein series E'^_-i(g,s', fa) for the special case n < 3. The case of n = 1 is well known. We will use our inductive formula to determine the cases n = 2 and n = 3. Actually, the case n = 3 is one of the most technical parts of our determination of the poles of Eisenstein series ■•‘'i /»)• First of all, we recall from [43] the results on the Siegel Eisenstein series. Theorem 4.4.1 (Kudla-Rallis [43]) For any holomorphic section fa in r,l{s), the (unnormalized) Siegel Eisenstein series E([{g,s] fa) enjoys the following properties: (a) -5; fs) M holomorphic for Re{s) > 0 except for s 6 {^-^, • • •, (h) At So = the residue representation i2eSs=s„(E’*((7,s; fa)) does not vanish. Here e'{n) = 1 ifn is even and 2 if odd. Actually, Kudla and Rallis’ result on the family of Eisenstein series jG"(g, .s; is much more general. They also proved that the normalized Eisenstein series is holomorphic for Re{s) < 0. 189 For n = 1, one has the following properties that will be used later to determine the poles of EH{g, s; fa) for n = 3. Corollary 4.4.2 Let any holomorphic section fa 6 7% (a). Then we have: (a) The Eisenstein series El{g,s‘,fa) has a zero at s = 0, and (b) The residue of E\{g,s\, fa) at s = 1 is ReSa=\{E\{g,s\fa)) - ReSa=i{j^fa{won{x)g)dx) where Wo = ^ ^ ^ and n{x) = ^ q j j • Proof: (b) is straightforward by considering the constant term along the standard Borel subgroup of 5"p(l). (a) is from Kudla and Rallis [44] or [45] since the normalized Eisenstein series E]*{g,s : fa) := 6^(5, l)E j («7,5 : fa) is holomorphic at s = 0, while the normalizing factor b^{s, 1) = + 1) has a simple pole at .s = 0. □ For n = 2, the analytic properties of the (unnormalized) Eisenstein series E'y{g, s; fa) can be described as follows. Theorem 4.4.3 For any holomorphic section fa in T\{-s), the (unnorm,alized) Eisen stein series E^{g, s; fa) is holomorphic for Re{s) > 0 except for s = 2 where it achieves a simple pole. Proof: According to formula (4.11) in Theorem 4.2.3, the constant term of E'({g, s; fa) along the maximal parabolic subgroup P^ is equal to ^lp^{^i('ti^s'),s-,fa) (4.44) 190 = l Since U'l’*{s) = f{fJf|C^4(^) Ul'*{s) = holoniorphic for Re{s) > 0, we can rewrite the above formula as Elp;im,(h,g'),s-,f.) = |i,r+V.(s') (4.45) + i',r'"% '(3)(/.)(y)^|^. The first term is always holomorphic as a function of s. For Re{s) > 0, the second term is holomorphic except for s = 1 and the third term is holomorphic except for .s = 1 and .s = 2. Hence the constant term achieves a simple pole at .s = 2. We are going to prove that the constant term is actually holomorphic at .s = 1. Taking the residue at .s = 1 of the constant term, we have ReSs=\ [Elp2{7n-i{tug'), •«;A)] (4.46) I C( By Corollary 4.4.2, the residue at .s = 1 of E]{g', s; o U'w*{s){fa)) is equal to / I 0 \ / 1 \ 0 1 1 Z g')dz]. (4.47) 0 1 1 V -1 0 j I 1 / We denote the integral by Ms{s){U^'*{s){fa)){g'). Note that U‘^'*{s) may be regarded as an 5p(2)-intertwining operator from If{s) to Then A4-(.s) is an 5p(2)-intertwining operator from It^T) to for generic values of .s. After normalizing by i.e. MZ{s) = the 191 modified operator M*{s) is holomorphic for Re{s) > 0. Hence the residue at s = 1 of E\{g',s-, o U'^'*{s){fs)) is equal to o U'i'*{s){fs)]{g'). In order to prove the holomorphy at s = 1 of the constant term p 2{m,i{ti,g'),s'^ fs), that is, the vanishingness at s = 1 of the residue, it suffices to prove the following identity of global intertwining operators: lim[P^-(»)(/.)I(j7') = Um[AI:(3) o %-(«)(/.)](/). (4.48) Since the residue Resa=i^{s) and the residue i?eSs=]C(-s — 1) are reciprocal. Because both sides of the above identity are eulerian, we shall first prove the corresponding local identity for each local place and then the global identity will follow the local one by the standard local-global argument. For each local place v, the local intertwining operator U'w*v{s) maps from to and similarly o U'^’*^{s) takes sections in to sections in Moreover, both of them take the unique nor malized spherical section in to the unique normalized spherical section in the respective space of induced representation. Let us consider a typical intertwining operator »(.;), which is defined by the following integral / 1 \ 1 M i M i f . m = c(7?i) j^/<( 1 V 1 \ and maps from to (l^iI1^21"^)- Further it is easy to check that the intertwining operator is holomorphic for Re{s) > 0 and takes the unique normalized spherical section in |* 2|~®) to that in The reason we introduce the operator My„{.s) is that we 192 have the following identity for Re{s) > 0: (4.49 ) This identity can be verified by straightforward computations of the integrals which define those intertwining operators and the relevant normalizing factors. Note that those intertwining operators VW* „(.s), and are well-defined, nonzero, and holomorphic for i2e(s) > 0. The above decomposition is valid for Re{s) > 0. For s = 1, VW* „(.;) becomes an endomorphism of By means of induction on stages, we have and the intertwining operator Al* „(1) can be interpreted as the canonical induction of the normalized standard intertwining operator from to itself, which is the identity map. In other words, (4.45) at s = 1 has following refined form: %«(!) = 0 %(1). (4.50) Next we return to prove the global version, identity (4.44). Without loss of gen erality, we assume that the section /, is eulerian, i.e. /, = ®vfa,v 6 /?(*’). Then there exists a finite set 5 of places of the number field F, so that /, = {(^vesfs,v) %"(«)(/.)(») = [II % .(')(/.,.)(».)I[II ves v^s 193 = [II mim o %(5)(/.,.)(gr«)][II m im 0 %(^)(0(!7«)]. u65 u^S Note that the products above are actually ones with finite number of local factors since U'i'*vis){f°^) and MZ„{s) o are the unique normalized spherical section in the respective induced space and for almost every local place v, g„ belongs to the maximal open compact subgroup Sp{2,0„). Thus each of those products is holomorphic in s for iîe(.s) > 0 and the limit lim*-».! can be exchanged with those products H- Therefore we obtain that, for g = G 5p(2, A), vÇS v^S = HI ,)b.)IHI ugS vftS Since the subspace of factorizable sections is dense in /if(-s), identity (4.44) follows. This proves the Theorem. □ Finally, we come to deal with the case n = 3, which is crucial for us to determine the order of the poles for the Eisenstein series E”_](< 7,s;/s). Theorem 4.4.4 For Re{s) > 0 and any holomorphic section fs in the (un normalized) Eisenstein series E.2(fir, s;/«) is holomorphic except for s = 1 ,|, and |, and at s = |, and | , the Eisenstein series El{g,s\ /,) achieves a simple pole. 194 Among the ingredients of our proof of this Theorem are: (1) the reducibility of degenerate principal series representations of Sp{7i) for n < 3, (2) the results of Kudla and Rallis about the poles of Eisenstein series of Siegel type, and (3) more techniques on intertwining operators. The proof of this theorem will be completed separately for the case of s | and the case of s = |. Those two cases will be treated in following two subsections. 4.4.1 Proof of Theorem 4.4.4 for the case of s ^ According to the constant term principal of the general theory of Eisenstein series, we will consider the constant term E^p 3{g,s] fa) of EUg-iS^fs) along the unipotent radical of the maximal parabolic subgroup Pf. Applying Theorem 4.2.3 to the case n = 3, the constant term can be expressed as follows: a'),+ In order to determine the analytic properties of the Eisenstein series Ei{g,s;fs) and ■Ë'2(ffî /«)) we have to modify the sections o (•?)(/«) and o (•!*)(/*)• For any section /* = [0„{js/s%] 0 [ fs,r = *2(/»)5 (4.51) / s , l = [0i)(?s/s,„,i] 0 [ 0 « e s ^ ° ^Wi,v{'^){fa,v))i (4.52) 195 f..z = <8 (»)(/.,)], (4.53) where / ° „ j and / “„ 2are the normalized JC3,„- spherical sections in the corresponding local representation spaces. Then the sections /j,r, and fa,2 are holomorphic for Re{s) > 0 and A'3,„-finite. Applying the results in section 4.2 and 4.3 to the case n = 3, the constant term E^p 3{g,.s] fa) can be expressed in terms of Eisenstein series of Sp{2) and the global zeta functions as follows: Elp3{m{U,g'),s;fa) = + ^; i;(/^)) + (4.54) Cl'S + 2) + D alf+ !)■ By means of Theorem 4.4.1 and 4.4.3, we are be able to conclude that the constant term E2 p 3{m{ti,g'),.r, fa) is holomorphic for Re{s) > 0 except for .s € A'3 = {1, |, |} . In fact, if we call the three terms in order term I, term II, and term III, then we see easily that for Re{s) > 0, term I only has a pole at s = |, term II only has a pole at .s = 1 and |, and term III has a pole at s € X3. Hence the constant term is holomorphic at s 0 X3. In addition, it is easy to see that the constant term 'S; fs) achieves a simple pole at .s = |, |. At s = |, each of those three terms achieves a simple pole and there are no cancelations among those three terms because the ‘exponents’ at t-[ of three terms are different, and at s = |, the first two terms are holomorphic while the third term achieves a simple pole. Therefore we have completed the proof of the Theorem for the case s ^ j. In the next subsection, we have to prove that the constant term E |p 3(m( pole at s = |! 4.4.2 Proof of Theorem 4.4.4 for the case of 5 = The difficulty to determine the order of the pole at 5 = | of the constant term is that (1) term I is holomorphic and both term II and term III have a double pole, (2) both term II and term III have the same ‘exponent’ at t-[, which implies that there are some cancelations between those two terms. We will work out such cancelations in this subsection. We need some more intertwining operators of 5'p(3). The analytic properties of those intertwining operators will be critical to our determination of the order of the pole at .s = i of the constant term E^pi{m{t-[,g'),s-,fs). For any section fg in JK-s), we will define 5p(3, A)-intertwining operators and Md{s) as follows: Mu{s){fg){g) := I fs{wunu{w,x,y,z)g)dwdxdydz (4.55) JA* Md{s){fg){g) := I fs{wdTid{v,w,x,y,z)g)dvdwdxdydz (4.56) JA^ Those integrals converge absolutely for the real part of s large and have meromorphic continuations to the whole complex plane of s. The Weyl group elements and Wd and the corresponding unipotent subgroups are described as follows: / 0 1 0 \ 1 0 w 0 X 0 1 1 y X 2 1 0 0 1 0 0 aiid nu{w,x,y,z) = W u = 0 0 1 1 - 1 0 0 1 \ 0 1 0 / —w -y 1 / 197 and 1 0 \ / 1 V w 2 X 0 \ f ° 0 1 0 1 y X 0 0 1 0 0 0 1 0 0 0 WJ : a n d iid(v, w,x,y,z) — - 1 0 0 1 0 0 1 0 — V 1 \ 0 0 1 0 / I — w ' -y 1 / Then for generic complex values of s, Mu{s) and Md(s) are 5p(3)-intertwining op erators in the following sense. Mu{s) : 7 f(.s ) -4- — -,.s — -) := Md{s) : ll{s) -4- I^{—s - -, -l,.s — -) := where the induced representations from the Borel subgroup are normalized and smooth (We always assume that the archimedean components are Koo- finite). By induction on stages, we can rewrite those principal series representations as follows: = ([l (8 7»dg%'^)(| ® 7^3 - ^)), (4.57) = det |-^ 0 ® - h )- (4.58) 'fg' 2' The canonical intertwining operator M-z from [|det|“^^ g) to [| det 1“^ ^ g) 7ndg^^^^’^^(||^|“^V^)j and the identity map of 7^(.s — |) together will induce a 5p(3, A)-intertwining operator denoted by Mu,d{s) from 7^(—1, —-s —|, -s —|) to P {—s~^, —1, .s—1). More precisely, Mu,d{s) can be expressed by following integral: for any section /, in 7nd^^|^)([| det Cg) 7ndg^(^'j^)(||^|^)] ® 7j(.s - 4)), Mu,d{-^){fs){9) := / fs{wu,dnu,d{z)g)dz, (4.59) JA 198 where the Weyl group element and the corresponding unipotent subgroup is as follows: \ \ aiid Hu,d(z) = 0 1 1 1 0 —z 1 1 / V 1 / The following lemma describes the relations among those intertwining operators just defined as above. Lemma 4.4.5 Wt have following identities of Sp{3, A)-intertwining operators: (a) For generic complex values of s, M.d{s) = Mu,d{s) o VWu(.s). (b) The normalized intertwining operators enjoy the same property: (4.60) Those operators are modified in the following way so that the normalized oper ators are holomorphic for Re{s) > 0 and take the unique normalized spherical sections to the unique normalized spherical sections in relevant spaces of induced representations: «•> - T a E îiS f C(2.s)C(.s + ])C(.s - ]) Proof: The identity in (a) holds since the integral defining the intertwining operator on one side coincides with that defining the intertwining operator on the other side when 199 they are absolutely convergent (i.e. for Re{s) large). The identity in (b) follows from (a) and the computations of those normalizing factors (c-functions). Our methods to determine the holomorphy of those intertwining operators are similar to those used in the previous section and we will omit the details here. □ The reason that we introduce those intertwining operators Ad„(s), A^u,d(.s) and Md{s) is made clear by the following lemma. Lemma 4.4.6 Let Sp{l) = Sp{Fe^ © Fe^), which is canonically embedded in Sp{3) via g t-4 m (l,l,(7), where m{t^,t 2,g) in GL{1) x GL{1) x Sp{l) C M |, the Levi factor of the standard maximal parabolic subgroup Pf of Sp{3). Then, for any section f s E Re{s) > 0, the following identities hold: (4 .61 ) M:(s)(f.){g) = (4.c2) where U^’*{s) is the Sp{2)-intertwining operator from /|(.s) to Ind^^^'^\\t 2\~"~^ @ 7’(.s — 1) for Re(s) > 0, which is defined in [43] by the following integral: = %(2t)((g + 1)^^ fs{w2n{x, y)gdn, (4.63) where the Weyl group element and the unipotent subgroup are / 0 1 ( \ X y \ 1 0 - -1*0*^ |andn(x,y) = 1 V 0 1 V -* 1 / Proof: The proof goes exactly the same as that of Lemma 4.4.5. We omit it here. □ 200 With all those lemmas stated above, we are able to determine the order of the pole at .s = I of the constant term E^ps{m{ti,g'),s-,fs). Again by the inductive formula (4.50) the constant term E2 p 3{m{ti,g'),s]fs) have the following expression: 2 + \h\^El{g\,rJs,,) C(-s + 2) + I Note that the first term \ti\^'^2E'f{g', s + i*{fa)) has at most a simple pole at .s = | by Theorem 4.4.3, while both the second and the third terms have the same ‘exponent’ for fi, the center GL{1) of the Levi factor Mf, and achieve a double pole at .s = | since the value representation of the Eisenstein series E^{g,s',fg) as .s = 0 does not vanish according to Kudla and Rallis’ argument for the ‘first term’ of regularized Siegel-Weil formula in [44]. In order to make the cancelation of the poles in both terms, we will consider the P/-constant term of f)C(2.f+1)' Since we concern the double residue, the factor |fi|* can be omitted. Then the P^- constant term evaluated at g' = mi{t2,g") of (4.60) equals + f(?+ |)«lf+I)'**'® ''"” * ~ h ‘'° 201 It suffices to prove the following two statements: (a) The sum of two terms with the exponent 1 for is = (4.65) + C(î + |)Ç (2 f^l)^' (»"' ■’ “ ? ■' ° % which has at most a simple pole at s = |. (b) The sum of three terms with the exponent 2 for < 2 is % («) := + (4.66) + V'}) + ^4('*’ - 2)(/».2)(” K1’î/”)))> which has an at most simple pole at s = The proof of Statement (a): According to Kudla and Rallis [43], for any holomorphic section /j,i in I^is) with Re{s) > 0, one has where the section is defined by = (».«.< ,,.,2.) ® ° which is a holomorphic and /f-finite section in 7^ (.s — |)- By means of (4.46), we have ■> Ui, - j)( /,,,) = . A ,,,. (4.69) 202 Let us set ri(s) : = • Note that the function rj(.s) has a double pole at s = Substituting those data into Ei{s) in (4.61), we deduce that Ei(.s) = ri(.s)[C(.s + ( Following Corollary 4.4.2, E]{g,s] fa) vanishes at s = 0. In order to prove that Ei{s) has at most a simple pole at s = 5 , we need more information about those two Eisenstein series E }{g",s- 5 ;/,, 1,2') and El{g",s - fa,2,1)- Actually, fa,i,2' = M*{s){fa) and fa,■2,1 = Mj{s){fa). They are holomorphic and is in P{—1, - s — .s — I) and P { —.s — —1, .s — ^), respectively, see (4.53) and (4.54). Moreover, one has M.* j{fa,i,2') = fa,2,i by Lemma 4.4.5 and 4.4.6. Since those two Eisenstein series live on the factor 5^p(l) of the Levi part of the maximal parabolic subgroup Pf and the intertwining operator Af* ,j(.s) is essentially A4*(.s) on the factor GL{2) of the Levi part M | of Pf, we consider the restriction of those two sections fa,},2' and fa,2,ï to the Levi part = GL{2) x 5p(l). According to our general assumption that those induced modules at the real archimedean place are Harish- Chandra modules (i.e. AT-hnite), we can separate the variables of the restriction of those two sections. In other words, for ( A ,1,2'((<71,< 72)) = (8 <^1,.((<7i,<72)) (4.71) t and /s.2,l((fifl,<72)) = '^MZ{s){^a,i) ^'a,i{{9u 92)) (4.72) t where the summation is finite and ^a,i is in [|det|“^ ^ Cx) 7 ( I I )] ■M:{s){(l)a,i) is in [| det 0 )], and is in I^(.9 - I). 203 Without loss of generality, we may assume that fa,i,2'{{9ii92)) = and both and more specifically as follows: (g'% -s - 2 » = ^«(1) • —2 > ^s) (4.73) (4.74) Plugging the above into Ei{s), we obtain the following: E i(s) = ri(.s)E j (<7",s — <^(,)[C(-s + 2)^»(1) + ((-s - 2)-^*(-^)(*^*)(^)]- (4-75) It is easy to see that Ei(.s) has at most a simple pole at s = | since ri(.s)E’(£f", s — 4; <^') has at most a simple pole at s = 4 and [({s + |)<î('»(l) + ((.s — |)A4*(.s)( can be proved to be holomorphic at s = 4. The last claim is equivalent to the identity: In fact, at s = 4, both (j>L and MZ{\){^i) are in [|det|~’ (gl At each local place u, [| det |“’ (g is irreducible and the normalized intertwining operator M*{\) is the identity map of [| det |“^ Thus we have „)(^fi,«). Without loss of generality, we may assume that is factorizable, i.e. <^1. = [®u6S<^i,„] Cg) J . Then = Hm[n )((».,.)(m,)|[n x:,.(»)«j(s...)l ^ *-^2 VÇ.S V^S = in .w:,.(l)(*i,.)(m..)i[n x;,.(i)(4,„)(«,,.)] ves ^ viS ^ ves v^s = This proves the global identity: and therefore proves that has at most a simple pole at 5 = |, i.e. Statement (a). The proof of Statement (b): Now we have to prove that E2{s), which is + 2) pi/// „ , 1 E'iis) = E-i{g",s + -;/s,i,o ) + 1) has at most a simple pole at .s = |. To this end, we need a lemma, which will be proved in section 4.6 of this chapter. Lemma 4.4.7 Let v be any local place of the totally real number field F. (a) The normalized Sp{2,F„)-intertwining operator ^ 0 from /|,„ (|) to „{0) is not surjective. (b) Let M l := 0 and M 2 := Then M i = M 2 as Sp{3, Fv)-intertwining operators from 7^ to ® 7^(0)), where the intertwining operator Ml,g „{s + |) is defined by following integral ( h 0 0 0 \ ( 0 0 0 \ C(f_+I) 0001 0 1 0 X C(« + 0 0 /2 0 0 0 h 0 / I 0 0 0 1 / for sections fs 6 7ndpV ’ (|ti| ' 0 7 |„(i)). From this Lemma, we have a global results about sections /, Proposition 4.4.8 For any section fs in 7|(.s) holomorphic at s = we have lim fs,2 = lim Ul^[s){fsa) = hm (4.76) 205 as sections in (g) /^(O)). Proof: First we shall prove the identity: lim,_^r /*,2 = linig_».L U'i^{s){faa)- The idea to prove this global identity is that we prove the local version of this identity for each local place v first and then the global identity will follow from the standard local-global argument. According to [37] and [?], for each local place v, the local Sp{2, F„)-module is a direct sum of two irreducible submodules denoted by Vo and V%, that is, = Vo ® Vi. We assume that the submodule Vo is generated by ATg,«-spherical functions in 7^(0). By the Lemma above, o in not a surjective 5p(2)-intertwining operator from 7g to /^(O) and takes the normalized ATg,«-spherical section to the normalized ATg,«-spherical section. This implies that iloU i;/-)(lU \)) = V,. (4.77) In other words, for any section /»,« in 7g „(.s) which is holomorphic at .s = |, we always have fs,2,v 6 Vq. By the standard normalization, U'^’f^^{s — |) = — |) for each local place V, and — ~) = — |) globally. At s = j , the normalized intertwining operator C7^j*«(0) is the identity map when restricted to submodule Vq . In other words, we have, for each local place u, (4.78) Note that this is the local version of the identity we are going to prove. 206 For the global identity, without loss of generality, we may assume that the section fs,2 is factorizable, i.e., /,,2 = [®vesfs,2,v] 0 [0«ç$s/°2,«]- Then we have - i)(/.,j(y)= in %',.(« - h(/., 2,.)(ff.)]in ^ «6.5 v i S ^ Note that the second product is the one of finitely many terms since is in 5p(3, O^) for almost all v and Ul;*„{s — |) ( / ° 2,«) is the unique normalized spherical section in <8) /i,„(0)). It follows by taking the limit and by Lemma 4.4.7 that = III C^%(0)(/i,2..)(5.)][II ves v^s ~ [II /g,2,«W")][II f\;2,vi9v)] «65 «{?5 Since lim^_,.L = 1, we finally obtain the global identity: g) = fi g. For the identity: = /i g, we notice that = Ati(/i J and / i , 2,« = • ^2(/i,u). By Lemma 4.4.7, we have the local version of the identity: -^wo,«(^)(/^,],«) — /i.,2,«- By the same local global argument as above, the global identity follows. The Proposition is proved. □ The double residue D.ReSg^iE2{s) of E2{s) can be easily expressed as — [6' ^ + i; /.,.,o)] (4.79) Since the restriction to 5p(l) of /s ,2 and U^^{s — \){fa,2) are constant, one has /s,2(m(l,(jf")) = /5,2(1) and U^^(s - i)(/,,2)(m(l,fif")) = - \){fs,2){l)- Note 207 that Æe5,^t[C(2.s)] = |i2eSs=i[C(.s)] and i2es,^i[C(5 + |)] = [((.$)]. Thus, by the identity in Proposition 4.4.8, we have that D.Res^^LEiis) (4.80) It suffices to prove that = “• (4-si) Taking again the constant term along the Borel B\ of where + i) is defined as in Lemma 4.4.7. Note that + ^){fs,\,r){g") is constant. It is reduced to prove that -^-^•^s=^[C(-S + + -)(/«,l,r)(l) + C(-S — g)A,2(l)] is zero. In other words, we have to prove that A4*g(l)(/i]) = f\_2- But this is exactly the identity we have proved in Proposition 4.4.8. We are done. Theorem 4.4.4 is completely jjroved. 4.5 Poles of Eisenstein Series It comes to determine the poles of Eisenstein series /j) for general n. We recall that as in Proposition 4.2.2, i*_i is the restriction operator from 7"_i(s) to 208 InZl{s + |), and for generic complex values of s, o [7^^ (.s) is the operator from to 0 is the one from to - |), and the intertwining operator U2^{s) maps from /"-iC-®) to -s). According to Theorem (4.3.17), the modified intertwining operators • i*_i o 17”^ (.s), o % ( a ), and are holomorphic for iîe(.s) > 0, with ^ C(®~ t + 2)C(^^ t + l)C(^-f) pr C(2s - j + l) Let / s = [ 0 u 6 5 / s , u ] 0 [<0üç;s/s% ] be any holomorphic section in set that = \ \ • CC.,«(a)(A«)] 0 [0«f?s/°„,ol; (4.85) = (4.86) = ■ ^uij,uC^)(/a..)] ® (4.87) Then the sections /j,o, /«.i, and fa,2 are holomorphic for Re{s) > 0 and fiTn-finite in the corresponding image spaces of under the intertwining operators 17,"^ (.s), U'w, (-®)> and I7”j(.s), respectively. Now we can determine the poles of our Eisenstein series {g, .s; fa) for Re{s) > 0 and n > 4. First we need following Lemma, which will be proved in the next section. 209 Let U’^'*{s) := ■^r^U’^{s) be the normalized intertwining operator associated to Then U'^'*{s) is holomorphic for Re{s) > — | + 1, which is greater than or equals —1 when n > 4. Hence, for any holomorphic section fs in /«.2 = *n-i ° %*('^)(/«) is holomorphic for Re{s) > -1 . Lemma 4.5.1 For any even integer n > 4, the Eisenstein series EHZlig^-^ — fs) vanishes at s = ^ for any section fs = fs,2 in the image z*_i o % *(])(/»_,(])) in •^«-•2 (0 ) ofl'n-iil) 'fender the map 0 %*(&)- Based on the results about the location and the order of the poles of the Eisenstein series E',l_i{g,s] fs) for n < 3 and the above Lemma, the location and the order of possible poles of the Eisenstein series /,) for general n can be determined as follows. Theorem 4.5.2 Assume that n > 4. For any holomorphic section fs in the (unnormalized) Eisenstein series E ’^_■^{g,s; fs) enjoys the following properties: (a) Ell_-i(g,s; fs) is holomorphic for Re(s) > 0 except for (4.88) where e{n) = 1 if n is odd and e{n) = 2 if n is even. (b) Ell_i {g, .s; fs) achieves a simple pole at s = ^ and a double pole at s = and E ”_i{g, .s; fs) achieves at most a double pole at s € 210 Proof: First we prove the case n = 4. According to the constant term principal in the general theory of Eisenstein series, the analytic properties of E^{g,s\fs) can be determined by its constant term pt{g,s] fs) along the unipotent radical Nf of the maximal parabolic subgroup P^. By means of Theorem 4.2.3 and (4.81)-(4.86) above for the case n = 4, we can express the constant term E^pi{g, s; fs) in terms of Eisenstein series of Sp{3) and the c-functions c^^ (a) and c^^(s) as follows: + (4.89) 2 ’ C{s) C{s -|- 3) According to the Theorem of Kudla and Rallis in [43] the (unnormalized) Eisenstein series E^{g, s; fs,-i) is holomorphic for Re{s) > 0 except for s = 1,2, and at s = 1,2, E^{g,s] fs,\) achieves a simple pole (by taking the normalized spherical section). On the other hand, by means of Theorem 4.4.4, we conclude that the (unnormalized) Eisenstein series E^ig', s - | - i^ifs)) is holomorphic for Re{s) > 0 except for s = 1,2, and E^ig', s isifs)) achieves a simple pole at s = 1,2 (by taking the normalized spherical section); and the (unnormalized) Eisenstein series E^ig'^s — fs,2) is holo morphic for i2e(.s) > 0 with exception of s = 1,2,3, where E^ig'iS — 1; fs;i) achieves a simple pole by the same reason. According to the above Lemma, the third term of (4.88) |t]|“*+^E|(( 7',.s - ^1/ 3,2) holomorphic at .s = Hence the con stant term E^pi{rn-[{t-[,g'), s] fs) is holomorphic for Re{s) > 0 except for .s = 1,2,3. The existence of the poles of E^ pt (mi (ti, g'), .s; /,) at s = 1,2,3 can be determined in the following way: We number the three terms in (4.97 by their order and call 211 them term I, term II, and term III. Since for .s = 1,2,3, those three terms have different ‘exponents’ at the center GL{1) x of M^, there are no cancelations among those three terms. Thus the poles of fs) is determined by each term. Note that if fs is the normalized jiT^-spherical section in /^(.s), then all three sections «^(/s), /*,i, and fs,2 are simultaneously the normalized /fa-spherical sections in the corresponding degenerate principal series representations /^(.s + |), /|(.s), and 7|(s — i), respectively. At s = 3, only term III achieves a simple pole, so does fs). At s = 2, all three terms achieve a simple pole, so does E^pi{m\{ti,g'), s\ fs) since no cancelations will happen. Finally, at s = 1, term I achieves a simple pole, and both terms II and III achieve a double pole. We conclude that E^p^{mi{t^,g'),s; fs) achieves a double pole at .s = 1 just because there are no cancelations between term II and term III. This proves the theorem for the case n = 4. Next we will prove the theorem for the case of general n. Inductively, we assume that the theorem hold for the case of n — 1. The constant term of Eisenstein series ■®n-i idi •*) fs) along the maximal parabolic subgroup P f can be expressed, by Theorem 4.2.3 and 4.3.17, as follows: + (4.90) - V ) C(.5 + By the assumption of induction, we have 212 (a) E'nZlig'iS + l;ilifs)) achieves a simple pole at s = | and .s = a double pole at s = and has at most double pole at s = (b) EnZl{g',s — I', fa, 2) achieves a simple pole at s = ^ and .s = ^, a double pole at s = and has at most double pole at s = • • •, £ÜLzp+l_ Since C(2s) has a simple pole at s = 0, | and ("(s + has a simple pole at s = — and > 1, it is easy to see that the term E',lzl{g',-^ — i; /»,2) achieves a simple pole at s = ^ and s = |, a double pole at s = and has at most double pole at s = • • •, 4"-^)+^. Note that when n is even, | is not in the set X ^. However, the Lemma above implies the third term in (4.88) is holomorphic at s = | and so is the constant term E'^_^ pn (mi(iti,/) , .s; fs). On the other hand, by Theorem 4.4.1 or [43], the term E"zl (g\ -s; fa.\) has a simple pole at .s = | and at .s = • • •, ^ with e'(n) = 1 if n is even and e'{n) = 2 if n odd, and has a double pole at s = Therefore, by comparing the ‘exponents’ of | the constant term E"_-y pn{mi{t\,g'),s; fs) achieves a simple pole and, at s = pn(mi(ti, (/'), .s; fs) achieves a double pole since there are no cancelations among those three terms. At the value of .s other than those three: ^ and the constant term (m i(ti,g'),a; y«) achieves at most a double pole. This proves the theorem. □ Whether the constant term E’^_i pn{mi{ti,g'),s-, fa) achieves a double pole at a E • • •, is not clear to me at present. However, we can prove the following 213 proposition. Proposition 4.5.3 If the Eisenstein series E^(g,s;fa) does not vanish ai .s = 0 for a Ki-spherical function in /^(O), the Eisenstein series El^_-i(g,s', fa) (n > 5) achieves a double pole ai s G { • Proof: First we will prove the proposition for u = 5. Basically, we have to check that the Eisenstein series El{g,s;fa) has a double pole at s = According to the inductive formula for n = 5, we have 2 + \t,\^Et{g',s■Ja,^)4p4/ / f \((^ ~ 2) At s = |, the first term has a double pole with ii-exponent 4, the second term has a double pole with ii-exponent 4, and the last term has a double pole with ii-exponent 3 only if the Eisenstein series E^{g\s — \',fs,2) does not vanish at .s = |. Therefore, under our assumption, the constant term E^ ps (m ,(i,, g'), .s; fa) will achieves a double pole at s = This proves the case of n = 5. Inductively we assume that the proposition hold for the case of n — 1. Following the proof of Theorem 4.5.1, we know that, at s G • • •, the constant term fa) achieves a double pole since the difference of the ti-exponent of each term in the inductive formula. While at s = the first term has a double pole with -exponent n — 1, the second term has a double pole with -exponent 214 n — 1, and the last term in the inductive formula has double pole with -exponent 3. Since n > 5, there are no cancelations among those three terms. Thus the constant term /«) achieves a double pole at s = This proves the proposition. □ From the proof of the case of n = 5, we believe that, if the Eisenstein series E^{g, s; fs) vanishes at s = 0, the Eisenstein series E\{g, s; fs) will only have at most a simple pole at 5 = | since the first two terms should be canceled by a predicted first term identity of the regularized Siegel-Weil formula in the sense of Kudla and Rallis [43]. Now we can prove our main Theorem of this paper. For a given section fs-, there is a finite set 5 of places which including all archimedean ones, the normalizing factor (-s) i® defined to be (®) = n + % + l)Cti(® + - 1) n Ct>(2.s + j)] "gS ^ -2 Z j=l,j=„(2) and the normalized Eisenstein series E^’f^{g,s; fs) is defined by Theorem 4.5.4 (Main) Assume that F is a totally real number field and n > 3. For any holomorphic section fs € the normalized Eisenstein series enjoys the following properties: (a) The set of possible poles (of order at most two) of the normalized Eisenstein 215 series is , 71-1-2 71 71 — 2 p- 71 — 2 77 77 -|- 2. {— ",0, ,g ,— }; (b) The (normalized) Eisenstein series achieves a simple pole at s = ^ and a double pole at s = Proof: We assume first that u > 4. Since the normalizing factor (.s) is holomor- phic for Re{s) > 0 (n > 4), the normalization does not change the holomorphy of the Eisenstein series 5 ;/,). Thus the analytic properties of .s;/«) for Re{s) > 0 follow from that of s ;/*) as stated in Theorem 4.5.1. Next we consider the situation for Re{s) < 0. By the general theory of Eisenstein series [49], [2], and [53] one has a functional equation as follows: K-, k, »;/.) = (sr. K. (.')(/•))• By Theorem 4.3.17, we have, for Re{s) < 0, = <:-,(»K:_,(a,-«;%(«)(/.)) “ti-ll Since /*,o in A"„-finite and holomorphic as a section in .s), it follows from the first part of the proof that E'^'fi{g,—s‘,fs^o) has at most double poles at s E { — • ■ •, —^2^}. It is easy to check that the factor is holo morphic for Re{s) < 0. In fact, one has, from subsection 4.3.4 n - 2 _ yj Ct.(-g + 2 + ^)Cv{-^ + + 2 ~ ^) n"=i,j=n( 2) C«(~2-S +j) „gS Ct>(« + t + l)Cu(-5 + f)C«('5 + 2 - 1) n"=l,j=„(2) C«(2-S + i) because c"^(.s) = Therefore --s; fs,o) has at most double poles at .s e { — f - -, —^ } , and so does ( theorem for n > 4. Finally, for n = 3, the normalizing factor d2’^(.s) = + «)C«(25 + 1)] ugS ^ ^ ^ is holomorphic for Re{s) > 0 except for -s = |. According to section 4.4, the Eisenstein series E|(<7,.s;/,) is holomorphic for Re{s) > except for and |. Thus the normalized Eisenstein series E\'*{g,s\ fa) has simple poles at s = | and s = and has a double pole at s = |! The poles of 5; fa) for Re{s) < 0 can be determined by the same argument as above. The theorem is proved. □ 4.6 Proof of Two Lemmas This section is devoted to the proof of Lemma 4.4.7 and Lemma 4.5.1, which are very important to our determination of the poles of the Eisenstein series. The technical part of the proofs of those Lemmas are essentially the estimates of the dimensions of certain spaces of intertwining operators between degenerate principal series represen tations of 5p(n), which follows from Bruhat theory on the estimate of the dimension of certain space of quasi-invariant distributions on Sp{n) for both archimedean and non-archimedean cases. 217 let Fv be the local field of the totally real number field F corresponding to the local place v. Denote (?„ := Sp{n,Fv). As usual, denote by (7~(G„) the space of all smooth functions over G„ with compact support. Note that the space C^{G,i) has a canonically defined topology, called Schwartz topology [73]. Let E be any vector space over the complex number field C. A distribution of with value E is by definition a continuous linear functional of C^{Gn) with value in E. The space of all £l-valued distributions is denoted by C{C^{Gn)\ E). Following Corollary 4.1.2, one has 5p(n,F„) = [p :u p n u u [p:_^w2 Pn = Oo u u 02. (4.91) Let Dg := O2, fii := 0\, and flo := They are open subset of G„ and O, is a closed boundary orbit of fl,- for z = 0,1,2. Let := P,"_j D The following are the spaces of certain P,"_j X P"-quasi-invariant distributions, which we are going to study. Let (cr, E^) be the representation of G„_i on the space P n -li^)- Then we define: for (pi,P 2) E P,"_i X P", U'\puP2) := |a(pi)ii(P2)r^o-(<7«-i(pj’)). (4.92) This can be viewed as a representation of P,"_i x P" on the space := C (x) Then we define some spaces of P<,.-valued, P ”_] x P"-quasi-invariant distributions: T” := {r6P(Cr(C?„);P.) : {puP2) 0 T = U{pr,p2) • T}. (4.93) T \ O i ) := {Per(Cr(D,);P.) : {p^,p 2) o T = U(puP 2)-T}. (4.94) By Bruhat theory as in Chapter 5 of [73] for archimedean cases and Chapter 2 of [66] 218 for nonaxchimedean cases, one has following inequality: dimT" < dimT"(Oo) + dimT’*(C?i) + dimT"(C>2). (4.95) Further, by the ‘Fundamental Estimate’ of Bruhat theory, one has following estimate: dimT"(Oi)< (4.96) m > 0 where z"(0,; rn) is the dimension of the space of all -intertwining maps from the representation ^w(p) := I 2 o-(<7„_i(u;,pu;i )) (447) to the representation A(")(p) := ( p ) < ^ P r )]"^<^(i)(P)Am (j)), (4.98) where S... is the usual modular character of the group • • • as in previous Chapters. More precisely, we have ( a * * \ B * a * 7 s(! * 5 ) and for p G P(J), ‘5(0) (p) = |a|2|detB|2or(p„_i(p)) ^(%)(P) = |a|'^|detS|tA,„(p); (4.99) / A O * * \ O b * * P'W = ‘A"’ 0 \ 0 6 -1 / 219 and for p € P("), |det ’)) ^(i)(p) = |ôp|det>l|?A„(p); (4.100) and ^ a 0 0 0 0 0 \ * B * 0 * * 0 0 a 0 * 0 % = a~^ * 0 0 ‘P - 1 0 I 7 0 * S 1 and for p G n—2 $(*(2)(P)2)(p) = = I |a|-o r ^ I detdet P1B\^o2 (T(ÿ»_(<7n- 1\[W2PU ( wgpwg ^ ) ) |a|“ detPl"^. (4.101) Note that in the case of nonarchimedean field, all of the distributions have transversal order zero. In other words, one have in this case: dimT"(0,) To compute the dimension ?"(C*,;m) for m > 0 and i = 0,1,2, one need certain information about the representation A,„ of P^"j, which is by definition the rn-th symmetric power of Ai, and Aj is the representation of P^”j induced from the adjoint representation on the quotient space sp (n ) (4.103) ’ PÎÎ-1 + ’ where g denotes the complexification of the real Lie algebra of the corresponding real group G. 220 Let M("j be the reductive part of the subgroup P^ô)- Then we have = GL{l)xGL{n-2)xSp{l), Ml\^ = GL{n-l)xGL{l), (4.104) = GL{1) X GL{n - 2) x 5p(l). By restricted to the representation V(,) for z = 0,1 can be described as follows: V(0) = C© C”-2 ®C2, V(1) = C " - \ (4.105) More precisely, for z = 0, we let (a, B, .s) be an element in and (w, z, x) a vector in C © C ”“^ © C^, then, as an M(J,j-module, the action of on C © is (a, B,s) o (w, z,x) = (a~^w, a~^zB~^,sxa~^). (4.106) For z = 1, we let (A, b) be an element in GL{n — 1) x GL{1) and y a vector in C ""\ then we have (A, b)oy = ^A~^yb ~^. (4.107) Then, as M(|j-modules, the m-th symmetric power of V(,) for z = 0,1 can be described as Am(V(0)) = @p+,+r=mAp(C) A,„(V(d ) = 6-’"A,„(C”-*’*). (4.108) It follows then that i"(C?o; m) 0 implies the following equation — ^ = — — 2p — q — r. But this is impossible. We thus deduce that i’^{Oo',m) = 0 for all rn > 0. 221 Similarly, ^ 0 implies that ^ = 1 — m, which has no solution for n > 4 and m > 0. But for n = 3, it has a solution m = 0. In other words, we have a« = 3 ;m = o : Finally, it is sure that m) = 0 for m > 1 and z”(C?2;0) = 1 since O-iis the open orbit. Therefore we obtain the following Lemma. Lemma 4.6.1 (a) For any place v of F and n > 4, the space o c J ( o ) ) ) is of one dimension. (b) For any place v, the suhspace Tq j o/T^ consisting of distributions in with support off the open orbit O 2 is of at most one dimension. Proof: Part (b) follows from the inequality (4.109). For part (a), we need to figure out the correspondence between intertwining operators and distributions. In fact we have following isomorphism as vector space: M Tm ' ' ' where Tm := (Afopr)(y)(e) for y 6 C^(G„), which is a distribution in ZI(C^(G»); E^.). Here pr is the canonical projection from C^{G,i) onto the induced module given by pr{(p){g) ^ipg)\a{p)\~^dp, (4.111) 222 with the right Haax measure dp on P"_i. The quasi-invariance for P"_, x P" of the distribution Tm can be checked as follows: for any {p\,P 2) € P,"_i x P" and y, e (Pi,P2) 0 Tm [ = V>{P\PP2^)Hp)\~^dp){e) ^n-l = Hp\)\~^ M{p 2 ^ 0 pr{if)){e) = l«bi)*i {P2)\~^ = l « b l )h (P 2) r ^ O - ( < / n - l {P2^ ))T m { By Chapter 5 of [73], both vector spaces have the same dimension, and because the map, M. M- Tm, is injective, we conclude that the map, M i-)- Tmi gives the isomorphism as required. This proves part (a). □ It is important to mention that, in the archimedean case, we have restricted ourselves to (gœ, 7i^t,oo)-niodules. Theoretically, Bruhat theory gives an estimate of the dimension of the intertwining operators from one smooth induced representation to another smooth induced representation of group G„. According to the works of Casselman [16] and Wallach [72], those smooth degenerate principal series represen tations of Gn are of moderate growth and become the canonical extension of the corresponding A,j,oo-finite degenerate principal series representations with respect to (gn,oo, A'n.oo), and also any (g„,oo,/Ci,oo)-intertwining map can be extended to be a 223 unique G„-intertwining map. Therefore part (a) of Lemma 4.6.1 really means, in the archimedean case, that there exists essentially one (gn.oo) ^»,oo )-intertwining operator from (g„,oo, A:„,oo)-module to (g„,oo,/C.,oo)-module Indp^{\U\~^ Corollary 4.6.2 For any place v and n > 4, theimage z*_, o in / " l 2,„(0) of the representation under the-intertwining operator o ?/"’*„(I ) is irreducible. Proof: Suppose the image o is not irreducible, i.e., it is a reducible submodule in Since /,"I2,«(0) is completely reducible as a G„_i- module, the image can be written as a direct sum of two submodules, say, Lq © VI. Then one can construct two G„_i-intertwining operators Ao and Ai with following properties: (a) 7m(Ao) = Vo and ker(Ao) contains Vi, and (b) 7m(Ai) = Vi and /;er(Ai) contains Vo- Composing the intertwining operator o with A q and Ai, respectively, we obtain two different intertwining maps Aq o o and Ai o o from 7’*_i,„(^) to l < i | ^ Cg) 7"Tj^„(0). By Frobenius Reciprocity Law [15], [16], and [13], those two intertwining maps give two G„-intertwining maps from 7"_, „(i) to Indpn{\t]\~^ Pnl 2,v{^))- According to part (a) of the previous Lemma, those two different intertwining maps must be proportional to each other. But this is impossible since they have unisomorphic image Vo, Vi, respectively (one is spherical, but the other 224 is not). Therefore the image o is irreducible. Because the map oZ7”^*„(|) takes the normalized spherical section to the normalized spherical section, the image „(!)) must be generated by the spherical section. This proves the Corollary. □ Proof of Lemma 4.5.1: According to Corollary 4.6.2, we obtain the following global consequence that for any even integer n > 4, the image z*_, o in /^T'KO) of the /"_](^) under the 6'p(n, A)-intertwining map o 17”;*(|) is an irreducible submodule, which is generated by the spherical section / ? In order to 2 ’ prove that the Eisenstein series E^Zl{g,0', f i 2) vanishes for such sections g in the image o i7^;*(|)(/,”_i (|)), it suffices to prove the vanishingness of E’,lzl ( (4.112) Note that the Eisenstein series EnZl{g,s] fs) is holomorphic at s = 0. According to (4.83), one has »-l / \ _ C(^ ~ ^ + 2)C(.S - + 1)((.S - ^ ) yy C(2.S - j + 1 ) ...I*; + + + + ((2,+;) ' At s = 0, we have c|[,“^(0) = (—1)”“^ = —1 since n is even. Thus we get following functional equation: ^::2(!7,0;/I,2) = (4.H3) Therefore we have that EHzl{g,0; /?.,) = 0. Lemma 4.5.1 is proved. 2 ’ Proof of Lemma 4.4.7: 225 We are going to prove part (b) of Lemma 4.4.7 first. We have constructed two intertwining operators M 2 = and / f^{wxniwonog)dnidno (4.114) C(-s + 2) -/f? where wq and no are as in Lemma 4.4.7. As in the case of arbitrary n, those two inter twining operators M i and M 2 correspond to two distributions Tmi and Tm2 hi the space T^, all x Pf-quasi-invariant distributions on G3. Because the intertwining operator U^^„{s){fg) has a simple pole at s = ^ for any general holomorphic section fs in the distribution Tm^ corresponding to the normalized intertwining op erator must have support ofiF the open orbit P2W2P1 . In other words, the distribution Tm ^ belongs to the subspace T gw hich is defined in part (b) of Lemma 4.6.1. Since the intertwining operator M i is given by a convergent integral and its integrating variables wiuiwono lives in the ‘middle’ orbit PfwiP^, the distribution Tmi will supported in the closed subset P^wiPf — G3 — Thus we also have that Tm\ belongs to Tgj. By means of part (b) of Lemma 4.6.1, the subspace Tq , is of at most one dimension. There exists a nonzero constant c, so that Tmi = 0 • Tm.2 • Equivalently, we have that M i{f){e) = c • Af2(/)(e) for / in Lg„(1). Then for any g e G3, we have Mi{f){g) = Mi{go /)(e) = c • Mzig o /)(e) = c ■ M 2{f){g). (4.115) In other words, M i{f) = c • M 2{f)- Because both M i and M 2 take the normalized siiherical section f° in fg „(1) to the normalized spherical section in Ind^J'^'’^'’\\ti Cg 7^(0)), the nonzero constant c must be 1, that is, M i = M 2- This completes the proof of part (b) of Lemma 4.4.7. 226 From the above proof, the distribution Tm^ corresponding to the intertwining operator M 2 = belongs to the subspace Tq^. By the similar argument used in the proof of Corollary 4.6.2, the property that the subspace Tg ^ is of one dimension implies the irreducibility of the image o Since 7^„(0) is a direct sum of two submodules, part (a) of Lemma 4.4.7 follows. Therefore Lemma 4.4.7 is proved. 227 4.7 Notations Used in Chapter IV p.218 C (x ), p. 184 Cl (a), C.,rc(a), , p. 186 = 4.W , p 186 5pn, J9.155 p.214 e'{ji), p.l88 Er{9,s',fs), p.156 ^n-i,Pi"(<7,s;/s), P-157 £<2, p 201 e(n), p.209 fs,r, fs,i, fs,2, p .l9 4 -p .l9 5 , p.208 A,1,2', /s,2,l, P 201 Gn,vi p. 162 ri(s), p.202 p.156 C -I, p.157 7^(—1, —s — 5 — I), p.l97 .5 — —1, s — p 197 î"(0 j,m ), p.218 J„, p. 165 K„, K„.u, p.l55 r(C r(G » );^ ), p.217 Ai, Am, p.219 M;*, Nl\ p.151 M s, Ms, P - 1 6 8 M x{s), M w,y {s), p.l76 A4u(s), Md{s), p.l96 Aiz(g), p.190 My,v{s), p.l91 MuA»), p.198 V - , p.173 p.220 n{xn-2,z), p.l65 n{xn-2,x,y,z), p.167 n{x,y,z), n(x„-2), P-167 m-(z»_3,z), p. 173 , IV(X,W,V), N(X), N(W,Y), p. 175 Oi, p.217 p;*, p. 150 AVi> P-164 Pn-1,1, Pl,Ti-l, p. 164 Pi%_2, P.167 P\,n~2i P\,n-2, p.l72 P.173 C-1,1, p.176 Pn-Yu P.177 P"_-,\ p.180 ^0’„(-Ru;), p. 184 f(?), p.217 $(,), p.218 pr, p.221 0L-2, p.168 p.176 {cr,E„), p.217 Tn, p. 182 T”, T"(0,), p.217 Pm , p.221 Tg,„ p.221 p. 164 P.167 p.175 P.157 % ( ^ ) , P'168 U^(s), U'^<*(s), p. 199 V(0, P.219 W], W2, p. 152 Wo, p.l65 üJj, p.l65 K. p.173 w*, p.179 w', p.l85 Wj, p.l96 Wu,d, p.l98 p.182 %3, p.l95 %+, p.209 C H A PT E R V Degenerate Principal Series Representations of Sp{n) over a p-adic Field From the previous Chapter, the normalized Eisenstein series s;/,) attached to holomorphic sections /, in the degenerate principal series representation of Sp{n,A) are entire except for , n + 2 n n — 2 - n — 2 n n + 2 . s e x „ = {— ——,...,0,. } and have possibly an at most double pole for each s € X„. The residue representa tions of the family of Eisenstein series at those values of s G especially at those positive values of .s G will play important roles in our studies of the degree 16 standard L-function of GSp{2) x GSp{2) and the theta correspondence between the reductive groups of the relevant dual reductive pair. As showed in [43] and [51], the local components of those residue representations should in general be certain special factors in the Bernstein-Zelevinsky composition series of the relevant principal series representation of the reductive group. In our special case, we have to study the irre ducible quotient representations of for each place v of the underlying totally real number field F. 228 229 In this chapter, we only concern the case where the place v is finite. In other words, the underlying field is nonarchimedean. Prom now on in this chapter we drop the V from the subscriptions of all our notations. We are going to recall some well known facts on the representation theory of a p-adic reductive group. As in [23] and [37], we shall determine the actions of some special Hecke operators on the subspace of vectors in which are fixed under the action of the relevant parahori subgroup. Combining those computations with the standard argument of .Jacquet modules, we will prove our main theorem of this chapter, which is Theorem 5.0.1 (Main) The degenerate ■principal series representation I^is) has a unique irreducible quotient representation of S'p(4) for these values of s : either Re[s) < 0 or F[s) ^ 0, where F{s) = (1 — -|- 5.1 Basic Facts In this first section, we are going to introduce necessary notations, recall and prove some basic facts on the representation theory of p-adic groups. For general reference, see [5], [13], and [14]. By abusing the notation, we denote by Sp{n) the F-rational points of the sym- plectic group Sp{n) of rank n. Let P := P,"_, = M,"_j AT”_, be the standard maximal parabolic subgroup of Sp{n) with its Levi factor M”_i isomorphic to GX(n—l)x5p(l). 230 that is, for m G , one has a \ a /? m = m(a; a, /3,7 , J) = with a G GZ,(n — 1). 7 ^ / The modular function will be Jp(m(a; • • •) = |det(a)|^. Let % be a character of M ,"_i such that %(m) = |det(a)|. Then the normalized induced representation is defined to be the space of C-valued functions / over Sp{n) satisfying f{m7ig) = and the action of 5p(n) is defined by the right translation. Let O the ring of the integers in F, and V the maximal ideal in O. Let K = Sp{n,0) and K{ = ker{pi), where pi the canonical reduction map (mod V*) of K. Then we have a filtration of compact open normal subgroups of K: {Lg,;} <1 • • • <1 K2 <1 7ii <1 Ko = K- (5.1) Let Ip = P\^{P{OfV)), which is called the parahori subgroup associated to the parabolic subgroup P. The Hecke algebra associated to the parahori subgroup Ip is defined to the space of all smooth functions over Sp{n) which are bi-invariant under the action of the parahori subgroup Ip. The multiplication on the space is defined to the usual convolution over Sp{n). With this structure of multiplication, the space forms an associated algebra, which is generally non-commutative. As usual, we denote the Hecke algebra associated to the parahori subgroup Ip by 'H{Sp{n) j j Ip). 231 Lemma 5.1.1 In the p-adic group Sp{n), one has following decomposition with re spect to the maximal parabolic subgroup P and/or the parahori subgroup Ip: (1) Sp{n) = (2) Sp{n) = \J^^^^^^UpPw(k,i)Ip, and (S) K = [Jw^^^^ççipIpW(k,i)Ip, where Up = {u^(fc,/) E W5p(n) : fc,Z = 1,2,• • • ,ra — 1, k < I < k -{• 1}, and the Weyl group elements can be chosen as follows: ( 0 —In -k -l \ h+i 0 (5 .2 ) In—k — l 0 \ 0 4+1 and ( 0 — In-k-2 \ 4 : 2 » « W(k,k+\) - w[k]k+l) — (5 .3 ) In-k-2 0 (fc+2) \ 0 “’(fc,fc+l) / For a given representation (tt, V) of G and a unipotent radical N, one can define the .Jacquet module with respect to N as follows. Let V(ZV) :=< {7r(n)u — v : v £ V ,n Ç. iV} >, which is a subspace of V stable under the right translation of the subgroup N and its normalizer M = N g {N). Then the .Jacquet module Vyv is defined to be Vn := V/V(N). (5.4) It is clear that the quotient space is a M-module. We are going to state basic properties of representations of p-adic groups, which will be used later. The proofs of those properties can be found in [5], [14], [13], and [23]. 232 Theorem 5.1.2 (Frobenius Reciprocity) With the notations above, the following isomorphism of vector spaces holds Homsp(n){In-\ (a), /"-I(--s)) = HomMn_^ (/"_i Theorem 5.1.3 ([14]) Let {7ri,V„,) (^2, Kt2) he representations of Sp{n). Then the spaces and of Ip-fixed vectors can be canonically defined as representa tions of the Hecke algebra 'H{Sp{n)l flp). IfV,^, is generated by V jf as Sp{n)-module and every non-zero subrepresentation of 7T2 has non-zero Ip-fixed vector, then the following isomorphism of vector spaces holds HomSp(^,[tti, Trf) = I ,7T2 ). Corollary 5.1.4 The following isomorphism of vector spaces holds for all s. Proof: What we have to check is that both and satisfy the cor responding condition in theorem 3. By lemma 2.23 in [14], we only need to show that every nonzero submodule of -s) has a nonzero /p-fixed vector. Let V be any nonzero submodule of ,s). Since s) can be embedded into 233 for some unramified characterXa of the standard Borel subgroup from [7], 0, where /s„ is the Iwahori subgroup associated to Bn- Then yÉ Osince K \ C IBn^ and V ^ ' is stable under the natural action of the maximal open compact subgroup Kq. By means of Frobenius reciprocity law, we obtain that (yAi)Po ^ 0 for Po = -P n Ko- It follows then that ^ 0 because of the Iwahori decomposition Ip = iVf MqNq = K\Pq- □ Lemma 5.1.5 Every non-zero subquotient o / / ’,*_j(.s) has non-zero Ip-fixed vectors. Proof: Similar to the proof of the corollary, any sub quotient of 7’*_i (.s) is a sub quotient of some Ind^^'*\xa) for some unramified characterx* of the standard Borel subgroup Bn- By Borel [7], ^ 0 and then ^ 0, which is stable under the action of Ko- Because of the complete reducibility of 7 Ô'-i (’*)|a'o — can be embedded into By Frobenius reciprocity law, ^ 0. This is equivalent to that ^ 0. □ Theorem 5.1.6 ([23]) The map V 1-4 gives an isomorphism between the lattice of all Sp{n)-submodules (or subquotients) and the lattice of all'H{Sp{n)l jlp)- submodules (or subquotients, resp) in Next we are going to prove that the degenerate principal series 7,"_^(.s) has a unique irreducible quotient representation with the assumption that the dimension of the space 77omsp(„)(7’*_,(s),7”_ i(-s)) is one. 234 First of all, we recall a theorem of Waldspurger. Let ( W, <, >) be a non-degenerate symplectic vector space over a p-adic field and 5p(W) the group of isometries on {W, <, >). Choose such an element S in GL{W) that for w, vJ 6 W, < Sw, 6w' > = < w \w > and for an irreducible representation tt of 5p(W), define the representation tt"* as follows: for g 6 S'p(W), ■ïï\g) := n{5g5~'^). (5.5) Theorem 5.1.7 (Waldspurger [52]) If w is an irreducible admissible representa tion of Sp{W), then the representation tt* and the contragredient representation are equivalent. Theorem 5.1.8 For such a value s that the dimension of Hornsp(n){I]l_]{s),,m_^{—s)) is one, the degenerate principal series /^ * _ i(so ) has a unique irreducible quotient, up to isomorphism. Proof: We choose S in GSp{n) as follows: J = ^ ^ in GSp(n). It is not difficult to check that 5 satisfies the condition of Waldspurger’ theorem and the condition: for g e Sp{W), SgS-'^ 6 5p(W ). Now suppose that I"_j(so) has two irreducible quotients, say, X and V. This means that there exist two 5p(n)-intertwining projections: Px : -^,"_i(-So)— % and py : /,7_i(-So)— i-Y. (5.6) 235 Since the contragredient representation of /,"_i(5o) is So), the contragredient representation X'^, can be embedded into /»_](—So), respectively, i.e., (5.7) Let (tt, V) be a representation of Sp{n). The representation (tt^, V^) twisted by 5 is defined as follows: the underlying space = V and the group action is twisted by conjugation of 5, that is, for u G F and g G -S'p(n), one has 'K^{g)v = 7r(SgS~^)v. We claim that (5-8) In fact, we can construct an 5p(n)-intertwining map Ts in the following way: : CiW —^ CiW / ^ Ts(f) and for g G *S'p(n), define Ts{f){g) = f{g^) where g^ = SgS~^. It is not difficult to see that Ts is an isomorphism of vector spaces and for p G P,"_j and x G 5p(n), Ts{f)(ps) = f(p‘g‘) = Hp‘)r ’^K s‘) = H p )r’*Ts(f)(s) where a{p) is the p-adic absolute value of the determinant of the GL{n — l)-part of the Levi factor of p and Tt{x)Ts{f){g) = f(g^x^) = 7r'*(;c)/(p'*) = Ts{Tr^{x)f){g). This proves our claim. It is also evident that there exist two twisted (by 5) projections from onto X^, , respectively. Therefore by the theorem of Waldspurger, we have C l (4 X ' ^ and -> -> y ‘ = y '' 236 The assumption that dim f/'om5p(„)(/^_j(s),7^_i(—s)) = 1 implies that the two irreducible quotients X and Y must be equivalent. This proves our theorem. □ 5.2 Computations of Hecke Operators The Hecke algebra H = 'H{Sp{n)lflp) with respect to the parahori subgroup Ip is defined in section 5.1. It is well known that the representation ps of Sp{n,Fv) on induces a natural representation p^ of H on the subspace of Ip- fixed vectors in In^i{s). The representation is given as follows: for (j) E and fen, = j^f(h) 7 = a fixed generator v of T. (5.10) \ In-l / Then we define an element in the Hecke algebra n A = t^y^'l -.chilpjlp) and W = .ch{lpwtlp)en, (5.11) p{lp-jlp) p{IpW]Ip) where /t is a Haar measure on Sp{n) and w\ = to„_ 2,n-2- We are going to describe the actions of those two Hecke operators on the space /^(.s)^'’. 5.2.1 Hecke Operator A According to the decomposition in Lemma 5.1.1, any function (/> in /"_] (.s)^'’ is deter mined by its values (j){wk,i). It suffices to compute ps{A){(j)){w(k,i)) in order to describe 237 the action of the Hecke operator A on By the definition, one has PA^)W('>^(k,i)) = J^A{x)(j>{w[k,i)x)dx = [ ^{W(k,i)h'i)dh. (5.12) Jlp Consider the Iwahori decomposition of Ip with respect to P: Ip = NqMoN^, where = jV n /f, Mo = M f] K, and iVf = N~ D K }. It is easy to check that for the chosen 7 , 7 ~^iVf 7 C /p and 7 commutes with the factor 5^(1, 0) of Mq. Therefore we deduces from (2.3) that Ps{A){ Let B be the Iwahori subgroup associated to the lower standard Borel subgroup of GL{n - 1). Then we have GL{n - 1 ,0 ) = UweWann-i)^'^^ = ^w&Wann-i)^owB where B = UqTo and Wci(n-\) is the Weyl group of GL{n — 1). Since W(k,i)UoW^h) C P n A" P C Nip{No), and 7 "^P 7 C Ip, (5.14) denote pw = ix'{UqwB) for the Haar measure y.' on GL{n — 1), we obtain that Ps{A){(}>){wf^k,l)) = Y 1 (j)(w(k,i)Uwhno'i)dnodmo = (j>{w^k,i)'wno'y)dno W&WaL(n-l) = (l>iw{k,i)no'yw)dnQ, (5.15) ^^Wann-i) where 7^, = wjw~^. Because we have two types of the representatives and <^(ik,fc+i) for the double cosets Pw^k,l)Ip, or for Pw(t,()P, we have to consider the last 238 integral separately. Choosing the standard root data for GL{n — 1), we can identify ^GL(n-\) with Sn-\i the symmetric group on n — 1 letters. Proposition 5.2.1 In the case of W(k,k)> A; = 0,1, • • • ,n — 1, we have [ ^ fJ'wW j 7T~^z, ir~^Ui, • • •, 7r“ ^u„_fc_2,7r“ '‘j/))dn + [ Y1 f^w]q~^' where s' = s + and we denote u„_2 = (ui, • • • ,u„_jt_2, 0, • • •,0) and M X Z Un-2 y \ In—2 'v,n~2 n{x, z, Ui,---,Un-k- 2,y) = 1 y 1 /n-2 \ 1 / Proof: When u;(l) > n — A' — 1, 7„, commutes with N°k,k)- The integral / ^{w[k,k)no'rw)dno = q~^^'^^U{w{k,k)) '/No since No = N^k,k)^ where = iVo D w^^f.^Pw^k,k) and C Ip. When u)(l) < u — A — 1, we can choose w' E WaL(n-k-\) C Wox(n-i) so that w ^Jww' = 7 , w'~^Nqw' = Nq, and w' commutes with W(^k,k)- The integral we concerned will be reduced in the following way: / d>{w(k,k)no'1xo)dno = / d>i'W{k,k)w'w ^uqw'w' ^'^^w')duo J No J No = / (j){w^k,k)noj)dno. J No 239 Since the subgroup No D M" commutes with 7 , let N q = D iV", the last integral will be deduced as follows: / é{w{k,k)no'))dno JNq = , 7T ’u„_fc_2, 7T y))dn, where n'o = n(x, z,uj,---, u „ -k- 2 ,y) 6 Nq = Noll N\\ in other words, if set u „_2 (u i, • • •, U n -k - 2 , 0, , 0), we have / I X z Un-2 y \ /fi —2 —2 1 y n{x,z,uu---,u„.k- 2,y) = 1 ln-2 1 / This proves the proposition. □ Proposition 5.2.2 In the case ofw^k,k+i)> k = 0,1, - • ■ ,n — 2, we have PÀ^)i [ E ) P-w]q^' ci(u;(fc,fc+i)ni(7r"^a;,7r"'^z,7r“’ui,---,7r"’u„_fc_3,7r"^u))dni w [i) where s' = s + and we let u'„_ks = Un-k-a), K -k -2 = ( % ! ," ',u„-k-2), / 1 X z u:n—k —3 tn -k -3 *“n-fc-3 V Ik 0 1 0 1 In-k- h —ar 1 and ^ In -k-2 u0 “n-fc-2 1 a: <_fc_2 0 h 1 In—k—2 1 Ik \ - X 1 J Proof: When w{l) > n — k — 1, 'jw commutes with N^k,k+i) then the integral / (l>{‘f^(k,k+^)^o7w)dno = (|>{w^kM\)lw) J No When za(l) = n — A: — 1, 7 ,u commutes with the subgroup -^(i+i.fc+i)- Note that Wo = X n X n Let W" = •N(fc,fc+i) n Then the integral / = , çi(u;(fc,fc+])n2(7r"’a;,7r"’u i,---,7 r"’u„_i_2))f/u2. 241 Here if we denote = (ui, • • •, Un-k-2), any n" = ri2(x, ui, • • •, n„_fc_2) E N" = iVo n can be written as ^ ^n—k—2 0 K -k-2 1 X <_fc_2 0 h Jl2 {x,Ui, Un-k- 2 ) = In—k—2 Ik —X 1 Finally, when r«(l) < n — k — 2, we are able to choose an element w' £ WcL(n-k-2) C Hc,x(n-i) so that w'~^‘jww' = 7 , w ~^N qw' = iV o, and lo' commutes with w^k,k+^)- The integral considered is equal to / Hw{k,k+^)^olw)dno = / ^'l/wwyino J No J No J Nf\ Since the subgroup A/oflM” commutes with 7 , let iVp = the last integral will be deduced as follows: / H'^{k,k+^)^ol)dno J No = ^ (;6(u;(fc,fc+i)ni(7r"^c,7r"^2,7r"^ui, - •• ,7r"’u„_fc_3,7r"’u))dni, 242 where né = ni(x, z, Ui, • • •, Un-k-3, v) £ NÔ = N°^,k+ï) ^ can be written as, if we set <_fc_3 = («!,•• • ,n„_fc_3), / 1 In—k —3 ^ 1 V Ik 0 Ul{x,Z,Ui, - ■ ■ ,Un-k-3,v) = 1 0 1 In —k~3 Ik 1 This finishes the proof. □ Lemma 5.2.3 With the notations above and s' = 5 + ^ ^ , we have, ford> £ (a) If z £ O'" and x, y, U], • • •, Un-k-i 6 O, and D < k < n — 2, 4{w(k,k)n{7r~'^x, 7t"^z, , • • •, 7r"^u„_fc_2, 7t“V)) = ?"^^V('t«(it+i,fc+i)); (5.16) (b) If z £ O" and x, u, wi, • • •, Un-k-3 E O, and 0 < k < n — 3, (5.17) Proof: Note n(.c, z, Ui, • • •, u„_fc_ 2, y) is defined in Proposition 5.2.1. Since the 5p(l)- part of Mg = GL{3) x 5p(l) is included in wf!^^^^Pw(k,k)^Ip and acts by conjugation on n (æ,Z, Ui ,- • • ,U„_fc_2,y), 243 that is, 5p(l) transforms x and y, and keeps the other variables fixed, we obtain that • • ■, 7r“^u„_A,-2,7r"^y)) = ç^(tü(fc,fc)n(0,7T~^2,7r“‘u i,- - •,7r"^ti„_fc_2,7r“ ‘y))- (5.18) We choose, for 0 < A; < n — 2, an element p{un-k- 2,y) in W^k,k)^'^(k,k) H Ip, which is of form: (un-k- 2,y) X-e, (~ + » n (~T)• Then the conjugation of n(0,7r“^2, 7t“^Ui, • • • ,7r“^u„_fc_2,7r“^j/) by p(u„_t_ 2,y) has form: / I 0 0 \ In-2 0 -'iU n -k-2 -2 X2e.(7T ^Z) 1 0 fUn-k-2 -y 1 In-2 1 / (2.6) is then ecjual to ^(u;(&,k)n(0, tt 0, • • •, 0,0)) Since the right factor of this prod uct belongs to Ip. Applying the Bruhat decomposition of following type: (A ;) = ( f ;!)(-, i)(i f> we obtain that ^(u)(t,t)n(0,7r"'^z, 0, - - - , 0,0)) = ç”^^*+^^<^(u;(/t+i,A:+i)). This proves (a). For (b), we let, for 0 < fc < n — 3, _ / 7TU] 7TUn-k-3s / 7TU TT.f X —1| +«2 ( ^ ) X—ei+en-*-2( ^ ) " X—ei+en-fc-l ( ^ I ’ X —ei-£ n ( ^ )• It is easy to check that pi is in w'^|^|^_^_^<^Pw^k,k+'^) ^ nnd the conjugation of the element n(7r“hr,7r“^z,7T“’u i,• • • ,7r“’'u„_jt_3,7r“'u) by pi is a product of 244 and / 1 0 0 0 0 0 \ —k—3 0 4 —fc—3 4 \ 4- 0 1 / Because the last factor is in Ip, we know that • • • ,’’■ ^Un-fc-3,7T ’u)) <)^(W(t,k+1)M(0, 7T“^3r, 0, • • •, 0, 0)) g-2('+4^)^(u,(t+l^+2)) (5.20) The last equality results from the same Iwasawa decomposition as in the proof of (a). Thus the lemma is proved. □ L em m a 5.2.4 (1) Assume that z E O. When 0 < k < n — 2, we have (a) ÿ(u,,„,„(0.x-'.,0.--,0,0)) = { t l l o \ (b) (j){w(k,k)n{0, 7T-’z, 0, • • •, 0, Tf-^)) = î"(*'''^V(u^{fc,fc+i)). (2) When 0 < k < 7i — 3, the following identitij holds 245 Proof: Statement (l)(a) follows directly from the Bruhat decomposition similar to (1,2). For statement (l)(b), we have ifzeV] 1)) if z E Note that n>(jfc,fc)n(0,TT ^z,0, • • • ,0,7T ^) = X-e,+£„(-7r ^)x-2ei(7r ^z)w(k,k)- If ^ E "P, we consider the following Bruhat decomposition / —n 1 In-2 X-e,+e„(-?r ) = — 7T In-2 1 TT / Thus we obtain that ^(w(t,t)u(0,0,0,0, - - , 0,7r ’)) = q because W(n-2,n-\)W(k,k) = U;(Jt,fc+l)- If z E we have to consider the following Bruhat decomposition: X-e,+s„(-7T )X-2£,(7T z) v-1 1 -TT" WiX2£,(-7rz“’)Xe,+ï„(z ’). In-2 \ By the /^-invariance of we obtain that <^(W(t,t)»(0, TT ’z,0,0,---,0,7T 0) = Ç ',^(x_.,+..(-Z ^)îü(„-2,»-2)^«(fc,Jt))- Applying the Bruhat decomposition to the variable z , we can conclude easily that Now we are going to prove part (2). First of all, it is easy to find the following Bruhat decomposition in Sp{2m): Let / Im \ X Im G Sp(2m). Z 0 Im - ‘X \ 0 0 Im / Then Z is symmetric and this element of Sp{2m) can be written as / Im \ X Im (5.22) Z 0 Im V 0 0 Im / / Im Z \ ( Im z - ‘ Im X XZ-^^X -XZ-^ w(m) Im Im 'm / \ Im / 0 —/m \ ( 0 I . Because u;(fc,fc)n(0,7T“^z,7r~^,0, ••• ,0,7r“^y) = X-e,+e„(-7r“'y)x_2e,(7r~^2)X-s,-£2(-7r"*)u;(fc,fc), it is not difficult to see by means of the Bruhat decomposition above that Thus if y 6 T’, we have = (5.23) If y E we can assume that y = 1 and <^(w(t,t)M(0,7T-^z, 7T-\0 ,...,0 ,7T-^y)) = cj){w(k+i,k+2))- (5.24) This proves the lemma. □ 247 L em m a 5.2.5 (1) Assume 0 < fc < n — 3 and s' = s + (a) If z e O, then <^(tü(fc,fc+i)ni(7r " \ 7r"‘0,O, • ■ • ,0,0)) = (b) IfvE O, then (c) Ifz € then <;i(u;(fc,jk+i)«i(0 , 7r“’^ ,0 , • • • , 0 ,tt-’u)) = g-''(; 6(w(t+i,t+2)). (2) When Q < k Proof: (l)(b) follows in the same way as (l)(a) in Lemma 5.2.4. For (l)(a), we have W(k,k+l)%(7T"\7r-^Z, 0, • • • , 0,0) = X-2e, (-% " ^ z)x_g, (tt'^)u;(fc,fc+i). Applying the Bruhat decomposition (2,8) to the case JC = 0, we obtain that (7T-^ , 7T-^z, 0, - " , 0,0)) = ç"^(®+^V(îü(fc+i,fc+2))- (5.25) By the argument similar to that of (l)(b). Lemma 5.2.4, it is not difficult to find that <;6(W(t,t+1)TZ] (0 , 7T-^Z, 0 , - " , 0, TT'^U)) = 9" *V (w ^ (n -2,n -2 )X e,+ £ „(-2 " ’ î')î"(ik,fc+l))- By the similar argument, we conclude that <;6(W(k,k+1)Ml(0, 0, - " , 0, 7T-^u)) = g-('+4^)<^(u,(t+l^+2)) as long as u E O. For part (2), it is easily obtained, by the argument similar to that of part (2), Lemma 5.2.4, that = g-"('+'^)^(«;(t+2jt+3)). (5.26) 248 Therefore Lemma 5.2.5 is proved. □ Lemma 5.2.6 Let 3' = 5 + (a) WhenQ (b) When 0 < A; < n - 3, TT"', 0, - " , 0,0)) = q~^'^{w(k+\,k+2)) Proof: Since the proof is similar to the previous ones, we just sketch it for complete ness. It is easy to see that u;(fc,fc+i)7i2(’r"S0, • • • ,0,0) = }u;(fc,fc+i). Thus we have that çi(u;(fc,*+i)n2(7r“\ 0 , • • • ,0,0)) = g-('+^)^(w(t+i,k+i))- On the other hand, W(k,k+i)M2(0,7r-\0," ,0,0) = x-e,+e„(-7r"^)iO(fc,fc+i). Similarly, we get <6M t,t+])»2(0, 7T"\ 0, • • • , 0, 0)) = V(l«(n-2,,.-2)U^(A:,fc-t-l)) = g-(*+^V(u;(fc+i.fc+2)). ( 5 .2 7 ) We are done. □ Now we are going to use these lemmas to compute the integrals. Again, we will compute the integrals involving w^k,k) or «^(fc,fc+i) separately. Proposition 5.2.7 For 0 < k < n — 2, the integral is Jon-k+i ^i^{k,k)n{Tr~'^x,Tr~'^z,Tr~'^ui,--- ,TT~hin.k-2,'^~^y))dn + (1 - + g-('+4^)-("-*+:)]^(u;(&+i,t+i)) 249 + (5-28) + ?■"(! - ■where •we assume that = 0 if i or j > n — 1 end n(- • •) is defined as in Proposition 5.2.1. Proof: First of all, we decompose our integral into two integrals in the following way: Jon-k+x ^i'^ik,k)n{Tr~'^X, 7T"^2, 7T"^Ui, • • •, ■K~'^u„-k-2,T^~^y))dn = / / çi(n;{fc,fc)n(7r"'z,7r"^2,7r“'Mi,---,7r"^n„_fc_2,7r"’î/))f^”' VzgOX Jon-k + 5 “ ’ j^n-k+i d>{w{k,k)n{ir~^x, ■K~h, 7r"*ui, • • •, 7r“ ’u„_;t_2,7r“ ’j/))dn = fi + l2‘ (5.29) By Lemma5.2.3, 7i = For the integral/ 2, the domain of the integration may be cut into four subdomains: (1) ;c,j/;ni,---,n„_fc_2 E V; (2) .T OT y e and ni, • • •,Un-k-2 € V] (3) X, y e V and one of u i, • • •, u„-k-2 in ; (4) X OT y E and one of Ui, • • •, Un-k-2 in . Note that (3) or (4) occurs only when k < n — 2. The integral I2 can be deduced as follows: I2 = /(i) + f(2) + 1(3) + 1(4) where ^ ^(u;(k,k)n(0,7T-^z, 0, ' " , 0,0))dz 250 f = g / (j){w{k,k)n{'Jr ^x,ir *2,0, •••,7r ^y))dn (5.30) J('Z) J{2) I = q~^ I <^(iü(fc,fc)n(0,7r"*2,7r"*ui,---,7T“'‘u„_A:_2,0))(in J{3) J{3) I = I 9^{i«(fc,fc)n(7r"*x,7r"*2,7r"*ui,---,7r"*u„_fc_2,7r“*i/})c?n. J(4) J{4) Since the function $(g) := 4>{w(k,k)g) is invariant under the coadjoint action of the subgroup integrals are eventually reduced to the forms: - Ç-2) / = ?-^(l-ç-("-''-‘'^)^<^(u;(fc,fcp(0,7r-*2,7r-*,0,"-,0,0))d2 (5.31) = ?■'(! - g-")(l - ^ Thus the integral Lz equals to h = ç“* / + g-("-*:-:)(l _ Ç-2) + ?"^(1 - ^ fi(W(t,t)M(0, 7T-*2, 7T-* , 0, • • • , 0, 0))dz + g -\l - g-("-':-^))(l - q-'^) jT ^(u,(t,t)n(0, ^"* 2 , tt"* , 0, • • • , 0, tt"’ ))r/ 2 . According to Lemma 5.2.4, the above will = g-("-*+:)[g-i^(«,(t,&)) + (1 - g-*)g-('+4^) + g -("-t-i)(i _ g-^)g-('+4^),^(u;(&,&+i)) + 5"’(1 - g-("-"-''^)(l - Ç"'')Ç"''‘*+'^V(«^(fc+l,fc+2))- (5.33) 251 The last two terms will not occur when k = n — 2. This proves the Proposition. □ Proposition 5.2.8 For 0 < k < n — 3, the integral is TT~'^z, ir~'^ui, • • •, 7r"*n„_fc_3, TT~^v))dn-i = + (1 - + g-('+4^)-("-*-l))^(«;(t+l.&+2)) + (1 - (5.34) where we assume that = 0 if i or j > n — 1 and ni(---) is defined as in Proposition 5.2.2. Proof: We will go through the same procedure as the proof of the last proposition. / (j>{wik,k+i)ni{T^~^x, TT'^z, 7T"’ui, ■ • •, 7T"^n„_fc_3,7r“'‘u)dni J0n-k = / (j){wik,k+\)ni{Tr~^x,Tr~'^z,Tr~\i,-■ ■ ,Tr~^u„.k-3,T^~^v)dn^ jQn-k-\ J.çOX + ^ 7T~'^z, 7r"^wi, -, r"^u„_*_3, n~'^v)dnx = 7] + h . (5.35) By Lemma 5.2.3, 7i = _ ^-i),ji(t«(;j.^.,_j.+2)).The integral 7^ can be com puted as follows: The domain of the integration may be divided into three subdo mains: (1) X e and z , U n -k -3 , V E O] (2) Æ,ui, • • •, Un-k-3 E V and z ,v E O, and (3) X E V, z.,v E O, and one of ui, • • •, Un-k-3 is in O'^ 252 Since the function ^(ff) := ^(w(k,k+i)ff) is invariant under the coadjoint action of the subgroup Hip, we can deduce as in the case of W(/.,k) that /2 = g-i(l-ç-i)/ $i(u;(fc,fc+i)ni(7r"’,7r"^2:,7r"’w i,---,7r"’u„_fc_3,7r"’i;))dni JC)n—k—l + ^(^(k,k+i)^j(0, 7T"’ui, • • • , 7T“^U„_fc_3, 7T~'^v))dnj = q~^(l-q~^) J^^(w(k,k+i)ni(7r~\7T~'^z,0,---,0,0))dz (5.36) + <^(w(k,&+i)Mi(0, 7t“^z, 0, • • •, 0, TT~^v))dzdv + 9~^(1 - 7T“\ 0, • • • , 0, ■K~^v))dzdv. Notice that the reduction of the integration over subdomain (1) is made by the ac tion of the element which is in [^(fc!fc+i)-^^(fc.fc+i)] ^ We also have following identity 7T“’z, 7T”\ 0, • • • , 0, 7T"^u)) = TT~^Z, 7r“\ 0, • • •, 0, 0)) by means of the conjugation of X-e 2+e„-*-i which is in fl Ip. Then applying Lemma 5.2.5, we obtain that (2.23) = g-Z('+4^)-l(l - q-^) + g-("-':+1)^(u,(t,&+1)) + g-('+=^)-("-t)(l _ g-1)^(u,(t+1,&+2)) + g-(»+^)-(«-<=-l)(l - g-l)^(u,(t+U+2)) + (1 - 9-<”-''-">)(Z-"(*+^)-V(«^(fc+2,fc+3)). (5.37) The last term will not occur if k = n — 3. The proposition is proved. □ 253 Proposition 5.2.9 When 0 < k < n — 2, the integral J(^n—fe—1 + (1 _ (5.38) where we assume that <^(w(,j)) = 0 if i or j > n — 1 and ri2 (‘• •) is defined as in Proposition 5.2.2. Proof: The domain of the integration may be divided into three subdomains: (1) .•c,ui,---,u„_fc_2 G V; (2) X E O^-, and (3) X EV and one of , • • •, Un-k-2 G . So aijplying Lemma 5.2.6, we deduce that the integral will be . • • ,7r"’u„_fc_2))dn2 = + (1 - ?-')<^(«;(fc,fc+i)n2(7r-\0, . . .,0,0)) + (1 - g-("-':-2))g-l^(m(j,,t+l)»2(0, 7T-’, 0, ■ • • , 0, 0)) = ç-("-*-')4)(u,(&,t+i)) + (1 - ç-')ç-("+^V(î«(fc+iMi)) + (1 _ Q-("-'=-2))Q-(*+^)-V(u;(fc+,,fc+2)). (5.39) Again, the integration over the second subdomain is reduced by the conjugation of the element %„+,.(«:)...... X.._*_:+,n(un-t- 2), which is in [w^k]k+i)Pw(k,k+\)] H Ip. When k = n — 2, the last term will not occur. This proves the proposition. □ 254 By means of Proposition 5.2.1, 5.2.2, 5.2.7, 5.2.8, and 5.2.9, and the fact from [23] that E = „ » - i _ 1 and E flu, = 1 . • (5.40) tu(l) Theorem 5.2.10 With the notations above, the action of the Hecke operator A can be described as follows: Ps{A){(l>){w^k,k)) (^n-fc-l _ _ ç-(n-fc-2))ç-(s+^)-2 - 1 ) , - 1)(1 - g~^)(l - g-("-t-2))g-(.+4^) _ l)o«+4^-(«-*+l) / n _ ,,-fcN _(s+2±2.) + ------^ ( ^ n r r n j ------and Ps{A){(l)){w(k,k+i)) (^n-fc-2 _ _ g-(»-t-3)^g-(«+=±^)-l <^(î"{fc+2,fc+3)} - 1) (ç«-fc-2 _ i)(g _ l)[ç-("-^)(ç + 1) + g-('+4^)(2 + q-^ )] + ------ç(ç» -i'Z f)------Hw^k+u+2)) -1) 4w(t+i,t+i)) (çn-fc-2 _ l)g«+=f-("-'=) + (ç _ 1) + (ç« _ + _ 1) &+])), 255 where we assume that (j>{w({j)) = 0 ifi or j > n — 1. 5.2.2 Hecke Operator W By definition, the Hecke operator W is in form: W = ch{IpW \Ip) where = ïü(r,-2,n-2) is &s in Lemma 5.2.3. The action of W on the space I^isY ’’ of Jp-fixed vectors in /|(s) can be described in the same way as we did for the Hecke operator A. However, the computation here will be simpler. For any ^ € I^{s)^’’, by definition, we have Ps{y^)W{w[k,i)) = / ^{w^k,l)bwi)db = / (f>{w(k,i)monoWi)dmodnQ (5.41) Jlp JMoNo = Pw I f li-i 0 ^ 0 -1 tn-i 0 Wi = w f = 0 /.-I 1 0 \ 0 In-i J Thus we have »i-i» i- i . PÀ^)Wiw(k,i)) = Y if Y Pw] / Hw^k,i)noWi)dno. (5.43) .= 1 fNo We will compute the last integral for the case I = k and the case I = A;+l, respectively. The idea we are going to use to process the computation will be the same as that used in the computation of A. 256 In the case / = A;, & = 0,1, • • •, n — 1, we have 7J-1 . = . 71-1 = [ YL / ^{w(k,k)noW\)dno + X) t /^«,]9^(w’(fc,fc)W.). ti;(l)<7i—fc—1 7=71—k 777(1)=: We can easily obtain that / As before, we have ^ q n - k - i _i ^ 5”-’ - X / _ 1 X , - „-i _ 1 • (o.46) uj(l) Therefore we obtain the formula for W when I = k: Ps{y^){(f>)iw(k,k)) (5.47) In the case / = A; + 1, A; = 0,1, • • •, n — 2, we have similarly /)XM/)(<^)(w(&.k+i)) 257 = ^(w(k,t+])»o«'i)dno (5.48) i=l u,(l)=i '^^0 n—k—2 p p = ^{w^k,k+i)'>^oWi)dno + 4" ^ y l^’W j ^(lll(A:,fc+l)^Oll’n—fc—1 4” [ ^ l,fc)}) iu(l)=n—fc—1 n—fc / ^{u;(fc,fc+i)no'ioi)dno = q V(t"(fc+i,fc+2)) 4- (1 - g ^)<^(w(t,t+])), (5.49) JNo/No Hw{k,k+i)‘^oWn-k-i)dno = g~V(t"{/t,fc+i)) 4-(1 - g"^) Therefore we obtain the formula for W when / = A; 4-1 Theorem 5.2.11 Let W 6e t/ie Hecke operator in 'H{G//Ip) as defined in (5.11). Then the action o fW on the space / ”_t(-s)^'’ is given by Ps{yV)i (5.52) 258 for A: = 0,1, • • •, n — 1 and _ l)o“^ ps{'W){(l>){W(k,k+l)) = ---- ^ , . - 1 ---_ l-----<^(W(fc+l,fc+2)) «n—t—31 \2 + _ 1 (5-53) («n-l _ \ + — „„-î"Z 1— /or fc = 0,1, • • • ,n — 2. 5.3 Proof of the Main Theorem In this section, we shall apply the general results obtained in previous sections to the special case n = 4. In terms of the standard argument of Jacquet modules, we are able to prove that dimHom(/ 3(.s), 7g(—a)) is always one for all real values .s. 5.3.1 The Dimension of Homsp{4 ){Iz{s),li{—s)): In this subsection, we shall estimate the dimension of the space Ho7nsp{4){l3{s),I^{—.‘i)) by the standard argument of Jacquet modules. 259 Proposition 5.3.1 For any complex number s except s = 0, |,1 , and 2 (mod the dimension of the space Homsp( 4){l3{s), I^{—s)) is one. Proof: Applying the Frobenius reciprocity law, we obtain with x(m) = \det a(m)|. The Jecquet module has a filtration of M^-modules: n{^)N ‘ = -f«f 3 3 3 l{!f D 3 /{?;" 3 = 0 where is the image of the subspace /('J) under the Jacquet functor with respect to IV3 and /(*’•') := { / 6 Isi-s) : supp(/) C Note that C if Â: < i and I < j. The corresponding subquotients are ^j(3,3)yj(2,3)^^^ ^ ^.+3 = J»d%',(")(|de(A|'+&|c|-'+^)(81sp(,) (/(I-■')//(’-'))yv4 ^ ;»dg:'M(|o|'+2|6|2|c|-'+2) 0 (7(°'0));V4 = Considering the exponents of the Jacquet modules with respect to the center of M^, we conclude that 2 1 27T7 dim 7Tom 5p(4)(/^(.s),7^ (-s)) = 1 if .s ^ 0, - , - , 1,2 (mod j ^ ) ; 260 and 1 < dim Homsp{4){l3{s),I^{-s)) < ( dimiîom(x® © X^) if s = 0 (mod I dimHom(Indp^j^\\deiA\^\c\'^) (g) /n d |'’^^^(|o;[j),x^) + 1 if a = § (mod dimHom{Indp^l^\\detA\^\c\^),X^) + 1 if A = ^ (mod j^); dimHom{Indp^^^\\a\'^\detA\ 2) (g) 7nd|''^^^(|o:P),xO + 1 if s = 2 (mod j^ ); and when 5 = 1 (mod j ^ ) , we have 1 < dim7Tom5p(4)(/^(5), J^(-.s)) < dim7fom(7ndg^^^^(|ap|6p|cp) We shall determine the dimension of these ‘Hom’ spaces by using the results of Bernstein and Zelevinski [5]. Since /ndg’^^^(|o;|^) is irreducible, it is evident that eg) Ind‘^^^\\a\^) has no one dimensional quotient. Thus when 5 = 1 (mod j^ ) , dimciïom5p(4)(7|(-s),73(--5*)) = 1- When 5 = 1 (mod rewriting these induced modules of GL{3) or 5p(l) in terms of notations of Bernstein and Zelevinski, we obtain that In4^j^\\detAW \i)^In4^^\< [I],[| >,[|). Since the induced data is linked, the induced GIr(3)-module Indp^^^\< [^], [|] >, [|]) is reducible and has a unique one dimensional quotient x^- When 5 = 2 (mod g^), (|ap|deiA|^) 0 7»d§'^')(|ap) ^ ^([2], < [1,0] >) © 7n4"^')([l]). Hence the GL(3) x 5p(l)-module 7»dp^^\[2], < [1,0] >) ® 7nd|''^'^([l]) is reducible and has a unique one dimensional quotient x^ (8) 1 .5p(i)- 261 When .s = 1 (mod j^ ) , we have similarly that ® Jn4 ''<”(|a|') ^ /n4 '’‘” ([31, ].HI)[2 « / » 4 ''">{[1]) “ /nd“';<">((31,<[2,lj>). By the same reason, both of these two GL{3) x 5p(l)- modules are reducible and have a unique one dimensional quotient 0 l 5p(i). Therefore the dimension of the space dime Homsp{ 4)(l3{s),I^{—s)) is one for any complex number .s except s = 0, 1, or 2 (mod 1% ) . □ From the proof of the above proposition, we obtain Corollary 5.3.2 We have following estimates — n 1 — 2 (777'-'' 27T1 1 log 7^ 5.3.2 The Proof of the Main Theorem: For n = 4, we modify the Hecke operator as follows: + Q + l)-4- (5.54) By Theorem 5.2.10, we have Corollary 5.3.3 For (j) G I^(s)^'’, the action of the Hecke operator A! is /,,(^')(^)(«)(3,3)) = ( r ' + g-"-' + g-'-^) Ps{A'){ Ps{A'){(I>){w^2 ,2)) = (g - i)(g"*"^ +g"^)<^(w(3,3)) + (l - q~'^) + (g* + g”*"’ + g~’*)^^(^( 2,2))) /,.(^3 (,^)(u,(i,2)) = (g - l)(g'" + g-" + 2g-'-" + g-'-“)<^(«^(2,3)) + (g ~ i)g * ^<^(î^( 2,2)) + (g* +1 + g *)<^(^(i, 2)), = (g" - l)g-'-®<^(u;(3,3)) + (1 - q~'^?q~^~^4{w(2,3)) + (g^ — + g'"*)<^(^(2,2)) + (g^ - i)(g~^ + g~^)ç^(^{i,2)) + (g' + g*"’ + g’"*) /3s(^0('/')(^(o,i)) = (g" - i)g'""'^ + (g' - l)(g“" + g"“ + 2g— " + g— + (g — i)g~*~V(t"(i,i)) + (g* + g* ’ + /,.(y4')(,^)(u;(o.o)) = (g" - l)(g—" + g-'-l^(«,(2,2)) + (g" - l)(i - g-")(g-'-" + g— + (g"-i)(g—" + g“"M«^(i.i)) + (g^ — i)(g"^ + g~"*)^(w^(o,i)) + (g' + g® * + g® ")^(îx^{o,o))- It is easy to check that I^{sY^ is of 7-dimension by Lemma 5.1.1. A natural basis ill I^isY^ will be described below. Let S{Sp{A, F)) be the space of Schwartz-Bruhat functions on 5p(4, F) and chipu^f^ i^ip the characteristic function of Ipw^k,i)Ip- Then the functions belong to and enjoys the following properties: (5,6) Those functions are linearly independent and for any (j> E M = I ] (f>{'W{k,i)) that is, {^{k,i) : fc, / = 0,1,2,3, k < I < k + 1} is a basis in For convenience, we set Hecke operator can be described as follows: Ps{Â!){(j>^) = ç-*(l + ?■’ + + (? - + (? - + { Pa{-^){(j>2) = + (1 - + (g' - ps{^!){h) = (g* + g"*"'+ + (g - + (g'- i)(g"*"^ + g"‘')<î^5 + {(Y - l)(g-'-= + g— 1^7 ps{.A!){^Y) = (g* + 1 + g + g ^(g^ — l)(g + 1)<^5 + g-'(g' - i)(g + 1 + 2g-+' + q~^)ct>e + (g= - 1)(1 - g-")(g-'-" + g-*"®) Ps{-^){ Ps{'A'){^7) = (g* + g* ’ + q^~^)(j>7. 264 Similarly, for n = 4, we modify the Hecke operator VV as follows W := - 1)W. (5.58) By Theorem 5.2.11, we obtain Corollary 5.3.4 Let W he the Hecke operator defined as above. The action of W on I^{s)^^ can he described as follows: For any (j) € P À ^ ') W i‘W(3,3)) = - 1)<6(W(2,2)) /’«(W')(«î')(lü(2,3)) = (î - l)ç"V(«^(2,3)} + (ç - 1)(1- q~^) + (?■’ - Ps{yV'){ + (g — i)(i — ?~'}^(^(2,2)) + {q^ — q){w(i,\)) ^»(W')(Ç^)(W(1,2)} = (g-l)î"'V(u^(2,3)) + {?-l)(2-ç"^) + (ç - l)'V(n;(i,i)) + (ç - l)î'V(u^(0,i)) />s(W')( + {q^ — i)(i — î ”' + q^ — q^H'^(o,o)) p3{yv'){ P«(W')((^)(îü(0,0)) = (1 - ?~^)<^(î"(l,l)) + (1 — ~ 1)<^(^(0,1)) + — i)(i — g ^)H'^{o,o))- 265 Under the natural basis {<^1, (^2, <^3, (^4, <^5, (f>e^ <^7}, the action of the Hecke operator W' can be determined as follows: Ps{W){2) = (l-Ç-')<^2 + (l-< Z "')(l-î“")<^3 + (g-l)9-V 4 Pa(W')(^3) = {q^ — l)(f>i + q { l — q + g (l — 9 *)'^<^3 + (?^ — l) ç ^^5 Ps(W')(<^4) = (î^ — g)^2 + (5 “ 1)(2 — î~^)<^4 + — l)(ç ' — Q ^)<^5 + (1 — Ç "^^6 Ps(VV’')(<^5) = (î^ — ?)^3 + ( Ps{yV'){(l>6) = (? — 1)Ç'^ÇÎ'4 + [(î^ - + (5^ “ ?)]^6 + ( Ps(W')(<^7) = (?^ — 5'^)<^5 + î(Ç — + (3^ — 1)(1 — <7 ’)<^7- Now we choose a new basis • • • , ^ 7} in the space /K-s)^^, which is 7 ^ (j)i (spherical vector) (5.59) 1=1 X 2 = ^ 2 + <^3 + (1 + 3~^)(^4 + ^5 ) + (1 + 3 ^ + 3 '^)(<^6 + ^7 ) ^ 3 = %4 = 3”V4 + 3“'(l + 3"’)<^5 + 3"'(l + 3"’)"<^6 + 3"'(l + 3"’)(l + 3"’ +3"')<^7 %5 = 3“'’(l + 3"')(<7^5 + çi6) + 3”^(H -3"*)(l+3"V3"")<^7 %6 = ç-^(1 + 3~')<^6 + 3"^(H -3”’)(1 + 3"'+3"'')<^7 X 7 = 3"'‘(1 + 3”’)(1 + 3"^+3"')<^7. Under this new basis, the action of the Hecke operators A' and W can be written in nicer forms: Ps{A'){X,) = 3 " ( 1 + 3-^ + 3-')%i + (1 - 3-'-")X 2 + (3' - 3"^)X3 (5.60) 266 Pa{>A!){X2) — (ç * + Ç * * + 1)^2 + (î* ~ ^)-^3 + (1 - g-"')(i + + g')%4 + (g-'-" - 1)X, Pb {'À'){X3,) = (ç* + Ç * ^4-ç ^)X3 + {q — q ® ^)Xj + — 1)-X's /).(^0(^4) = (g' + l + g-')%4 + (g'+'-l)% = + (1 — g“* ')(g + 1 + g®)%6 + (g ® ^ — 1)%? = (g® + g®~* + g~*)-^5 + (1 — g'"*”^)-^6 + (g®~^ — g~'^)-^7 Ps{^'){X6) = (g® + g® ^ + l)X e + (g® ^ — g and = (g"-i)A:i (s.ei) /)&(W)(%g) = (g^ — l)%i + (g^ — 1)%2 — (g — l)g ^Xz Ps[y\^'){Xz) = (g^ — 1)X\ + (g ' — 1)%2 + (g^ — l)-^3 = (g^ — 1)X2 + (g — g“^)X 3 + (g — 2 + g ^)%4 = (1 — g ^)(l + g ’)Xs + (g ’’ — g ')X 4 + (g —l)Xs Pa{y^'){Xo) = (g — g + (g — 1)^5 + (g ^ — l)(g + l)Xe + (1 — g /5«(W')(X7) = (g^ — g ^)%5 + (g ^ — g)-^6- Theorem 5.3.5 Let X \ be the spherical vector in . We define a polynomial F{s) in g* as follows: F{s) := (1 - g“*"^)(l - g~®"^)(l - g"'^*"'^)(l + g*"’)(l + g®). Then is generated by X^ as a 'H{Sp{A)IJIp)-module at s if F{s) ^ 0. 267 Proof: Let < > be the ?i(5p(4)///p)-subm odule in According to (5.60), that ps{A'){Xi) is inside < > implies that (1 — q~^~^){X2 + q^Xs) €< X^ >. If F(s) 7^ 0, then X 2 + q^Xs € < X i > We consider the vector ps{yV'){X 2 + 9 ^X3), which is in < X\ > By the action of the Hecke operator ps{yV'), we obtain that (. Since the determinant • 1 q^ q + 1 - g®-^ 5®+^ + ή - q~^ that F(.s) ^ 0 implies that the determinant does not vanish and then X 2 and X3 are in the submodule < JAi >. Similarly, we consider the vectors ps{A'){X 2) and ps{A'){X 3), which are in the submodule < >. From the action of the Hecke operator A', it is easy to see that both vectors (1 — Ç ® ^)[(1 + Î ^ + q^)X4 — X5] and q{l — g ® '^)(%4 + q^^^Xs) belong to the submodule < >. Again, by F{s) ^ 0, we obtain that (1 + q~^ + q^)X4 — X 5 and X 4 + q^'^^Xs are in < X i >. The determinant l + î"’ + Finally, we come to the vectors pa{A'){X 4) and /),(A')(%5). The action of A' tells that (1 - + 1 + ?*)^6 - and (1 - + g-'X?) belong to the submodule < X\ >. Because (1 — g“®“') 7^ 0, we get that both (g + 1 + q^)Xe~X7 and Xe + q^~^X7 are in < X\ >. Similarly, the corresponding determinant 268 is equal to ( X& and are also in < Xi > This proves the theorem. □ Corollary 5.3.6 For the value of s such that F{s)F{—s) ^ 0, is irreducible. In other words, when s is real, /^(s) is irreducible except for s = ±1,±2, ±3. Proof: According to Corollary 3.5 in [23], I|(.s) is irreducible if and only if the unique (up to a scalar) spherical vector X i generates both and I^{—s)^^ as 'H(5p(4)///f>)-modules. Thus the Corollary follows from theorem 9. □ Now we are able to prove that the dimension of Homsp{ 4){l3{-s),l3{—s)) is one when F{s) ^ 0. Proposition 5.3.7 For the value ofs such that F{s) ^ 0, the dimension of the space ffomsp( 4)(I^(s), I^(—s)) is equal to one. Proof: By means of Corollary 1, we have Homsp(4){lt{s),l3{-.9)) = Homn{Sp{4)//Jp){lii-^y'’^ (5.62) Since F{s) ^ 0, the spherical vector X-[ generates the whole module Isi-s)'’’ under the action of the Hecke algebra H{Sp{A)//Ip). Let Ms and AC be any two intertwining maps from /^(.s)^^ to I^{—s)^'‘. It is clear that any nonzero intertwining map will take a nonzero spherical vector to a nonzero spherical vector since the spherical vector Xi generates the whole module I^{s)^’‘. Thus we obtain that both Ad,(X] ) and 269 J^a{X\) are nonzero spherical vectors in Because the subspace consisting of the spherical vectors is of one dimension, there exists a constant c(s) so that MaiXi) = c(fi)Ms(Xi). Therefore we have M s = c{s)Ms since X\ generates the whole module This proves the proposition. □ The main theorem now follows directly from Proposition 5.3.1 and 5.3.7 and Theo rem 5.1.8 in the first section. The case remains to consider is Re(.s) > 0 and F(fi) = 0, which will be included in our further study. We also believe that our argument and results hold for Sp(n). 5.4 Notations Used in Chapter V O, p.227 P, p.228 F, p.227 K, Ki, p.227 Pi, p.227 Ip, p.227 n{Sp[n)Uip), p.m = p.228 Vn , V{N), p.228 y ^ . p.229 C, isY<’, p.229 IB n , p.230 p.231 A, p.233 w , p.233 Wi = 2,n—2 , p.233 p st p.233 A\ p.258 W', p.261 , p.263 p.262 hk.l), p.260 >, p.256 n{- • ■), p.235 »i(- p.237 »2( -.), p.237 p, p.233 No, p.234 Pw, p.234 X, p.256 B ibliography [1] J. Arthur, Eisenstein series and the trace formula, in Proceedings of Sym posia in Pure Math., Vol. 33, Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 253-274. [2] .1. 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Sup. 13(1980), pp 165-210.pi)„{Ti„,T2„)))X‘ = ^ (m i, m2|«i, «2)P(7«i, /«g, fh7* ,2 ; %), f=0 7T2, p i (8) p i ) will be determined if we have suffi cient acknowledge about those ramified local zeta integrals Zv{s, VF2,,, both archimedean and non-archimedean. It is our goal here to study those ramified local zeta integralsvesfa,v] in we denote that