Schubert Eisenstein Series and Poisson Summation for Schubert

Home , Character group

arXiv:2107.01874v2 [math.NT] 19 Jul 2021 eodato sprilyspotdb S rn M 918,an 1901883, DMS grant grant. NSF 2017R1A2B2001807 by supported partially is author second CUETESNTI EISADPISNSMAINFOR SUMMATION POISSON AND SERIES EISENSTEIN SCHUBERT .Introduction 1. h rtato a atal upre yNF2018R1A4A102359 NRF by supported partially was author first The phrases. and words Key 2010 representations Induced of space 3.1. Schwartz The setting 3. unramified The 2.7. Quasi-characters 2.6. Measures 2.5. spaces Function 2.4. Examples 2.3. The 2.2. spaces Braverman-Kazhdan 2.1. Preliminaries 2. Acknowledgments Outline of representations 1.5. integral On 1.4. Conjectures conjecture summation 1.3. Poisson The series Eisenstein 1.2. Schubert Generalized 1.1. ahmtc ujc Classification. Subject Mathematics sals h ojcueo upadtefis uhri aycase many in author first a the varieties and Schubert Bump to of conjecture related the closely establish schemes th prove certain We for formula. Sa c formula summation Ngô, and Poisson their the Lafforgue, revisit of We Kazhdan, generalizations Braverman, lishing general. of in program true the is to same c the meromorphic conjectured have and series eters Eisenstein Schubert these that proved eeaeEsnti eist atclrShbr ait.I th In variety. Schubert particular a to series Eisenstein generate Abstract. Y P cuetEsnti eisaedfie yrsrcigtesummatio the restricting by defined are series Eisenstein Schubert 1.3 cuetEsnti eis oso umto ojcue Schu conjecture, summation Poisson series, Eisenstein Schubert and ONJ HI N AC .GETZ R. JAYCE AND CHOIE YOUNGJU 1.4 Y CUETVARIETIES SCHUBERT P,P ′ ( F ) rmr 17;Scnay1F5 11F85. 11F55, Secondary 11F70; Primary L Contents -functions 1 eevdspotfo .CoesNRF Choie’s Y. from support received d n R 07122087 The 2017R1A2B2001807. NRF and 0 aeo GL of case e niutosi l param- all in ontinuations elrdsamda estab- at aimed kellaridis s. duei orfieand refine to it use nd netr n eaeit relate and onjecture oso summation Poisson e 3 over nade- a in n etvarieties. bert Q they 20 20 19 19 18 17 17 14 11 11 10 10 8 9 7 2 2 2 YOUNGJU CHOIE AND JAYCE R. GETZ

3.2. The Schwartz space 20 3.3. The Fourier transform 26 3.4. The unramiﬁed setting 29 3.5. The adelic setting 31 4. The Poisson summation formula on X (F ) 32 P ∩Mβ0 5. Proofs of Theorem 1.1 and Theorem 1.2 38 6. On the poles of degenerate Eisenstein series 45 References 47

1. Introduction In this paper we prove the Poisson summation conjecture of Braverman-Kazhdan, Laf- forgue, Ngô, and Sakellaridis for a particular family of varieties related to Schubert varieties (see Theorem 1.2). As an application we prove (in many cases) the functional equations of Schubert Eisenstein series conjectured to exist by Bump and the first author in [BC14]. We begin by recalling the definition of the Schubert Eisenstein series and then move to a discussion of the Poisson summation formula.

1.1. Generalized Schubert Eisenstein series. Let F be a global ﬁeld with ring of adeles

AF , and let G be a (connected) split reductive group over F. We let

T ≤ B ≤ P ≤ G

be a maximal split torus, a Borel subgroup, and a parabolic subgroup of G. Moreover we let

T ≤ M ≤ P

be the Levi subgroup of P containing T . Let NG(T ) be the normalizer of T in G and let

W (G, T ) := NG(T )(F )/T (F ) be the Weyl group of T in G. Finally we let

ab k+1 ωP : M −→ Gm be the isomorphism of (2.1.11). Here M ab = M/M der is the abelianization of M, where Qder is the derived group of an algebraic group Q. × × × k+1 × Let AGm < F∞ be the usual subgroup (see (2.6.1)) and let χ :(AGm F \AF ) → C be a character. For s ∈ Ck+1 we deﬁne

k si χs(a0,...,ak) := χ(a0,...,ak) |ai| . i=0 Y We form the induced representation

G IP (χs) := IndP (χs ◦ ωP ), SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 3 normalized so that it is unitary when s ∈ (iR)k+1. Using the Bruhat decomposition of G one has a decomposition of the generalized ﬂag variety

P \G = P \P wB. w∈W (M,Ta)\W (G,T ) Here we have used the same symbol w for a class in W (M, T )\W (G, T ) and for a represen- tative of that class. Let Xw be the (Zariski) closure of the Schubert cell P \P wB in P \G. It is a Schubert variety. χs The Schubert Eisenstein series attached to a section Φ ∈ IP (χs), g ∈ G(AF ) and w ∈ W (G, T ) is deﬁned as

χs χs SEw(g, Φ ) := Φ (γg). γ∈Xw(F ) χs It converges absolutely provided that Re(s) lies in a suitable cone. The function SEw(g, Φ ) is no longer left G(F )-invariant. However, it is invariant under the stabilizer of the Schubert cell Xw under the natural action of G on P \G. Since this stabilizer contains B, it is a parabolic subgroup, and it is often larger than B. In [BC14] Bump and the first author posed the following questions: (a) Does the Schubert Eisenstein series admit a meromorphic continuation to all s? (b) Do a subset of the functional equations for the full Eisenstein series continue to hold for the Schubert Eisenstein series? (c) Is it possible to find a linear combination of Schubert Eisenstein series which is entire? We generalize the definition of Schubert Eisenstein series in higher rank in a manner that

can be specialized to the deﬁnition in the G = GL3 case treated in [BC14]. We then answer all of these questions in great generality. We refer to Theorem 1.1 and the subsequent remarks for details. Before proceeding, let us examine the setting considered in [BC14] and isolate the change

in viewpoint that is the germ of our work. Let G = GL3 and let B < GL3 be the Borel subgroup of upper triangular matrices. From the point of view of this paper the most interesting Schubert variety in this setting is that attached to the Bruhat cell Bs1s2B where

1 1 s1 = 1 , s2 = 1 . 1 1 In this case using standard facts on the Bruhat ordering one has

(1.1.1) Bs1s2B = Bs1s2B ∐ Bs1B ∐ Bs2B ∐ B.

Let R be an F -algebra. Then ab c (1.1.2) Bs1s2B(R)= def ∈ GL3(R) . g h n o Indeed, the set on the right is the R-points of an irreducible closed subscheme of GL3 that

contains the irreducible closed subscheme Bs1s2B, hence we deduce equality by considering 4 YOUNGJU CHOIE AND JAYCE R. GETZ

dimensions. If we let ab c ab c P2,1(R) := def ∈ GL3(R) , P1,2(R) := e f ∈ GL3(R) h g h n o n o then

(1.1.3) Bs1s2B = P2,1s1s2P1,2. Indeed, it is easy to see that each Bruhat cell on the right of (1.1.1) is contained in

P2,1s1s2P1,2, and it is clear on the other hand from (1.1.2) that P2,1s1s2P1,2 ⊆ Bs1s2B. The natural map

P2,1 × P1,2 −→ Bs1s2B

is a lift of the Bott-Samelson resolution of the image Xs1s2 of Bs1s2B in B\SL3. This was the point of departure for the arguments in [BC14]. Instead of pursuing Schubert varieties and the Bott-Samelson resolution to study higher rank analogues of the left hand side of (1.1.3), we generalize and study the right hand side directly. Let us step back to consider the situation for a general reductive group G. Consider ′ the closure P wB in G. It is the union of Bruhat cells ∪w′≤wPw B, where ≤ denotes the Bruhat order. The group G acts on itself on the left, and we let Q be the stabilizer of the closed subscheme P wB ⊂ G (i.e. the algebraic subgroup of G sending P wB to itself). This is a closed algebraic subgroup of G [Mil17, Corollary 1.81]. The group Q contains P, and hence is a parabolic subgroup. Thus rather than studying P wB, for any parabolic subgroup P ′ with P ≤ P ′ ≤ Q we can study generalized Schubert Eisenstein series built from sums over P ′wB. In fact, there is nothing special about B, and it is more interesting from an automorphic perspective to replace B by the largest possible group that preserves P wB under right multiplication. Thus we could work with

(1.1.4) P ′γH

where P ≤ P ′ ≤ G are a pair of parabolic subgroups, γ ∈ G(F ) and H is an arbitrary algebraic subgroup of G. From the automorphic point of view this may be the most important situation. However, not all Schubert cells are the image in P \G of a set of the form (1.1.4). Indeed, Schubert cells are often nonsmooth, whereas the image of any set of the form P ′γH in P \G is a smooth subscheme (this follows from lemmas 2.4 and 2.5). In order to treat Eisenstein series indexed by sets of the form Y and P wB simultaneously we work with an arbitrary (locally closed) subscheme Y ⊆ G that is stable under left ′ ◦ der multiplication by P . Let XP := P \G be the Braverman-Kazhdan space associated to P and G. Let

◦ (1.1.5) YP = Im(Y −→ XP ). SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 5

◦ To be more precise, the set theoretic image of Y → XP is locally closed by Lemma 2.4. This ◦ set is the underlying topological space of a subscheme YP of XP . For some examples when ◦ G = SLn we refer to §2.3. The subscheme YP ⊆ XP is quasi-aﬃne. Let XP be the aﬃne ◦ closure of XP and let

(1.1.6) YP,P ′ ⊆ XP

be the partial closure of YP in XP constructed in (2.2.6) below. We observe that there is a ab natural action of M on YP,P ′ that preserves YP (see (2.1.5)). Without essential loss of generality we assume G is simple and simply connected. We additionally make the following technical assumption:

(1.1.7) P is maximal in P ′.

Under (1.1.7) there is a unique parabolic subgroup P ∗ < P ′ with Levi subgroup M that is not equal to P. Thus one might think of P ∗ as the opposite parabolic of P with respect to P ′. In §3 we deﬁne Schwartz spaces

S(YQ,P ′ (AF )) for Q ∈{P,P ∗} together with a Fourier transform

(1.1.8) FP |P ∗ : S(YP,P ′ (AF ))−→S ˜ (YP ∗,P ′ (AF )).

The Schwartz space S(YQ,P ′ (AF )) is contained in the set of restrictions to YP (AF ) of functions ∞ ◦ in C (XP (AF )). Let H ≤ G be a subgroup, and consider the action of H on G by right multiplication. Assume that Y is stable under the action of H. Then the Schwartz spaces ab S(YP,P ′ (AF )) and S(YP ∗,P ′ (AF )) are preserved under the action of M (AF ) × H(AF ) of (3.3.2) and the Fourier transform satisﬁes a twisted equivariance property by Lemma 3.4. ∗ G Let IP ∗ (χs) := IndP ∗ (χ ◦ ωP ). The ∗ indicates that we are inducing χ ◦ ωP , not χ ◦ ωP ∗ . ab The group M acts on YP and YP ∗ on the left, and hence we obtain Mellin transforms

′ S(YP,P (AF )) −→ IP (χs)|YP (AF )

1/2 −1 f 7−→ fχs (·) := fχs,P (·) := δP (m)χs(ωP (m))f(m ·)dm, ab ZM (F ) (1.1.9) ∗ ∗ ′ ∗ S(YP ,P (AF )) −→ IP (χs)|YP ∗ (AF )

∗ ∗ 1/2 −1 ∗ f 7−→ fχs (·) := fχs,P (·) := δP ∗ (m)χs(ωP (m))f(m ·)dm. ab ZM (F )

Here δQ is the modular quasi-character of an algebraic group Q. The fact that the Mellin ∗ transform fχs (resp. fχs ) is absolutely convergent for Re(s0) large (resp. Re(s0) small) is built into the deﬁnition of the Schwartz space. 6 YOUNGJU CHOIE AND JAYCE R. GETZ

χs Remark. We will write Φ for a section of IP (χs) that is not necessarily a Mellin transform

of an element f ∈ S(YP,P ′ (AF )). We take the analogous convention in the local setting and ∗ when IP (χs) is replaced by IP ∗ (χs).

For f1 ∈ S(YP,P ′ (AF )), f2 ∈ S(YP ∗,P ′ (AF )) we deﬁne generalized Schubert Eisenstein series

EYP (f1χs ):= f1χs (y), y∈M ab(F )\Y (F ) (1.1.10) X P E∗ (f ∗ ):= f ∗ (y∗). YP ∗ 2χs 2χs ∗ ab y ∈M X(F )\YP ∗ (F ) These sums converge absolutely for Re(s0) suﬃciently large (resp. small). To help motivate this deﬁnition we point out that Lemma 2.1 implies that

ab P (F )\Y (F ) −→ M (F )\YP (F ) is a bijection, and the same is true if we replace P by P ∗.

Theorem 1.1. Let f ∈ S(YP,P ′ (AF )). Assume that (1) F is a function ﬁeld, or (2) F is a number ﬁeld and Conjecture 1.3 is valid. ∗ ∗ Then EYP (fχs ) and EYP ∗ (FP |P (f)χs ) are meromorphic in s. Moreover one has ∗ ∗ E (f )= E (F ∗(f) ). YP χs YP ∗ P |P χs Conjecture 1.3 below states that certain normalized degenerate Eisenstein series have only

ﬁnitely many poles. It is known if G = SLn and in several other cases (see the remark after the statement of the conjectures below). Remarks.

(1) Let Pmax be the stabilizer of P wB under the left action of G. Then one expects a family

of functional equations and analytic continuation for EP \P wB(fχs ) analogous to the family of functional equations for the degenerate Eisenstein series on a Levi subgroup Mmax of Pmax attached to the parabolic subgroup P ∩ Mmax < Mmax. In the special case where P

case where P is not necessarily maximal in Pmax.

(2) We have already seen that one can choose the data so that YP,P ′ is a lift to XP of a partial closure of a Schubert cell. Schubert varieties admit analogues in any Kac-Moody group, and it would be interesting to generalize Theorem 1.1 to this setting. The key observation here is that, unlike Kac-Moody groups and their flag varieties, which in general are infinite-dimensional, the Schubert cells in the flag varieties of Kac-Moody groups are finite-dimensional. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 7

(3) It is an intriguing problem to investigate whether generalized Schubert Eisenstein series can be used to produce integral representations of automorphic L-functions. See §1.4 below for more details. For example, one could try to generalize the famous doubling method of Piatetski-Shapiro and Rallis [GPSR87].

(4) For some information about the possible poles of EYP (fχs ), see Corollary 5.5.

1.2. The Poisson summation conjecture. To prove Theorem 1.1 we prove new cases of a seminal conjecture due to Braverman and Kazhdan [BK00]. The conjecture was later discovered independently by Lafforgue [Laf14] and refined by Ngô[Ngô14]. It was partially set in the framework of spherical varieties by Sakellaridis [Sak12]. Here a spherical variety X for a reductive group G over F is a normal integral separated G-scheme X of finite type

over F such that XF admits an open orbit under a Borel subgroup of GF . The conjecture can be roughly formulated as follows. Assume that X is an aﬃne spher- sm ical variety with smooth locus X . Then there should be a Schwartz space S(X(AF )) < ∞ sm C (X (AF )) and a Fourier transform

FX : S(X(AF )) −→ S(X(AF ))

satisfying a certain twisted-equivariance property under G(AF ) such that for f ∈ S(X(AF )) satisfying certain local conditions

f(x)= FX (f)(x). sm sm x∈XX(F ) x∈XX(F ) We refer to this (somewhat vaguely stated) conjecture as the Poisson summation conjecture. The original motivation, explored in [BK00, Ngˆo14, Ngˆo20], is that it implies the meromorphic continuation and functional equation of Langlands L-functions in great generality. By the converse theorem [CPS99] this would imply Langlands functoriality in great generality.

Remark. We highlight the possibly confusing convention that functions in S(X(AF )) need sm not be deﬁned on all of X(AF ), only on X (AF ). One expects that for each place v el- ∞ sm ements of S(X(Fv)) are functions in C (X (Fv)) that are rapidly decreasing away from sm the singular locus (X − X )(Fv) and have particular asymptotic behavior as one approaces sm (X − X )(Fv). This was conjectured in [Ngˆo20, §5] in a special case and is expected to be true in general.

The only case of the Poisson summation conjecture that is completely understood is the

case where X is a vector space. For the aﬃne closures XP of the Braverman-Kazhdan space ◦ XP much of the conjecture is known [GL21, GH20, GHL21, JLZ20, Sha18]. There are some additional examples in [GL19, GH20, Gu21]. However the cases that are known are still very limited. 8 YOUNGJU CHOIE AND JAYCE R. GETZ

In order to prove Theorem 1.1 we prove the Poisson summation conjecture for YP,P ′. We do not know if YP,P ′ is aﬃne, but it is clearly quasi-aﬃne. We also do not know whether it is always spherical under the action of a suitable reductive subgroup of H, but this is true in many cases [Gae21, HY20]. In Theorem 1.2 we state our Poisson summation formula in an imprecise form. The precise ab form is given in Theorem 5.2 below. Let KM be the maximal compact subgroup of M (AF ).

Theorem 1.2. Let f ∈ S(YP,P ′ (AF )). Assume (1) F is a function ﬁeld,

(2) F is a number field, Conjecture 1.3 is valid, and f is KM -finite, or (3) F is a number field and Conjecture 1.4 is valid. One has

∗ f(y)+ ∗ = FP |P ∗ (f)(y )+ ∗∗ . ∗ y∈XYP (F ) y ∈XYP ∗ (F ) The sums over y and y∗ are absolutely convergent.

In the theorem the contributions marked ∗ and ∗∗ are certain boundary terms coming from residues of auxilliary degenerate Eisenstein series. Using Lemma 3.10 one can choose many f so that these contributions vanish. Theorem 1.2 is our main theorem in the context of Poisson summation formulae. It is a vast generalization of the Poisson summation formula for Braverman-Kazhdan spaces of maximal parabolic subgroups in reductive groups [BK02] which in turn is a vast generalization of the Poisson summation formula for a vector space.

Remark. In the degenerate case H = G, Theorem 1.2 reduces to the Poisson summation for-

mula for the Braverman-Kazhdan space XP . In this special case under suitable assumptions on f the formula was proved in [BK02]. When G = Sp2n and P is the Siegel parabolic it was proved for general test functions in [GL21].

1.3. Conjectures 1.3 and 1.4. Let Mβ0 be the simple normal subgroup of the Levi subgroup M ′ of P ′ deﬁned in (2.2.2) below. For any topological abelian group A we denote by A the set of quasi-characters of A, that is, continuous homomorphisms A → C×. For Q ∈{P,P ∗} let S(X (A )) be the Schwartz space of (3.3.3). For any Q∩Mβ0 F b \× × (m, f , f ,χ,s) ∈ M (A ) × S(X (A )) × S(X ∗ (A )) × A F \A × C 1 2 β0 F P ∩Mβ0 F P ∩Mβ0 F Gm F s let χs := χ|·| and form the degenerate Eisenstein series

E(m, f1χs )= f1χs (xm)

x∈(P ∩Mβ )\Mβ (F ) (1.3.1) X0 0 ∗ ∗ ∗ E (m, f2χs )= f2χs (xm). x∈(P ∗∩M )\M (F ) Xβ0 β0 SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 9

∗ They converge for Re(s) large enough (resp. Re(s) small enough). Here f1χs and f2χs are ′ the Mellin transforms of (1.1.9) in the special case P = Mβ0 .

Let K ≤ Mβ0 (AF ) be a maximal compact subgroup. The following conjecture appeared in the statements of theorems 1.1 and 1.2 above:

\× × Conjecture 1.3. For each character χ ∈ AGm F \AF there is a ﬁnite set Υ(χ) ⊂ C such that if E(m, f ) has a pole for any K-ﬁnite f ∈ S(X (A )) then s ∈ Υ(χ). χs P ∩Mβ0 F In fact we expect the following stronger conjecture to be true:

Conjecture 1.4. There is an integer n and a ﬁnite set Υ ⊂ C depending only on Mβ0 such that if E(m, f ) has a pole for any K-ﬁnite f ∈ S(X (F )) and χ ∈ A \F ×\A× then χs P ∩Mβ0 Gm F χn =1 and s ∈ Υ.

In §6 we prove Conjecture 1.3 (or more accurately extract it from the literature) when Mβ0

is SLn and Conjecture 1.4 when P ∩ Mβ0 is a Siegel parabolic in the symplectic group Mβ0 . We point out that the natural analogues of conjectures 1.3 and 1.4 for general sections of

IP (χs) are false. It is important that one uses Mellin transforms of elements of the Schwartz space. 1.4. On integral representations of L-functions. In this subsection we make some comments on how one might try to use Theorem 1.1 to study integral representations of L- functions. Though it is purely speculative we include this discussion because it allows us to pose several questions in the study of algebraic homogenous spaces that are of independent interest. Let a subgroup H ≤ G act on G via right multiplication, and assume that Y is stable under the action of H. For example, we could take Y = P ′γH for some γ ∈ G(F ). Assume that Y admits an open H-orbit. Let ϕ be a cusp form in a cuspidal automorphic representation π of H(AF ). If one wanted to use Theorem 1.1 to study integral representations of L-functions one could try to investigate expressions of the form

(1.4.1) ϕ(h)EYP (R(h)fχs )dh. ZH(F )\H(AF ) where R(h)fχs (y) = fχs (yh). If convergent, (1.4.1) admits a functional equation because

EYP (R(h)fχs ) admits a functional equation by Theorem 1.1. The problem is deciding whether or not (1.4.1) or some variant of it yields any new L-functions. Ignoring all questions of convergence, the integral above unfolds into a sum

(1.4.2) ϕ(h)fχs (y0h)dh. ab Hy (F )\H(AF ) y0∈M (FX)\YP (F )/H(F ) Z 0 ab where Hy0 is the stabilizer of y0 ∈ (M \YP )(F ) in H. Due to the conjectures of Sakellaridis and Venkatesh [Sak12, SV17] one expects this expression to unfold into a sum of Eulerian ab integrals when M \YP is spherical as an H-scheme. This motivates the following questions: 10 YOUNGJU CHOIE AND JAYCE R. GETZ

ab (1) When is M \YP spherical as an H-scheme? ab (2) If M \YP is spherical, what is the stabilizer in H of a point in the open H-orbit? (3) If H and a spherical subgroup H′ ≤ H are ﬁxed, can one classify the possible spherical ′ ab ′ embeddings of H \H obtained from M \YP as P, P , G, and Y vary?

We can also pose analogous questions when H is not necessarily reductive by replacing H by a Levi subgroup as in [Gae21, HY20]. Finally, it is of interest to consider the questions above in the Kac-Moody setting mentioned in Remark 2 after Theorem 1.1.

1.5. Outline. We outline the paper and give some indication of the proofs. We begin with some comments on the underlying geometry we are considering in §2. We then deﬁne the

Schwartz space of YP,P ′ locally and adelically in §3. We also construct a Fourier transform and prove that the Fourier transform preserves the Schwartz space. To accomplish this we reduce the question to a similar statement on an auxilliary Braverman-Kazhdan space and then use the methods developed in [GL21, GHL21]. We then prove a Poisson summation formula for an auxilliary Braverman-Kazhdan space in §4. This formula is proved in [BK02] for a different definition of the Schwartz space with some restrictions on the test functions involved. It was obtained for arbitrary test functions in [GL21] in a special case. In Theorem 4.4 below we deduce it for all Braverman-Kazhdan spaces attached to maximal parabolic subgroups. We point out that it is most natural in our setting, and perhaps even necessary, to work with sections that are not finite under a

maximal compact subgroup of G(F∞) when F is a number ﬁeld (see §3.1). Hence, some care is required in applying well-known results on Eisenstein to our setting. Indeed, our sections need not even be standard, so we cannot even use Lapid’s work in [Lap08]. In §5 we prove theorems 1.2 and 1.1. The procedure is to ﬁrst prove Theorem 1.2 by reducing it to a summation formula on an auxilliary Braverman-Kazhdan space so that we can apply the results of §4. We then use Theorem 1.2 to deduce Theorem 1.1. Finally, in §6 we verify Conjecture 1.3 (restated below as Conjecture 1.4) in certain cases.

Acknowledgments. We appreciate the encouragement and questions of D. Bump. An- swering his questions led to a generalization of our original main result and simpliﬁcations in exposition. The authors thank S. Leslie for answering questions about parabolic subgroups and M. Brion for answering several questions about Schubert cells (and in particular for observing Lemma 2.5). We also thank D. Ginzburg and F. Shahidi for encouragement, S. Kudla for answering a question on Eisenstein series and M. Hanzer for explaining how to derive Theorem 6.4 from her paper [HM15] with G. Muic. The ﬁrst author thanks H. Hahn for her constant support and encouragement and for her help with editing and the structure of the paper. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 11

2. Preliminaries 2.1. Braverman-Kazhdan spaces. We work over a field F. Let G be a connected split semisimple group over F. We only consider parabolic subgroups of G containing a fixed split torus T ; for such a subgroup P we write NP for its unipotent radical. We fix a Levi subgroup M of G containing T and write

(2.1.1) M ab := M/M der.

For all parabolic subgroups P of G with Levi subgroup M we have a Braverman-Kazhdan space

◦ der (2.1.2) XP := P \G. We observe that

der der (2.1.3) P = M NP

◦ where NP is the unipotent radical of P. The scheme XP is strongly quasi-aﬃne, i.e.

◦ aff ◦ (2.1.4) X := X := Spec(Γ(X , O ◦ )) P P P XP ◦ is an affine scheme of finite type over F and the natural map XP → XP is an open immersion [BG02, Theorem 1.1.2]. Strictly speaking, in loc. cit. Braverman and Gaitsgory work over an algebraically closed field, but their results hold in our setting by fpqc descent along Spec(F ) → Spec(F ). A convenient reference for fpqc descent is [Poo17, Theorem 4.3.7].

Lemma 2.1. The torus M ab is split. The maps

M(F ) −→ M ab(F ) and G(F ) −→ (P der\G)(F ) are surjective.

Proof. In [GHL21, Lemma 3.2 and Corollary 3.3] this is proved in the special case where P is a maximal parabolic. The same proof implies the more general statement here.

ab ◦ We have a right action of M × G on XP given on points in an F -algebra R by ◦ ab ◦ XP (R) × M (R) × G(R) −→ XP (R) (2.1.5) (x, m, g) 7−→ m−1xg.

ab This action extends to an action of M × G on XP . We now discuss the Pl¨ucker embedding of P der\G as explained in a special case in [GHL21]. We assume for simplicity that G is simply connected. Let T ≤ B ≤ P be a Borel subgroup,

let ∆G be the set of simple roots of T in G deﬁned by B and ∆M the set of simple roots of T in M deﬁned by B ∩ M. Let

(2.1.6) ∆P := ∆G − ∆M 12 YOUNGJU CHOIE AND JAYCE R. GETZ be the set of simple roots in ∆G attached to P. For each β ∈ ∆G we let

B ≤ Pβ ≤ G be the unique maximal parabolic subgroup containing B that does not contain the root group attached to −β. ∗ ∗ As usual let X (T ) be the character group of T. For each β ∈ ∆P let ωβ ∈ X (T )Q be the fundamental weight dual to the coroot β∨, in other words,

∨ 1 if α = β (2.1.7) ωβ(α )= 0 otherwise.  ∗ Here α ∈ ∆G. Since G is simply connected, the fundamental weight ωβ lies in X (T ). There is an irreducible representation

(2.1.8) Vβ × G −→ Vβ of highest weight −ωβ. Here G acts on the right. Let

◦ (2.1.9) V := Vβ, V = (Vβ −{0}). βY∈∆P βY∈∆P Choose a highest weight vector vβ ∈ Vβ(F ) for each β.

Lemma 2.2. There is a closed immersion

◦ ◦ (2.1.10) PlP : XP −→ V given on points by

PlP (g) := (vβg).

It extends to a G-equivariant map PlP : XP → V. The character ωβ extends to a character of M for all β and the map

ab ∆P (2.1.11) ωP := ωβ : M −→ Gm βY∈∆P ab ab is an isomorphism. If we let M act via ωP on V then PlP is M -equivariant. In particular for (m, g) ∈ M ab(R) × G(R) one has

−1 (2.1.12) PlP (m g)=(ωβ(m)vβg).

Here

∆P Gm := Gm. βY∈∆P We will use similar notation below without comment. In the introduction, we identiﬁed ∆P with the set {0, 1,...,k}. To ease notation, for any subset ∆ ⊂ ∆G let

◦ (2.1.13) V∆ = Vβ and V∆ := (Vβ −{0}). βY∈∆ βY∈∆ SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 13

Proof. Denote by Pβ the unique maximal parabolic subgroup of G containing B that does not contain the root group attached to −β. Then the stabilizer of the line spanned by vβ in

Vβ is Pβ [Jan03, §II.8.5]. Since P ≤ Pβ we deduce that PlP is well-deﬁned, that ωβ extends

to a character on M, and that (2.1.12) holds. Since V is aﬃne the map PlP tautologically ab extends to a M × G-equivariant map Pl : XP → V. We are left with checking that PlP is a closed immersion and that (2.1.11) is an isomorphism.

∆P ab We first check that (2.1.11) is an isomorphism. The homomorphism Gm → M given on ∨ points by x 7→ β β (x) is a section of ωP . Thus it suffices to show that the image of this section is all of M ab. Since the image is closed and M ab is irreducible, it suffices to check Q ∆P ab ab that |∆P | = dim Gm is equal to dim M . The group M is isogenous to the center ZM of M which is contained in the subgroup of T on which all the β ∈ ∆M = ∆G − ∆P vanish.

Thus ZM has at most dimension |∆P |.

To check that PlP is a closed immersion we ﬁrst point out that we have a commutative diagram

◦ PlP ◦ XP V (2.1.14) Pl P \G P PV β∈∆P β Q where PlP is induced by PlP . We claim that PlP is a closed immersion. Since P \G is proper and PV is separated the map Pl has closed image. It is an immersion provided β∈∆P β P that P is the stabilizer of the image of (vβ) in PVβ by [Mil17, Proposition 7.17]. The Q β∈∆P stabilizer of the image of (v ) is β Q

(2.1.15) Pβ. β∈\∆P But (2.1.15) is a parabolic subgroup containing B. By considering the root groups contained in (2.1.15) we deduce that it is P . This completes the proof of the claim.

Extend ωP to a character of P in the canonical manner. Since the stabilizer of the image of (v ) in PV is P we deduce that the stabilizer of (v ) is contained in P and hence β β∈∆P β β equal to ker(ω : P → G∆P ). Since (2.1.11) is an isomorphism ker(ω : P → G∆P ) = P der. Q P m P m ◦ Thus the map PlP is an immersion of XP onto the orbit of (vβ)[Mil17, Proposition 7.17]. The left vertical arrow in (2.1.14) is the geometric quotient by the action of M ab, and the

∆P right vertical arrow is the geometric quotient by Gm . In view of the equivariance property (2.1.12) and the fact that (2.1.11) is an isomorphism, we deduce that the image of PlP is

closed, and hence PlP is a closed immersion.

Example. Let G = SL3 and P = B, the Borel of upper triangular matrices. The represen- 3 2 3 tations Vβ1 and Vβ2 are just the standard representation Ga and ∧ Ga. If we choose (0, 0, 1) 14 YOUNGJU CHOIE AND JAYCE R. GETZ

and (0, 1, 0) ∧ (0, 0, 1) as our highest weight vectors then a Pl b =(c, b ∧ c) . B c Under the Pl¨ucker embedding the image of XB is the cone C whose points in an F -algebra R are given by 3 2 3 C(R)= {(v1, v2) ∈ R × ∧ R : v1 ∧ v2 =0}.

2.2. The YP . We assume that we are in the situation of the introduction. Thus Y ⊆ G is a subscheme whose stabilizer under the left action of G contains a parabolic subgroup P ′ > P. We assume (1.1.7), that is, P is maximal in P ′. Moreover we let P ∗ ≤ P ′ be the unique parabolic subgroup with Levi subgroup M that is not equal to P. There is a unique simple ′ root β0 in ∆G − ∆M such that the root group attached to −β0 is a subgroup of P but not P. Let M ′ be the unique Levi subgroup of P ′ containing M. The set of simple roots of T in M ′ with respect to the Borel B is

(2.2.1) ∆M ′ = {β0} ∪ ∆M . Since G is simply connected, it follows from [BT72, Proposition 4.3] that the derived group M ′der is simply connected. Hence it is a direct product of its simple factors. There is a unique decomposition

′der β0 (2.2.2) M = Mβ0 × M

where Mβ0 is the unique simple factor containing the root group of β0. It then is the unique ∨ simple factor containing β0 (Gm). Let NP denote the unipotent radical of P.

Lemma 2.3. The group Tβ0 := T ∩ Mβ0 is a maximal torus of Mβ0 and B ∩ Mβ0 is a

Borel subgroup of Mβ0 . The group P ∩ Mβ0 is the unique maximal parabolic subgroup of Mβ0

containing B ∩ Mβ0 that does not contain the root group attached to −β0, and M ∩ Mβ0 is a

Levi subgroup of P ∩ Mβ0 . We have N = N ∩ M P ∩Mβ0 P β0 der der M ∩ Mβ0 =(M ∩ Mβ0 ) , (2.2.3) der der P ∩ Mβ0 =(P ∩ Mβ0 )

der der β0 ′ P =(P ∩ Mβ0 )M NP Proof. The ﬁrst two sentences follow from [ABD+66, Propositions 1.19 and 1.20, §XXVI]. Since N is the maximal connected normal unipotent subgroup of P ∩M it is clearly P ∩Mβ0 β0 contained in N , the maximal connected normal unipotent subgroup of P. Thus N ≤ P P ∩Mβ0 N ∩ M . On the other hand the intersection of N ∩ M < (M ∩ M )N = P ∩ M P β0 P β0 β0 P ∩Mβ0 β0 and M ∩ M is the identity. It follows that N = N ∩ M . β0 P ∩Mβ0 P β0 Let Q◦ denote the neutral component of an algebraic group Q. One can check the identities

der ◦ der (M ∩ Mβ0 ) =(M ∩ Mβ0 ) , SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 15

der ◦ der (P ∩ Mβ0 ) =(P ∩ Mβ0 )

der der ◦ β0 ′ P =(P ∩ Mβ0 ) M NP by observing that the groups on either side are connected and checking the corresponding identity on the Lie algebras using well-known facts about the decomposition of parabolic der subalgebras into root spaces. Thus to complete the proof it suﬃces to prove that M ∩Mβ0 der der ′der β0 and P ∩ Mβ0 are connected. The group M < M = Mβ0 × M is connected and is the direct product

der der β0 (2.2.4) M =(M ∩ Mβ0 ) × M

der In particular M ∩ Mβ0 is connected. We also conclude from (2.2.4) that P der is the semidirect product of (M der ∩ M )N β0 P ∩Mβ0 β der β der and M 0 N ′ . It follows that P ∩ M 0 is (M ∩ M )N , which is connected. P β0 P ∩Mβ0 Let ◦ pr : G −→ XP be the natural map. In the following it is convenient to denote by |X| the underlying topological space of a scheme X. As usual, a subset of a topological space is locally closed if it is open in its closure.

◦ Lemma 2.4. The set pr(|Y |) ⊂|XP | is locally closed.

Proof. Let Y be the schematic closure of Y in G. Then Y is stable under left multiplication by P der. The map pr is a geometric quotient [Ray67]. In particular the underlying map of ◦ topological spaces is a quotient map. It follows that pr(|Y |) is closed in |XP |. The restriction of pr to |Y | is again a topological quotient map, so pr(|Y |) is open in pr(|Y |). It is easy to check that the closure of pr(|Y |) is pr(|Y |).

◦ As in the introduction, YP is the subscheme of XP with underlying topological space pr(|Y |), given the reduced induced subscheme structure.

Lemma 2.5. If Y = P ′γH for some subgroup H ≤ G and γ ∈ G(F ) then the subscheme ◦ YP ⊂ XP is smooth.

Proof. The space Y is smooth, and pr is a locally trivial ﬁbration. Since Y is left P der- invariant pr restricts to a locally trivial ﬁbration pr : Y → YP .

Lemma 2.1 implies the following lemma:

Lemma 2.6. The map

Y (F ) −→ YP (F ) is surjective. 16 YOUNGJU CHOIE AND JAYCE R. GETZ

Using notation from (2.1.6) and (2.1.13) let

−1 ◦ X ′ : = Pl (V × V ) ⊆ X P,P P β0 ∆P ′ P (2.2.5) −1 ◦ XP ∗,P ′ : = Pl ∗ (V −1 × V ) ⊆ XP ∗ . P β0 ∆P ′

Thus XP,P ′ and XP ∗,P ′ are subschemes of XP and XP ∗ , respectively. Let

YP,P ′ := YP ⊆ XP,P ′ (2.2.6) YP ∗,P ′ := YP ∗ ⊆ XP ∗,P ′ be the closures of YP in XP,P ′ and YP ∗ in XP ∗,P ′ , respectively. The natural map

(2.2.7) N −→ N /N ′ P ∩Mβ0 P P is an isomorphism. For all y ∈ Y (F ) we have a morphism

◦ (2.2.8) ιy : X −→ YP P ∩Mβ0 characterized by the requirement that the diagram

Mβ0 Y (2.2.9)

◦ ιy X YP P ∩Mβ0 commutes, where the top arrow is given on points by m 7→ my and the vertical arrows are the canonical surjections.

Lemma 2.7. Let Ξ ⊂ Y (F ) be a set of representatives for P ′der(F )\Y (F ). One has

◦ ◦ YP (F )= ιy(X (F )) and YP ∗ (F )= ιy(X ∗ (F )). P ∩Mβ0 P ∩Mβ0 y∈Ξ y∈Ξ a a β ′der 0 ′ Proof. With notation as in (2.2.2) we have Mβ0 M NP = P . The ﬁbers of the canonical β ′der 0 ′ projection M (F )NP (F )\Y (F ) → P (F )\Y (F ) are Mβ0 (F )-torsors, so

Mβ0 (F )Ξ

β0 is a set of representatives for M (F )NP ′ (F )\Y (F ). Moreover no two elements of Ξ are in der the same Mβ0 (F )-orbit. By Lemma 2.6 P (F )\Y (F ) = YP (F ). By Lemma 2.3 the ﬁbers of the natural projection

β0 der M (F )NP ′ (F )\Y (F ) −→ P (F )\Y (F )= YP (F )

der der der are (P ∩ Mβ0 )(F )-torsors. Again by Lemma 2.3 we have P ∩ Mβ0 = (P ∩ Mβ0 ) and we deduce the ﬁrst equality of the lemma. To obtain the second equality we replace P by P ∗ and argue by symmetry. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 17

2.3. Examples. Let G = SLn,n> 2. We generalize the example of (1.1.1) to higher rank

in two diﬀerent manners. Let sj ∈ SLn(F ), 1 ≤ j ≤ n − 1, be the matrix that is the identity matrix with the rows ej and ej+1 replaced by −ej+1 and ej. Let

γ1 = s1s2 ··· sn−1.

This is a Coxeter element. Let R be an F -algebra. Let Bn ≤ SLn be the Borel subgroup of upper triangular matrices and let

′ g x P (R):= {( b ) ∈ SLn(R):(g, b) ∈ GL2(R) × Bn−2(R)} b x H(R):= { g ∈ SLn(R):(g, b) ∈ GL2(R) × Bn−2(R)}

′ Then Bnγ1Bn ⊂ P γ1H ⊆ Bnγ1Bn, and

(2.3.1) Bnγ1Bn(R)= {g ∈ SLn(R): gij = 0 if i > j +1} .

′ If we take P = Bn, then P is maximal in P and

∗ a x × (2.3.2) B (R) := bcz ∈ SLn(R): a, c ∈ R ,d ∈ Bn−2(R) . n d As another example, taken o J ǫ (2.3.3) γ2 = (−1) 1 where J is the matrix with 1’s on the antidiagonal and zeros elsewhere and ǫ ∈ {0, 1} is chosen so the matrix has determinant 1. Write

g x Pa,b(R) := {( h ) ∈ SLn(R):(g, h) ∈ GLa(R) × GLb(R)} . Then

Bnγ2Bn ⊂ Pn−1,1γ2P1,n−1 ⊆ Bnγ2Bn Choose positive integers a, b such that a + b = n − 1. Then we may take

g1 x y × P (R) := g2 z ∈ SL (R):(g ,g , a) ∈ GL (R) × GL (R) × R . a n 1 2 a b In this case n o

∗ g1 y × P (R) := x g2 z ∈ SL (R):(g ,g , a) ∈ GL (R) × GL (R) × R . a n 1 2 a b 2.4. Function spaces.n Let X be a quasi-projective scheme over a local ﬁeld Fo. We denote by C0(X(F )) the space of complex valued continuous functions on X(F ). Assume that F is nonarchimedean. In this case we denote by (2.4.1) C(X(F )) the space of locally constant compactly supported functions on X(F ), also denoted by ∞ Cc (X(F )). If X is smooth, then we set ∞ S(X(F )) = C(X(F )) = Cc (X(F )). For certain nonsmooth schemes X we will deﬁne S(X(F )). 18 YOUNGJU CHOIE AND JAYCE R. GETZ

Now assume that F is archimedean. In this case we deﬁne C(X(F )) as follows. The set

X(F ) = ResF/RX(R) is a real algebraic variety, and hence an aﬃne real algebraic variety [BCR98, Proposition 3.2.13, Theorem 3.4.4]. In particular there is a closed embedding of real algebraic varieties

X(F ) ֒−→ Rn (2.4.2)

for some n. We define C(X(F )) to be the space of restrictions of the usual Schwartz space S(Rn) to X(F ). This space is independent of the choice of embedding and is naturally a Fréchet space [ES18, §3]. Thus C(X(F )) ≤ C0(X(F )). We observe that if X is not smooth then C∞(X(F )) is not defined when F is archimedean. If X is smooth then we set

S(X(F )) = C(X(F )) ≤ C∞(X(F ))

∞ We have S(X(F )) ≥ Cc (X(F )) but the inclusion is strict in the archimedean case when X(F ) is noncompact. We will also deﬁne S(X(F )) for certain nonsmooth X.

× 2.5. Measures. Let F be a global ﬁeld. We ﬁx a nontrivial character ψ : F \AF → C and ′ a Haar measure dx = ⊗vdxv on AF . We assume that dxv is self-dual with respect to ψv. We × × let d x be the Haar measure on AF given by

′ ζv(1) ⊗v dxv. |x|v We ﬁx, once and for all, a Chevalley basis of the Lie algebra of G with respect to T. For every root of T in G this provides us with a root vector Xα in each root space, and hence isomorphisms

Ga −→ Nα where Nα is the root group. This in turn provides us with a Haar measure on Nα(AF ) for all α. As a scheme (but not a group scheme) the unipotent radical of any parabolic subgroup

P with Levi subgroup M is a product of various Nα. Thus we obtain a Haar measure on

NP (AF ). We use this normalization so that factorization of intertwining operators holds (otherwise it only holds up to a constant depending on the choice of Haar measures). der We ﬁx a Haar measure on M (AF ). We give M(AF ) the unique Haar measure such that, ab upon endowing M (AF ) with the quotient measure one has that

ab × ∆P (2.5.1) ωP : M (AF ) −→ (AF ) is measure preserving. This is independent of the parabolic P with M as its Levi, as diﬀerent choices will just replace various ωβ with their inverses. Each of the measures we ﬁxed above factors over the places of F into a product of local measures, normalized so that the local analogue of (2.5.1) is measure preserving. We use these local measures when working locally below. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 19

2.6. Quasi-characters. Let

× × ∆P × χ := χβ :(AGm F \AF ) −→ C βY∈∆P

∆P be a quasi-character. If s =(sβ) ∈ C we let

sβ χs = χβ|·| . βY∈∆P We deﬁne a subgroup

× (2.6.1) AGm ≤ F∞ in the following manner. The global ﬁeld F is a ﬁnite extension of F0, where F0 = Q or

F0 = Fp(t) for some prime p. Let Z ≤ ResF/F0 Gm be the maximal split subtorus. When F is a number ﬁeld we take AGm to be the neutral component of Z(R). Thus AGm is just R>0 × embedded diagonally in F∞. When F is a function ﬁeld we choose an isomorphism

Z−→˜ Gm

Z and let AGm be the inverse image of t .

2.7. The unramified setting. For a nonarchimedean place v of the global field F let Fv be the residue field of Fv and qv := |Fv|. Let S be a finite set of places of F including the infinite S places. Upon enlarging S if necessary, we can choose a reductive group scheme G over OF ∼ dβ (with connected fibers) and affine spaces Vβ = G S for some integer dβ > 0 (isomorphism aOF S over OF ) equipped with homomorphisms

G −→ Aut(Vβ)

S over OF whose generic fibers are the representations G → Aut(Vβ). By abuse of notation, ′ write G for G. For any of the subgroups M, P, P , etc. of GF we continue to use the same letter for their schematic closures in G. Upon enlarging S if necessary we assume that these groups are all smooth. We assume moreover that the groups whose generic fibers were reductive (resp. parabolic, etc.) extend to reductive group schemes (resp. parabolic group S schemes, etc.) over OF . We also assume that ωP induces an isomorphism on OFv -points for S all v 6∈ S, the highest weight vectors vβ ∈ V (F ) lie in Vβ(OF ), and their image in Vβ(Fv) is again a highest weight vector for GFv of weight −ωβ. Finally, we continue to denote by Y S ′ the schematic closure of Y in G. It is a scheme over OF , and the action of PF on YF extends to an action of P ′ on Y. Under the assumptions above (which are no loss of generality for S large enough) and we are considering the local setting over Fv for v 6∈ S we say that we are in the unramified setting. 20 YOUNGJU CHOIE AND JAYCE R. GETZ

3. The Schwartz space of YP,P ′ (F ) For all but the last subsection of this section F is a local ﬁeld.

3.1. Induced representations. Consider the induced representations:

G ∗ G (3.1.1) IP (χs) := IndP (χs ◦ ωP ) and IP ∗ (χs) := IndP ∗ (χs ◦ ωP ).

χs χs Let Φ be a section of IP (χs). Assume F is archimedean. We say Φ is holomorphic (resp. meromorphic) if s 7→ Φχs (g) is holomorphic as a function of s ∈ C∆P for all g ∈ G(F ) and characters χ : (F ×)∆P → C×. If F is nonarchimedean with residue ﬁeld of cardinality

χs χs sβ −sβ q we say that Φ is holomorphic if Φ (g) ∈ C[{q , q : β ∈ ∆P }] for all g ∈ G(F ) and characters χ : (F ×)∆P → C×. Similarly we say it is meromorphic if for all there is a

sβ −sβ χs p ∈ C[{q , q : β ∈ ∆P }] such that p(s)Φ (g) is holomorphic. Fix a maximal compact subgroup K ≤ G(F ) such that the Iwasawa decomposition holds: G(F )= P (F )K. We then say that Φχs is standard if the restriction of the function (s,g) 7→ Φχs (g) to C∆P × K is ∗ independent of s. We take the analogous conventions regarding sections of IP ∗ (χs). Let E denote the ring of entire functions on C∆P when F is archimedean and C[{qsβ , q−sβ :

× ∆P × β ∈ ∆P }] when F is nonarchimedean. For a ﬁxed character χ : (F ) → C there is

an obvious action of E on the C-vector space of holomorphic sections of IP (χs) preserving the subspace of K-finite sections. As an E-module, the subspace of holomorphic K-finite sections is generated by the subspace of standard K-finite sections. This allows us to apply results in the literature stated for K-finite standard sections to sections that are K-finite and merely holomorphic. We will use this observation without further comment below.

3.2. The Schwartz space. We define P,P ∗ and P ′ as in the introduction. Thus P ∩ P ∗ is the Levi subgroup M of the parabolic subgroups P and P ∗. Moreover P and P ∗ are maximal ′ (proper) parabolic subgroups of P . To define the Fourier transform FP |P ∗ we first apply an intertwining operator to certain functions on YP (F ) to arrive at functions on YP ∗ (F ). We then twist by certain operators that we recall in this section. ab ′ Suppose we are given λ ∈ X∗(M ). Let s ∈ C. We define

′ 0 (3.2.1) λ!(s ): C(YP ∗ (F )) −→ C (YP ∗ (F ))

′ 1/2 −1 s′ λ!(s )(f)(x)= δP ∗ (λ(a))f(λ(a )x)ψ(a)|a| da. × ZF Here ψ : F → C× is the local factor of the global additive character ﬁxed in §2.5 and da is the Haar measure on F. In the special case where P is maximal and P ′ = Y = G this reduces to [GHL21, (4.2)], where it was denoted by λ!(µs). The same operator is denoted by s′ ′ λ!(ηψ ) in [BK02, Sha18]. To extend the domain of deﬁnition of λ!(s ), let Φ ∈ S(F ) satisfy SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 21

∞ Φ(0) = 1 and Φ ∈ Cc (F ). Here Φ(x) := F Φ(y)ψ (xy) dy. Deﬁne the regularized integral

R a ′ b ′ reg b 1/2 −1 s (3.2.2) λ!(s ) (f)(x) := lim Φ δP ∗ (λ(a))f(λ(a )x)ψ(a)|a| da. |b|→∞ F × b Z This limit is said to be well-deﬁned if the integral

a 1/2 −1 s′ Φ δP ∗ (λ(a))f(λ(a )x)ψ(a)|a| da F × b Z ′ reg is convergent provided |b| is large enough and the limit in the deﬁnition of λ!(s ) (x) exists ′ and is independent of Φ. Thus if the integral deﬁning λ!(s ) is absolutely convergent then ′ reg ′ λ!(s ) (f) = λ!(s )(f). In particular this is the case if f ∈ C(YP ∗ (F )). To avoid more proliferation of notation we will drop the superscript reg. Let L(s′, χ), ε(s′, χ, ψ), and ε(s′, χ, ψ)L(1 − s′, χ−1) γ(s′, χ, ψ)= L(s′, χ) be the usual Tate local zeta function, ε-factor, and γ-factor attached to a quasi-character χ : F × → C× and a complex number s′ ∈ C. These factors are denoted ε′(s′, χ, ψ) in [GJ72]. We use a hat to denote the dual group of an F -group (more precisely the neutral component ab ′ 1 c of the Langlands dual). Let N be a 1-dimensional representation of ZM = M with s ∈ 2 Z c c attached to it. The action of ZM is given by a character λ : ZM → Gm, which we may ab c identify with a cocharacter λ : Gm → M . Let

′ ′ aN (χs) := L (−s , χs ◦ λ) and µN (χs) := γ(−s , χs ◦ λ, ψ).

We let

(3.2.3) N be N ∨ (on which M ab acts via −λ) with thee real number −1 − s′ attached to it. ab More generally, assume N = ⊕iNi is a ﬁnite-dimensional representation of M with each c c Ni 1-dimensional. We let the ZM -action on Ni be given by λi and assume each Ni is equipped with a complex number si. Deﬁne

(3.2.4) aN (χs):= aNi (χs), i Y ℓ

(3.2.5) µN (χs):= µNi (χs). i=1 Y

We also deﬁne N := ⊕iNi. We will in fact only consider N = ⊕i∈I Ni where the parameters ∨ attached to Ni are all of the form (si,λi) where λi is an integer multiple of β0 . Here β0 is the simple root of (e2.2.1). Wee will therefore abuse notation and allow ourselves to again write 22 YOUNGJU CHOIE AND JAYCE R. GETZ

∨ λi for the integer ni such that λi = niβ0 . Assume that λi > 0 for all i. Then we can order the Ni so that s s i+1 ≥ i λi+1 λi for all i. Choosing such an ordering we deﬁne

(3.2.6) µN := λ1!(s1) ◦···◦ λℓ!(sℓ).

Theorem 3.3 below explains why it is reasonable to use the symbol µN in both (3.2.5) and (3.2.6).

Let nP be the Lie algebra of the unipotent radical NP of P and let nP be its Langlands dual. Let b (3.2.7) nP |P ∗ := nP /nP ∩ nP ∗ . Let {e, h, f} ⊂ m be a principal sl -triple (here m denotes the Langlands dual of m, the b2 b b b e Lie algebra of m). Consider the subspace nP |P ∗ ≤ nP |P ∗ annihilated by e. It admits a decomposition b b

e b b (3.2.8) nP |P ∗ = ⊕iNi

where the N are 1-dimensional eigenspaces for the action of Zc. i b M ∨ c We observe that ZM acts via a power of β0 on Ni. We assign each Ni the real number si that is half the eigenvalue of h, and deﬁne

−1 (3.2.9) aP |P (χs)= a ((χs) ), aP |P ∗ (χs)= abne (χs). bn^e P |P ∗ P |P ∗ These factors enjoy the symmetry property

(3.2.10) aP |P (χs)= aP ∗|P ∗ (χs), aP |P ∗ (χs)= aP ∗|P (χs)

by the discussion on passing to the opposite parabolic contained in [GHL21, §4.2].

Lemma 3.1. There is an ε > 0 depending only on P and P ′ such that for any character

× ∆P × χ :(F ) → C the function aP |P (χs) is holomorphic and nonzero for

Re(sβ0 ) ≥ −ε.

∨ Proof. Consider the set of parameters {(si,λiβ0 )} attached to nP |P ∗ . It suﬃces to check 1 that for each i one has si ≥ 0 and λi > 0. Since si is 2 the highest weight of a certain sl2 representation with respect to the usual Borel subalgebra hh, ei itb is nonnegative. It is also clear that λi > 0.

′ ′ As above, let M be the unique Levi subgroup of P containing M and deﬁne Mβ0 as in

∗ ∗ (2.2.2). The closed immersion NP ∩ Mβ0 → NP induces a bijection

∗ ∗ ∗ NP (F ) ∩ Mβ0 (F )−→ ˜ NP (F ) ∩ NP (F )\NP (F ). SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 23

Thus the usual unnormalized intertwining operator restricts to deﬁne an operator

∞ ∞ (3.2.11) RP |P ∗ : Cc (YP (F )) −→ C (YP ∗ (F ))

given by

(3.2.12) RP |P ∗ (f)(g) := f(ug)du. N ∗ (F )∩M (F ) Z P β0

We will use the same notation whenever RP |P ∗ (f) is deﬁned (e.g. for more general smooth functions or via analytic continuation). In [Sha18, §4] Shahidi proves that this agrees with the operator deﬁned by Braverman and Kazhdan. χs A section Φ of IP (χs) is good if it is meromorphic and if the section

χs RP |QΦ (g)

aP |Q(χs) is holomorphic for all g ∈ G(F ) and Q ∈ {P,P ∗} (recall our conventions regarding meromorphic sections from §3.1). We deﬁned adelic Mellin transforms in (1.1.9) above. We use the obvious local analogues

of this notation. We write fχs for the Mellin transform of any function f : YP (F ) → C or ◦ f : XP (F ) → C such that the integral defining the Mellin transform is absolutely convergent or obtained by analytic continuation from some region of absolute convergence. Assume F is nonarchimedean. Let K ≤ M ab(F )×G(F ) be a compact open subgroup. ∞ ◦ Let Cβ0 (XP (F )) be the space of K-finite f ∈ C (XP (F )) such that for Re(sβ0 ) sufficiently large the integral defining the Mellin transform fχs converges absolutely and defines a good

section. We deﬁne the Schwartz space of YP,P ′ (F ) to be the space of restrictions to YP (F ) of functions in Cβ0 (XP (F )):

∞ ′ (3.2.13) S(YP,P (F )) = Im(Cβ0 (XP (F )) −→ C (YP (F ))).

Before explaining the archimedean analogue of this deﬁnition, let us write KGm for the × × × maximal compact subgroup of F . Say that two quasi-characters χ1, χ2 : F → C are s equivalent if χ1 = χ2|·| for some s ∈ C. Then the set of equivalence classes of quasi- × characters of F is in natural bijection with KGm . Thus we sometimes write KGm for a set of representatives of the quasi-characters of F × modulo equivalence. In the archimedean setting we ﬁx the following sets of representatives:b b

{1, sgn} if F = R KG := m m {z 7→ z : m ∈ Z} if F = C.  (zz)1/2 b For extended real numbers A, B ∈ {−∞} ∪ R ∪ {∞} with A < B let

∆P (3.2.14) VA,B := {s ∈ C : A< Re(sβ) < B for β ∈ ∆P }. 24 YOUNGJU CHOIE AND JAYCE R. GETZ

For functions φ : C∆P → C∆P and polynomials p on C∆P let

(3.2.15) |φ|A,B,p := sup |φ(s)p(s)| s∈VA,B (which may be infinite). Assume F is archimedean. The action (2.1.5) induces an action ab ∞ ◦ ab of U(m ⊕ g) on C (XP (F )). Here U(m ⊕ g) is the universal enveloping algebra of the complexification of the Lie algebra mab ⊕ g of M ab × G, viewed as a real Lie algebra. ∞ ◦ ab Let Cβ0 (XP (F )) be the set of all f ∈ C (XP (F )) such that for all D ∈ U(m ⊕ g) and × ∆P × each character χ :(F ) → C the integral (1.1.9) defining (D.f)χs converges for Re(sβ0 ) large enough, is a good section, and satisfies the following condition: For all real numbers ∗ A

We observe that it is indeed possible to choose pP |Q(s) as above (independently of η).

This follows directly from the definition of aP |Q(ηs). Since we have defined Cβ0 (XP (F )) for archimedean F we can and do define S(YP,P ′ (F )) as in (3.2.13).

′ In the archimedean case the seminorms |·|A,B,pP |Q,Ω,D give S(YP,P (F )) the structure of a Fr´echet space as we now explain. The seminorms |·|A,B,pP |Q,Ω,D give Cβ0 (XP (F )) the structure of a Fr´echet space via a standard argument. See [GH20, Lemma 3.4] for the proof in a special case, the proof in general is essentially the same. Using Mellin inversion, one

checks that with respect to this Fr´echet structure evaluating a function in Cβ0 (XP (F )) at a ◦ point of XP (F ) is a continuous linear functional on Cβ0 (XP (F )). Thus the C-linear subspace

I ≤ Cβ0 (XP (F )) consisting of functions that vanish on YP (F ) is closed subspace. Restriction of functions to YP (F ) induces a C-linear isomorphism

′ (3.2.16) Cβ0 (XP (F ))/I −→ S(YP,P (F )) and thus we obtain a Fr´echet topology on S(YP,P ′ (F )) by transport of structure. This deﬁnition is inspired by [ES18, §2.2]. The set of seminorms giving S(YP,P ′ (F )) its topology are

|f|A,B,pP |Q,Ω,D := inf{|f|A,B,pP |Q,Ω,D : f ∈ Cβ0 (XP (F )) and f|YP (F ) = f}

for f ∈ S(YP,P ′ (F )). Let H ≤ eG act on G viae right multiplication,e and assume that H ab stabilizes Y. Then (2.1.5) restricts to an action of M (F ) × H(F ) on YP (F ). This induces ab an action of M (F ) × H(F ) on S(YP,P ′ (F )) that is continuous in the archimedean case.

Remarks. ◦ (1) We have deﬁned the Schwartz space in terms of restrictions of functions on XP (F ) in ◦ order to take advantage of the transitive action of G(F ) on XP (F ). SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 25

(2) The space YP,P ′ (F ) plays no role in the definition of S(YP,P ′ (F )). However, it should be possible to give a characterization of S(YP,P ′ (F )) as the space of smooth functions on YP (F ) that have particular germs as one approaches the boundary YP,P ′ (F ) − YP (F )[Ngô20, §5]. As in the special cases treated in [GL21, GH20, GHL21], we have defined the Schwartz space to be the space of smooth functions with sufficiently well-behaved Mellin transforms. This is reasonable because we can obtain information on f from its Mellin transforms via Mellin inversion as in the proof of Lemma 3.2 below.

We now discuss the problem of bounding functions in S(YP,P ′ (F )). Since we have an

embedding PlP : XP → V of XP into an aﬃne space we will phrase our bounds in terms of this aﬃne space. Let K ≤ G(F ) be a maximal compact subgroup such that the Iwasawa decomposition G(F )= P (F )K

holds. If we are in the unramiﬁed setting in the sense of §2.7 we take K = G(OF ). The group K does not act on YP (F ) in general.

The group G(F ) acts on each Vβ(F ). For each β ∈ ∆P choose a K-invariant norm |·|β on the F -vector space Vβ(F ). As a warning, for F = C we have |cv|β = cc|v|β for c ∈ F der and v ∈ Vβ(F ) (this is the “number theorist’s norm”). If we write x = P (F )mk with (m, k) ∈ M ab(F ) × K then by Lemma 2.2 one has

−1 (3.2.17) |PlPβ (x)|β = |PlPβ (m)|β = |ωβ(m)| .

The inverse here appears because G is acting on Vβ on the right. Choose rβ ∈ R so that

rβ 1/2 (3.2.18) |ωβ(m)| = δP (m). βY∈∆P Recall the deﬁnition of V from (2.1.8) and V ◦ from (2.1.13). β0 ∆P ′

Lemma 3.2. Let f ∈ S(YP,P ′ (F )). For suﬃciently small α > 0 there is a nonnegative Schwartz function Φ ∈ S(V (F ) × V ◦ (F )) such that f β0 ∆P ′

α−rβ |f(x)| ≤ Φf (PlP (x)) |PlPβ (x)|β . βY∈∆P If F is archimedean, then Φf can be chosen continuously as a function of f.

Proof. Let log q log q i − π , π if F is non-archimedean IF := iR if F is archimedean.  Let c ∈ R be chosen so that c dx is the standard Haar measure on F , where dx is ψ >0  ψ normalized to be self-dual with respect to ψ. Here the standard Haar measure is the Lebesgue measure if F = R, twice the Lebesgue measure if F = C, and the unique Haar measure giving 26 YOUNGJU CHOIE AND JAYCE R. GETZ

1/2 OF measure |d| if F is nonarchimedean, where d is a generator for the absolute diﬀerent. Then let

cψ log q if F is nonarchimedean

cψ (3.2.19) cF :=  if F = R  2  cψ 2π if F = C  For suitable continuous functions f : YP (F ) → C the Mellin inversion formula states that

|∆P | cF ds (3.2.20) f(x)= fηs (x) |∆ | ∆P (2πi) P σ+IF b ∆ Z η∈K P XGm

∆P for suitable σ ∈ R . By [BB11, §2] this formula holds whenever the integral deﬁning fηs is ∆P absolutely convergent for all η ∈ KGm and Re(s)= σ and

b (3.2.21) |fηs (x)|ds < ∞. ∆P σ+IF b ∆ Z η∈K P XGm

By Lemma 3.1 we have that aP |P (χ) is holomorphic for Re(sβ0 ) ≥ −ε for some ε > 0

independent of the character χ. It follows that, for f ∈ S(YP,P ′ (F )), (3.2.20) holds for σ = (−ε/2,..., −ε/2). Writing x = P der(F )mk with (m, k) ∈ M ab(F ) × K the above becomes

|∆P | 1/2 cF ds (3.2.22) f(mk)= δP (m)ηs(ωP (m))fηs (k) . ∆P |∆P | σ+iI ∆ (2πi) Z F η∈Kb P XGm To obtain the bound and the continuity statement from this expansion one uses the same argument as that proving [GH20, Lemma 3.5].

3.3. The Fourier transform. To ease notation let

be µP := µn ∗ (3.3.1) P |P µ (χ ):= µbe (χ ) P s nP |P ∗ s

where the operator (resp. function) on the right is deﬁned as in (3.2.6) (resp. (3.2.5)). By the same argument proving [GHL21, Theorem 5.9 and Proposition 5.10] we obtain the following theorem:

Theorem 3.3. The map

FP |P ∗ := µP ◦ RP |P ∗ : S(YP,P ′ (F )) −→ S(YP ∗,P ′ (F )) SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 27 is a well-deﬁned isomorphism, bicontinuous in the archimedean case. Moreover the diagram

FP |P ∗ S(YP,P ′ (F )) S(YP ∗,P ′ (F ))

∗ (·)χs (·)χs µ (χ )R ∗ P s P |P ∗ IP (χs) IP ∗ (χs) commutes for all χ : F × → C× and s ∈ C×.

The commutativity of the diagram must be understood in the sense that one has an identity of meromorphic functions

∗ ∗ ∗ FP |P (f)χs = µP (χs)RP |P (fχs ). Let H ≤ G be a subgroup, and consider its action on G via right multiplication. Assume that Y is stable under the action of H. For

ab (m, h, x1, x2) ∈ M (F ) × H(F ) × YP (F ) × YP ∗ (F ) and (f1, f2) ∈ S(YP,P ′ (F )) × S(YP ∗,P ′ (F )) let

−1 −1 (3.3.2) L(m)R(h)f1(x1)= f1(m x1h), L(m)R(h)f2(x2)= f2(m x2h) be the left and right translation operators. It is easy to see that L(m)R(h) preserves

S(YP,P ′ (F )) and S(YP ∗,P ′ (F )).

Lemma 3.4. One has

F ∗ ◦ L(m)R(h)= δ ∗ (m)L(m)R(h) ◦F ∗. P |P P ∩Mβ0 P |P

ab Proof. The operator µP is M (F ) × H(F )-equivariant. Thus the lemma follows from the

deﬁnition (3.2.12) of RP |P ∗ . We have Schwartz spaces

(3.3.3) S(X (F )) and S(X ∗ (F )) P ∩Mβ0 P ∩Mβ0 deﬁned as in [GHL21, §5.2] and a Fourier transform

(3.3.4) F ∗ : S(X (F )) −→ S(X ∗ (F )) P ∩Mβ0 |P ∩Mβ0 P ∩Mβ0 P ∩Mβ0 deﬁned as in [GHL21, Theorem 5.9]. It is an isomorphism, bicontinuous in the archimedean

case. These facts are a special case of our construction of the Schwartz space S(YP,P ′ (F ))

and the Fourier transform FP |P ∗. One simply replaces (3.3.5) (G,P,P ′,Y ) by

(3.3.6) (Mβ0 ,P ∩ Mβ0 , Mβ0 , Mβ0 ).

◦ Recall that for each y ∈ Y (F ) we have a map ιy : X → YP deﬁned as in (2.2.8). P ∩Mβ0 28 YOUNGJU CHOIE AND JAYCE R. GETZ

Proposition 3.5. For each y ∈ Y (F ) one has a map

∗ ι : S(Y ′ (F )) −→ S(X (F )) y P,P P ∩Mβ0

f 7−→ f ◦ ιy that ﬁts into a commutative diagram

FP |P ∗ S(YP,P ′ (F )) S(YP ∗,P ′ (F ))

∗ ∗ ιy ιy FP ∩M |P ∗∩M β0 β0 S(X (F )) S(X ∗ (F )). P ∩Mβ0 P ∩Mβ0

∗ If F is archimedean then ιy is continuous. Proof. We recall that Langlands dual groups are contravariantly functorial with respect to morphisms of reductive algebraic groups G → H with normal image, and behave as expected with respect to Levi and parabolic subgroups. For precise statements see [Bor79, §1]. In particular the commutative diagram

M ∩ Mβ0 P ∩ Mβ0 Mβ0

M P ∩ M ′ M ′ of inclusions of subgroups induces a commutative diagram

\ \ M ∩ Mβ0 P ∩ Mβ0 Mβ0

d M P\∩ M ′ M ′ where the horizontal arrows are inclusions of subgroups. c c Consider the representation

n ∗ = n P ∩Mβ0 |P ∩Mβ0 P ∩Mβ0 of M\∩ M . We regard it as a representation of M via the quotient map M → M\∩ M . β0 b b β0 Choose a principal sl2-triple in m. Its image under the quotient map c c m −→ m\∩ m b β0 is a principal sl -triple in m\∩ m . Using our comments on dual groups at the beginning of 2 β0 b the proof one checks that the quotient map

n ∗ −→ n ∗ P |P P ∩Mβ0 |P ∩Mβ0 is an isomorphism of M-representations that restricts to a bijection b b e e nP |P ∗ −→ nP ∩M |P ∗∩M . c β0 β0

b b SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 29

∗ In view of these observations it is easy to check that the map ιy is well-deﬁned, the diagram ∗ is commutative, and ιy is continuous when F is archimedean.

Corollary 3.6. One has

FP |P ∗ ◦FP ∗|P = Id.

Proof. In [BK02] Braverman and Kazhdan prove that F ∗ ◦F ∗ is the P ∩Mβ0 |P ∩Mβ0 P ∩Mβ0 |P ∩Mβ0 identity. Thus the corollary follows from Proposition 3.5.

3.4. The unramified setting. We assume that F is nonarchimedean and is unramifed over its prime field, that ψ is unramified, and that we are in the unramified setting in the sense of §2.7. Let ∞ ½0 ∈ Cc (YP (F )).

be the characteristic function of the image of Y (OF ) in YP (F ). We deﬁne the basic function

(3.4.1) bYP,P ′ : YP (F ) −→ C

∞ to be the unique function in C (YP (F )) that is ﬁnite under a compact open subgroup of M ab(F ) such that

(bYP,P ′ )χs = aP |P (χs)(½0)χs for Re(s ) suﬃciently large. As explained in (3.3.5) and (3.3.6), the spaces X are β0 P ∩Mβ0 ′ special cases of YP,P , so bXP ∩M is deﬁned. β0 Lemma 3.7. der ∨ × ∗ Assume that y ∈ P (F )β0 (F )Y (OF ). Then ιy(bYP,P ′ )= bXP ∩M . β0

Proof. We have already explained the relation between n ∗ and n ∗ as repre- P ∩Mβ0 |(P ∩Mβ0 ) P |P \ sentations of M and M ∩ Mβ0 in the proof of Proposition 3.5. This relationship implies that a (χ )= a ((χ ) ). b P |P s P ∩Mβ0 |P ∩Mβ0 β0 sβ0 c Deﬁne rβ as in (3.2.18). Arguing as in the proof of Lemma 3.2 we obtain the following lemma:

Lemma 3.8. There are constants α,c > 0 independent of the cardinality of the residue ﬁeld q such that if q>c then

α−rβ |b (x)| ≤ ½ ◦ (Pl (x)) |Pl (x)| . YP,P ′ Vβ (OF )×V∆ (OF ) P Pβ 0 P ′ βY∈∆P

Remark. The claim on the independence of the residual characteristic is important because we will require this result for all but ﬁnitely many places of a global ﬁeld.

Proposition 3.9. ′ ∗ One has bYP,P ′ ∈ S(YP,P (F )). Moreover FP |P (bYP,P ′ )= bYP ∗,P ′ . 30 YOUNGJU CHOIE AND JAYCE R. GETZ

∨ Proof. The characters of ZMc that appear in nP |P ∗ are all of the form λβ0 with λ ∈ Z>0 by the proof of Lemma 3.1. Let b V = n ∗ (λ)= n (λ) λ P |P Mβ0 ∩P be the λβ∨-isotypic space and let 0 b b

rλ : Mβ0 −→ Aut(Vλ) be the corresponding representation. It is irreducible [Sha88, Proposition 4.1] [Lan71]. Let c triv : (F ×)∆P → C× be the trivial character. Recall the comments on the relationship between n ∗ and n from the proof of Proposition 3.5. Let π be the trivial representation P |P P ∩Mβ0

of Mβ0 (F ). The Gindikin-Karpelevic formula implies that b b ∨

∗ L(λsβ0 ,π,rλ ) ½ ∗ ½ (3.4.2) RP |P (( 0)trivs )=( 0)trivs ∨ L(1 + λsβ0 ,π,rλ ) Yλ where the product is over all λ ∈ Z≥1 such that Vλ =6 0 [Sha88, (2.7)] [Lai80, Proposition ∨ 4.6]. Here the L-functions are Langlands L-functions and rλ is the dual of rλ. In more detail,

π determines a Langlands class c ∈ Mβ0 (C) by the Satake isomorphism, and r∨(c) −1 L(s,π,rc∨) = det I − λ . λ Vλ qs 1/2 In fact, if σ : SL → M is a principal SL then c = σ q [Gro98, §7]. 2 β0 2 q−1/2 Consider nP |P ∗ (λ). As a representation of a principal sl2-triple in m it decomposes into c a direct sum of irreducible representations in natural bijection with the Ni in (3.2.8) that eb b appear in nP |P ∗ (λ). The dimension of the corresponding irreducible representation is 2si +1, where 2si is the h-eigenvalue on Ni as above. We conclude that

b ∨ −1−si−λsβ0 −si−λsβ0 −1+si−λsβ0 L(λsβ0 ,π,rλ ) 1 − q 1 − q 1 − q ) ∨ = −s −λs 1−s −λs ··· s −λs L(1 + λs ,π,r ) 1 − q i β0 1 − q i β0 1 − q i β0 β0 λ i Y −1−s −λs 1 − q i β0 = s −λs 1 − q i β0 i Y e where the product is over Ni in (3.2.8) that appear in nP |P ∗ (λ). Thus ∨ ∗ L(λsβ0 ,π,rλ ) aP |P (trivs) (3.4.3) ∨ = b . L(1 + λsβ0 ,π,rλ ) aP |P (trivs) Yλ We deduce for all unramiﬁed χ : F × → C× that

aP |P ∗ (χs) ∗ ½ ∗ ½ (3.4.4) RP |P (( 0)χs )= ( 0)χs . aP |P (χs)

′ It follows immediately that bYP,P ′ ∈ S(YP,P (F )). For unramiﬁed χ −1 aP ∗|P ∗ ((χs) ) (3.4.5) µP (χs)= aP |P ∗ (χs) SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 31 where µP (χs) is deﬁned as in (3.3.1). Hence −1

aP ∗|P ∗ ((χs) ) ∗ ½ ∗ ½ (3.4.6) µP (χs)RP |P (( 0)χs )= ( 0)χs . aP |P (χs) Applying Theorem 3.3 we have

∗ −1 ∗ ½ ∗ ∗ ½ ∗ ∗ (3.4.7) FP |P (bYP,P ′ )χs = aP |P (χs)µP (χs)RP |P (( 0)χs )= aP |P ((χs) )( 0)χs

∗ Combining this with Mellin inversion we have FP |P (bYP,P ′ )= bYP ∗,P ′ . Let ̟ be a uniformizer for F . For our use in the proof of Theorem 1.1 below we require the following lemma:

M(OF )×H(OF ) Lemma 3.10. Let (α,λ) ∈ C × Z6=0. For any f ∈ S(YP,P ′ (F )) α ∨ −λ M(OF )×H(OF ) Id − L(β0 (̟ )) f ∈ S(YP,P ′ (F )) 1/2 ∨ λ δP (β0 (̟ )) ! and

α ∨ −λ λ Id − L(β0 (̟ )) f = (1 − αχβ0 (̟ ))fχs . δ1/2(β∨(̟λ)) P 0 ! !χs

3.5. The adelic setting. Now let F be a global ﬁeld. We let

′ (3.5.1) S(YP,P ′ (AF )) = ⊗v|∞S(YP,P ′ (Fv)) ⊗ ⊗v∤∞S(YP,P ′ (Fv)) where the restricted direct product is taken with respect to the basic functions of (3.4.1). b Here when F is a number ﬁeld the hat denotes the projective topological tensor product and when F is a function ﬁeld it is the algebraic tensor product. The tensor product of the local Fourier transforms induces an isomorphism

(3.5.2) FP |P ∗ : S(YP,P ′ (AF )) −→ S(YP ∗,P ′ (AF )). Here we are using Proposition 3.9. The following is the global analogue of Proposition 3.5:

Proposition 3.11. For each y ∈ Y (AF ) one has a map

∗ ′ ιy : S(YP,P (AF )) −→ S(XP ∩M0 (AF ))

f 7−→ f ◦ ιy that ﬁts into a commutative diagram

FP |P ∗ S(YP,P ′ (AF )) S(YP ∗,P ′ (AF ))

∗ ∗ ιy ιy FP ∩M |P ∗∩M β0 β0 S(X (A )) S(X ∗ (A )). P ∩Mβ0 F P ∩Mβ0 F 32 YOUNGJU CHOIE AND JAYCE R. GETZ

Proof. Let K = v Kv ≤ G(AF ) be a maximal compact subgroup. The element y =(yv) ∈ Y (A ) has the property that y ∈ K for almost all v. Thus the proposition follows from P F Q v v its local analogue Proposition 3.5 and the corresponding statement for basic functions in Lemma 3.7.

′ ∗ ′ Lemma 3.12. For f ∈ S(YP,P (AF )) (resp. f ∈ S(YP ,P (AF )) the integrals defining fχs ∗ (resp. fχs ) converge absolutely for Re(sβ0 ) sufficiently large (resp. sufficiently small).

Proof. This follows from the estimates in lemmas 3.2 and 3.8.

Lemma 3.12 implies that the Mellin transforms (1.1.9) deﬁne maps

′ (·)χs := (·)χs,P : S(YP,P (AF )) −→ IP (χs)|YP (AF ) ∗ ∗ ∗ ∗ ′ (·)χs := (·)χs,P : S(YP ,P (AF )) −→ IP (χs)|YP (AF )

for Re(sβ) suﬃciently large (resp. suﬃciently small). These Mellin transforms will be used in the following sections.

4. The Poisson summation formula on X (F ) P ∩Mβ0 The Poisson summation formula on X (F ) was established under some local assump- P ∩Mβ0 tions on the test functions involved in [BK02] with a slightly diﬀerent deﬁnition of the Schwartz space. In this section we establish it in general following the arguments of [GL21].

To ease notation, for this section only we assume that P ≤ Mβ0 , which implies Mβ0 = G. This amounts to assuming that G is simple and P is a maximal parabolic subgroup. Thus

X = X = X P ∩Mβ0 P P,PIG where I ∈ G(F ) is the identity. The construction of the Schwartz space and the Fourier transform given in the previous section reduces to the construction of [GHL21] in this case.

We observe that ∆P = {β0} in the setting of this section. Thus we drop ∆P and β0 from notation when no confusion is likely. × × × ∗ For a quasi-character χ : AGm F \AF → C ,s ∈ C, f1 ∈ S(XP (AF )) and f2 ∈ S(XP (AF )) we have degenerate Eisenstein series

E(g, f1χs ):= f1χs (γg) γ∈P (F )\G(F ) (4.0.1) X ∗ ∗ ∗ E (g, f2χs ):= f2χs (γg). ∗ γ∈P X(F )\G(F ) s Here χs = χ|·| . It is well-known that these converge absolutely for Re(s) large enough (resp. small enough). For the proof of absolute convergence in a special case see [GL21, Lemma 6.5]; the proof generalizes to our setting. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 33

Let

(4.0.2) aP |P (χs) := aP |P (χvs). v Y \× × Lemma 4.1. Let χ ∈ AGm F \AF . The function aP |P (χs) is holomorphic and nonzero for Re(s) > 0. It admits a meromorphic continuation to the plane. Moreover there is an integer n n depending only on G such that aP |P (χs) is holomorphic if χ =16 .

Proof. The ﬁrst claim follows from the same remarks proving Lemma 3.1. Since aP |P (χs) is a product of (completed) Hecke L-functions the second two assertions are clear.

Theorem 4.2 (Langlands). Let f ∈ S(XP (AF )) be ﬁnite under a maximal compact subgroup × × × of G(AF ). For a character χ : AGm F \AF → C the Eisenstein series E(g, fχs ) has a meromorphic continuation to the s-plane and admits a functional equation

∗ ∗ ∗ (4.0.3) E(g, fχs )= E (g, FP |P (f)χs ).

If Re(z)=0 then the order of the pole of E(g, fχs ) at s = z is bounded by the order of the

pole of aP |P (χs) at s = z.

Proof. To ease translation with the manner the theory is usually phrased, let Q be the unique ∗ parabolic subgroup containing B that is conjugate to P , and let MQ be the Levi subgroup ∗ −1 −1 of P containing T. Choose w ∈ G(F ) so that wP w = Q and wMw = MQ. There is an isomorphism

∞ ◦ ∞ ◦ j : C (XP ∗ (F ))−→ ˜ C (XQ(F )) f7−→˜ (x 7→ f(w−1x)).

χs For suitable sections Φ ∈ IQ(χs) and g ∈ G(AF ) we can then form the Eisenstein series

χs χs (4.0.4) EQ(g, Φ )= Φ (γg). γ∈Q(XF )\G(F ) χs Let K be a maximal compact subgroup of G(AF ). If Φ is K-finite and holomorphic for χs Re(s) sufficiently large, then EQ(g, Φ ) is absolutely convergent for Re(s) sufficiently large.

We observe that for f ∈ S(XP (AF )) one has

∗ ∗ ∗ (4.0.5) E (g, fχs )= EQ(g, j(fχs )) for Re(s) suﬃciently small.

By deﬁnition of the Schwartz space, for any f ∈ S(XP (AF )) the section fχs is a holomor-

phic multiple of the product of completed Hecke L-functions aP |P (χs). With this in mind, the meromorphy assertion is proven in [BL19, Lan76] for K-ﬁnite functions f. These references also contain a proof of the functional equation

∗ E(g, fχs )= EQ(g, j RP |P (fχs ) ) 34 YOUNGJU CHOIE AND JAYCE R. GETZ

which implies by (4.0.5) that

∗ ∗ (4.0.6) E(g, fχs )= E (g, RP |P (fχs )).

We have

∗ ∗ ∗ (4.0.7) RP |P (fχs )=(FP |P (f))χs by Theorem 3.3 and the argument of [GL21, Lemma 6.2]. Thus the functional equation stated in the theorem follows from (4.0.6).

As mentioned above for any f ∈ S(XP (AF )) the Mellin transform fχs is a holomorphic multiple of aP |P (χs). Thus the last assertion of the theorem follows from the fact that Eisenstein series attached to K-ﬁnite holomorphic sections are themselves holomorphic on the unitary axis [Art05, Theorem 7.2], [Lan76].

Let C(χs) be the analytic conductor of χs, normalized as in [GL21, (5.7)]. Using notation as in (3.2.14) and (3.2.15) one has the following estimate:

Theorem 4.3. Assume that F is a number ﬁeld, that Conjecture 1.3 is true, and f ∈

S(XP (AF )) is K-ﬁnite. Let A < B be real numbers and let p ∈ C[x] be a polynomial such

that p(s)E(g, χs) is holomorphic in the strip VA,B. Then for all N ≥ 0 one has

−N |E(g, fχs )|A,B,p ≪N,f C(χs) .

Proof. The argument is the same as that proving [GL21, Theorem 6.3].

ab Let KM < M (AF ) be the maximal compact subgroup and let

r r 2 1 (2π) 2 hF RF 1/2 if F is a number ﬁeld dF eF (4.0.8) κF :=  hF 1/2 if F is a function ﬁeld  dF (q−1) log q

where r1 and r2 are the number of real (resp. complex) places, hF is the class number, RF is

the regulator, dF is the absolute discriminant, eF is the number of roots of unity in F , and

in the function ﬁeld case F has ﬁeld of constants Fq. For complex numbers s0 let

1 2 if Re(s0)=0, (4.0.9) w(s0)= 1 otherwise.  We now prove the following special case of Theorem 1.2:

Theorem 4.4. Let f ∈ S(XP (AF )). Assume that (1) F is a function ﬁeld,

(2) F is a number field, Conjecture 1.3 is valid, and f is KM -finite, or (3) F is a number field and Conjecture 1.4 is valid. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 35

One has

w(s0) ∗ ∗ f(x)+ Res E (I, F ∗(f) ) κ s=s0 P |P χ−s ◦ χ F x∈XP (F ) s0∈C X X Re(Xs0)≥0

∗ w(s0) ∗ = FP |P (f)(x )+ Ress=s0 E(I, fχs ). κF ∗ ◦ χ s ∈C x ∈XP ∗ (F ) 0 X X Re(Xs0)≥0

× × ∗ Here the sum on χ is over characters of AGm F \AF . The sums over x and x are absolutely convergent.

It is clear that if f is KM -ﬁnite and Conjecture 1.3 is valid then the double sum over χ and

s0 has ﬁnite support. The same is true if F is a function ﬁeld or Conjecture 1.4 is valid. This is how we are using these assumptions here and below.

Remark. Before proving Theorem 4.4 we clarify its meaning. Assume F is a number ﬁeld and ∞ ∞ Conjecture 1.3 is valid. For any s0 ∈ C and f ∈ S(XP (AF )) consider the linear functional

∞ (4.0.10) f∞ 7−→ Ress=s0 E(I, (f∞f )χs ).

It is deﬁned on the dense subspace of S(XP (F∞)) consisting of K∞-ﬁnite functions. The proof of the theorem shows that this linear functional is continuous with respect to the

Fr´echet topology on S(XP (F∞)). Hence it extends to all of S(XP (F∞)). For f∞ that are not

K∞-ﬁnite this is the manner the expression Ress=s0 E(I, fχs ) is to be interpreted in this paper. ∗ ′∗ We take the obvious analogous conventions regarding the meaning of Ress=s0 E (I, fχs ) for ′ f ∈ S(XP ∗ (AF )) that are not K∞-ﬁnite.

Proof of Theorem 4.4. Let K∞ ≤ G(F∞) be a maximal compact subgroup. Assume ﬁrst ∞ that f = f∞f where f∞ ∈ S(XP (F∞)) is K∞-ﬁnite. Let

π π i − log q , log q if F is a function field, and IF := iRh i if F is a number field.  By Mellin inversion and Theorem 4.3 in the number field case ds (4.0.11) f(x)= E(I, f ) χs c2πi x∈X◦ (F ) χ σ+IF XP X Z for σ sufficiently large and a suitable constant c. Here the sum over χ is as in the statement

of the theorem. Since f is K∞-ﬁnite, the support of the sum over χ is ﬁnite. The constant c may be computed as follows. It depends on our choice of Haar measure on [M ab], which is × × normalized to correspond to the measure on AGm F \AF described in §2.5. This is induced as explained in §2.5 from a measure on AF that is self-dual with respect to ψ. The induced 36 YOUNGJU CHOIE AND JAYCE R. GETZ

measure on F \AF is independent of the choice of ψ. Thus using [Wei74, §VII.6, Proposition 12] and a choice of self-dual measure one obtains

r r 2 1 (2π) 2 hF RF 1/2 if F is a number ﬁeld × × dF eF meas(AGm F \AF )=  hF 1/2 if F is a function ﬁeld.  dF (q−1)

meas(A F ×\A×)  Gm F This implies c = κF . Notice that in the function field case κF = log q because of the measure of IF . We shift contours to Re(σ) very small to see that this is ds 1 (4.0.12) f(x)= E(I, fχs ) + Ress=s0 E(I, fχs ) ′ κF 2πi κF x∈X◦ (F ) χ Re(s)=σ χ s ∈C XP X Z X X0 ′ where now σ is sufficiently small. Here the support of the sum over χ and s0 is finite by

assumptions (2) or (3) in the number field case and the fact that E(I, fχs ) is rational in the sense of [MW95, IV.1.5] in the function field case [MW95, Proposition IV.1.12]. In the number field case the bound required to justify the contour shift is provided by Theorem 4.3. We now apply the functional equation of Theorem 4.2 and Mellin inversion to deduce the identity

∗ 1 ∗ (4.0.13) f(x)= FP |P (f)(x )+ Ress=s0 E(I, fχs ). κF x∈X◦ (F ) x∗∈X◦ (F ) χ s ∈C XP XP ∗ X X0

Since all elements of the Schwartz space S(XP (AF )) are K∞-finite in the function field case ab assume for the moment that F is a number field. Write KM∞ = KM ∩ M (F∞). We assume ∞ ∞ ∞ without loss of generality that f = f∞f where f∞ ∈ S(XP (F∞)), f ∈ S(XP (AF )). We additionally either assume (2) and that f∞ transforms according to a particular character η

under KM∞, or we assume (3). Since K∞-ﬁnite functions are dense in S(XP (F∞)) by the

standard argument [War72, §4.4.3.1] we can choose a sequence {fi : i ∈ Z≥1} ⊂ S(XP (F∞))

of K∞-ﬁnite fi such that fi → f in the Frech´et topology on S(XP (F∞)). Under assumption

(2) we additionally assume that the fi transform under KM∞ by η. We observe that the ∞ support of the sum over χ and s0 in (4.0.13) and its analogues when f is replaced by fif are contained in a finite set independent of i under either assumption (2) or (3). In fact, the ∞ ∞ finite set can be taken to depend only on the KM -type of f . ∞ ∞ It is clear from Lemma 3.2 that for each fixed f ∈ S(XP (AF )) the map

Λ1,f ∞ : S(XP (F∞)) −→ C

∞ f∞ 7−→ f∞(x)f (x) x∈X◦ (F ) XP

is continuous on S(XP (F∞)). The same is true of

Λ2,f ∞ : S(XP (F∞)) −→ C SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 37

∗ ∞ ∗ f∞ 7−→ FP |P ∗(f∞)(x )FP |P ∗(f )(x ) x∗∈X◦ (F ) XP ∗

since the Fourier transform is continuous. Thus Λj,f ∞ (fi) → Λj,f ∞ (f) for as i → ∞ for j ∈{1, 2}. Finally consider

∞ Λf ,s0 : S(XP (F∞)) −→ C

f∞ 7−→ Ress=s0 E(I, fχs ).

′∞ ∞ Using Lemma 3.10 we can choose an f ∈ S(XP (AF )) such that

∞ ′∞ ′∞ Λf ,s0 =Λ1,f − Λ2,f .

′∞ In more detail one uses Lemma 3.10 to choose f so that the contribution of Ress=s0 E(I, fχs ) to (4.0.13) is unchanged, but the contribution of the other residues vanish.

∞ Thus Λf ,s0 is continuous. We deduce that (4.0.13) is valid for all KM -ﬁnite f under assumption (2) and for all f under assumption (3). We now return to the case of a general global ﬁeld. To obtain the expression in the theorem from (4.0.13) we use the functional equation and holomorphy assertion of Theorem 4.2 and the observeration that for any meromorphic function f on C and any s0 ∈ C one has

(4.0.14) Ress=s0 f(s)= −Ress=−s0 f(−s).

\× × Lemma 4.5. Let χ ∈ AGm F \AF . Let n be the maximal order of the pole at s0 of E(I, fχs ) ∞ ∞ as f ranges over K∞-ﬁnite elements of S(XP (F∞)). For each f ∈ S(XP (AF )) there are continuous linear functionals

∞ Λi((·)f ): S(XP (F∞)) −→ C such that if f∞ is K∞-ﬁnite then

n−1 −1/2 ∨ −s0 i ∞ ∨ ∞ (4.0.15) δP (β0 (t)) |t| (log |t|) χ(t)Λi(f∞f ) = Ress=s0 E(I, (L(β0 (t))f∞f )χs ). i=0 X As usual, when F is a function ﬁeld we give S(XP (F∞)) the discrete topology.

j 1 ∞ (−(s−s0) log |t|) Proof. We have |t|s = j=0 j!|t|s0 for s in a neighborhood of s0. On the other hand if f is K -ﬁnite then we can write ∞ ∞ P n−1 (−1)ii!Λ (f f ∞) E(I, (f f ∞) )= i ∞ + g(s) ∞ χs (s − s )i+1 i=0 0 X where g(s) is holomorphic in a neighborhood of s0 and Λi(f∞f∞) ∈ C. Then the expression

(4.0.15) is valid for K∞-ﬁnite f∞. 38 YOUNGJU CHOIE AND JAYCE R. GETZ

∞ The fact that Λi((·)f ) extends to a continuous functional on all of S(XP (F∞)) is tau- tological in the function ﬁeld case. In the number ﬁeld case it follows from a variant of the

∞ proof of the continuity of the functional Λf ,s0 in the proof of Theorem 4.4.

5. Proofs of Theorem 1.1 and Theorem 1.2

For f ∈ S(YP,P ′ (AF )) consider the sum

(5.0.1) f(y). y∈XYP (F ) Lemma 5.1. The sum (5.0.1) is absolutely convergent.

Proof. Let |·|β = v |·|β,v, where |·|β,v is the norm on Vβ(Fv) ﬁxed above (3.2.17). Using lemmas 3.2 and 3.8 and the notation in these lemmas we see that for α> 0 suﬃciently small Q ◦ there is a Schwartz function on Φ ∈ S(Vβ (AF ) × V (AF )) such that the sum is dominated 0 ∆P ′ by

α−rβ (5.0.2) Φ(ξ) |ξ|β < ∞ ◦ ξ∈XV (F ) βY∈∆P where V ◦ is deﬁned as in (2.1.9).

Recall the function w(s0) from (4.0.9). The following theorem is the precise version of Theorem 1.2:

Theorem 5.2. Let f ∈ S(YP,P ′ (AF )). Assume (1) F is a function ﬁeld,

(2) F is a number field, Conjecture 1.3 is valid, and f is KM -finite, or (3) F is a number field and Conjecture 1.4 is valid. One has

w(s0) ∗ ∗ ∗ ∗ f(x)+ Ress=s0 E (I, ιy(FP |P (f))χ−s ) ′der κF x∈YP (F ) y∈P (F )\Y (F ) χ s0∈C X X X Re(Xs0)≥0

∗ w(s0) ∗ = F ∗(f)(x )+ Res E(I, ι (f) ) P |P κ s=s0 y χs ∗ ′der F x ∈YP ∗ (F ) y∈P (F )\Y (F ) χ s0∈C X X X Re(Xs0)≥0

× × where the sums on χ are over all characters of AGm F \AF . Moreover the sums over x and ∗ x and the triple sums over (y,χ,s0) are absolutely convergent.

Remark. We point out that conjectures 1.3 and 1.4 are assertions about functions in S(X (A )), P ∩Mβ0 F whereas f lies in S(YP,P ′ (AF )). SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 39

Proof. We use Lemma 2.7 to write

∗ f(x)= f(ιy(γ)) = ιy(f)(γ). (5.0.3) ◦ ◦ x∈YP (F ) y γ∈XP ∩M (F ) y γ∈XP ∩M (F ) X X Xβ0 X Xβ0 Here and throughout the proof all sums over y are over P ′der(F )\Y (F ). By Proposition 3.11 we have ι∗(f) ∈ S(X (A )). There is a natural map (M ∩ M )ab → M ab and hence y P ∩Mβ0 F β0 ab ab ′ (M ∩ Mβ0 ) (AF ) acts on S(YP,P (AF )). With respect to this action ιy is (M ∩ Mβ0 ) (F )-

equivariant. In particular if (2) is valid the assumption that f ∈ S(YP,P ′ (AF )) is KM -ﬁnite ab implies ιy(f) is ﬁnite under the maximal compact subgroup of (M ∩ Mβ0 ) (AF ). Hence applying Theorem 4.4 we see that the above is

w(s0) ∗ ∗ ∗ − Res E (I, (F ∗ (ι (f))) ) s=s0 P ∩Mβ0 |P ∩Mβ0 y χ−s κF y χ s0∈C ! X X Re(Xs0)≥0

∗ w(s0) ∗ + F ∗ (ι (f))(γ)+ Res E(I, ι (f) ) . P ∩Mβ0 |P ∩Mβ0 y s=s0 y χs ◦ κF y γ∈X ∗ (F ) χ s0∈C ! P ∩Mβ X X 0 X Re(Xs0)≥0

By Proposition 3.11 this is

w(s0) ∗ ∗ ∗ ∗ − Ress=s0 E (I, ιy(FP |P (f))χ−s ) κF y χ s0∈C ! X X Re(Xs0)≥0

∗ w(s0) ∗ ∗ + ιy(FP |P (f))(γ)+ Ress=s0 E(I, ιy(f)χs ) . ◦ κF y γ∈X ∗ (F ) χ s0∈C ! P ∩Mβ X X 0 X Re(Xs0)≥0

By Lemma 2.7

∗ ∗ (5.0.4) ιy(FP |P ∗(f))(γ)= FP |P ∗(f)(x ). y ◦ ∗ ∗ γ∈XP ∗∩M (F ) x ∈YP (F ) X Xβ0 X This completes the proof of the identity in the theorem. For the absolute convergence statement in the theorem, we observe that (5.0.3) and (5.0.4) are absolutely convergent by Lemma 5.1. By the argument above and the functional equation in Theorem 4.2 we see that

f(ιy(γ)) ◦ γ∈XP ∩M (F ) Xβ0

∗ 1 ∗ = F ∗ (ι (f))(γ)+ Res E(I, ι (f) ) P ∩Mβ0 |P ∩Mβ0 y s=s0 y χs κF ◦ χ s ∈C γ∈XP ∗∩M (F ) 0 Xβ0 X X 40 YOUNGJU CHOIE AND JAYCE R. GETZ

for all y. We deduce that the sum over y in 1 Res E(I, ι∗(f) ) κ s=s0 y χs F y χ ! X X sX0∈C is absolutely convergent as well. The support of the sum over χ is ﬁnite and under assumptions (1) or (2) it depends only on ∞ the KM -type of f. Under assumption (3) it depends only on the KM ∩M(AF )-type of f. With ′ this in mind, for any ﬁxed s0 ∈ C we can use Lemma 3.10 to choose an f ∈ S(YP,P ′ (AF ))

with the same KM -type as f so that

∗ ′ ∗ (5.0.5) Ress=s0 E(I, ιy(f )χs ) = Ress=s0 E(I, ιy(f)χs ). y χ ! y X X sX0∈C X We deduce that the right hand side of (5.0.5) is absolutely convergent, and the same is true

if we replace f by FP |P ∗ (f) by Proposition 3.5 and Theorem 4.2.

Let (f1, f2) ∈ S(YP,P ′ (AF )) × S(YP ∗,P ′ (AF )). Recall the deﬁnition of the generalized Schu- bert Eisenstein series E (f ) and E∗ (f ∗ ) of (1.1.10). Using lemmas 3.2 and 3.8 and YP 1χs YP ∗ 2χs standard arguments one obtains the following lemma:

Lemma 5.3. For f1 ∈ S(YP,P ′ (AF )) one has

1/2 −1 δP (m)|χs(ωP (m))f1(m x)|dm < ∞ ab ab M (AF ) x∈M X(F )\YP (F ) Z

for Re(sβ0 ) sufficiently large. In particular, EYP (f1χs ) converges absolutely for Re(sβ0 ) suf- ∗ ∗ ficiently large. Similarly, for f ∈ S(Y ∗ ′ (A )) the series E (f ) converges absolutely 2 P ,P F YP ∗ 2χs for Re(sβ0 ) sufficiently small.

With notation as in Lemma 2.2, let

β0 ab (5.0.6) A := ker ωβ0 : M −→ Gm

ab ∨ β0 ′ Thus M is the internal direct sum of β0 (Gm) and A by Lemma 2.2. Recall that M is the unique Levi subgroup of P ′ containing M. Applying Lemma 2.2 with P replaced by P ′ we see that the canonical map M ab → M ′der restricts to induce an isomorphism

(5.0.7) Aβ0 −→˜ M ′ab.

× β0 We have a Haar measure on AF and we endow A (AF ) by declaring that the product β0 ∨ × ab measure on A (AF )β0 (AF ) is our Haar measure on M (AF ). As usual let × 1 × × + × × − × (AF ) := {t ∈ AF : |t| =1}, (AF ) := {t ∈ AF : |t| > 1}, (AF ) := {t ∈ AF : |t| < 1}. For algebraic F -groups G we set

[G] := G(F )\G(AF ) SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 41 and deﬁne

1 × × 1 ± × × ± [Gm] := F \(AF ) , [Gm] := F \(AF ) .

× 1 We endow (AF ) with the unique measure so that

× × 1 |·| : AF /(AF ) −→˜ Im(|·|) ⊆ R>0 is measure preserving. Here in the number ﬁeld case Im(|·|) = R>0 with its usual Haar dt measure t (where dt the Lebesgue measure on R). In the function ﬁeld case the image is discrete and endowed with the counting measure. The following technical lemma will be used in the proof of Theorem 1.1:

Lemma 5.4. Let f ∈ S(YP,P ′ (AF )). Assume (1) F is a function ﬁeld,

(2) F is a number field, Conjecture 1.3 is valid and f is KM -finite, or (3) F is a number field and Conjecture 1.4 is valid.

\× × ∆P ′ \× × For every χ ∈ (AGm F \AF ) , χ ∈ AGm F \AF and s0 ∈ C

1/2 ∨ ∗ ∨ × ′ (5.0.8) δP (β0 (t)m)|χs(tωP (m))Resz=s0 E(I, ιy(L(β0 (t)m)f)χz )|dmd t 1 β y [Gm] ×A 0 (AF ) X Z

is ﬁnite. Moreover for Re(sβ0 ) suﬃciently large

1/2 ∨ ∗ ∗ ∨ ∗ × (5.0.9) δ (β (t)m)|χ (tω (m))Res E (I, ι (L(β (t)m)f) ′ )|dmd t P 0 s P z=s0 y 0 χz − β y [Gm] ×A 0 (AF ) X Z converges. The expression

1/2 ∨ ∗ ∨ × ′ (5.0.10) δP (β0 (t)m)χs(tωP (m))Resz=s0 E(I, ιy(L(β0 (t)m)f)χz )dmd t − β y [Gm] ×A 0 (AF ) X Z ∆P originally deﬁned for Re(sβ0 ) large, admits a meromorphic continuation to s ∈ C . Here all sums over y are over P ′(F )\Y (F ).

1 Proof. The group [Gm] is compact. Using this observation and the argument proving Lemma

5.1 we see that for all f1 ∈ S(YP,P ′ (AF )) and f2 ∈ S(YP ∗,P ′ (AF )) the integrals

1/2 ∨ ∨ −1 × δP (β0 (t)m) |f1((β0 (t)m) x)|dmd t 1 β [Gm] ×[A 0 ] Z x∈YP (F ) (5.0.11) X 1/2 ∨ ∨ −1 ∗ × δP (β0 (t)m) |f2((β0 (t)m) x )|dmd t, 1 β [Gm] ×[A 0 ] ∗ Z x ∈XYP ∗ (F ) are convergent. ∞ ∞ × β0 Let (f∞, f ) ∈ S(YP,P ′ (F∞))×S(YP,P ′ (AF )), let (t, m) ∈ AF ×T (AF ), and let y ∈ Y (F ). ′ \× × Arguing as in the proof of Theorem 5.2 for each χ ∈ AGm F \AF and s0 ∈ C there is an 42 YOUNGJU CHOIE AND JAYCE R. GETZ

′∞ ∞ f ∈ S(YP,P ′ (AF )) such that (5.0.12) ′∞ ∨ −1 ∗ ∨ ′∞ ∗ ι (f f )((β (t)m) x) − F ∗ ι (L(β (t)m)(f f ))(x ) y ∞ 0 P ∩Mβ0 |P ∩Mβ0 y 0 ∞ ∗ x∈XP ∩M (F ) x ∈XP ∗∩M (F ) Xβ0 X β0 ∗ ∨ ∞ ′ = Resz=s0 E(I, ιy(L(β0 (t)m)(f∞f ))χz )

1/2 ∨ ′der Multiplying both sides of (5.0.12) by δP (β0 (t)m)χs(tωP (m)), summing over y ∈ P (F )\G(F ) 1 β0 and then integrating over [Gm] × [A ] we arrive at (5.0.13)

1/2 ∨ ′∞ ∨ −1 δP (β0 (t)m)χs(tωP (m)) ιy(f∞f )((β0 (t)m) x) 1 β [Gm] ×[A 0 ] Z y x∈XP ∩M (F ) X Xβ0

∗ ∨ ′∞ ∗ × − F ∗ ι (L(β (t)m)(f f ))(x ) dmd t P ∩Mβ0 |P ∩Mβ0 y 0 ∞ ∗ x ∈XP ∗∩M (F ) ! X β0 1/2 ∨ ∗ ∨ ∞ × ′ = δP (β0 (t)m))χs(tωP (m)) Resz=s0 E(I, ιy(L(β0 (t)m)(f∞f ))χz )dmd t 1 β [Gm] ×[A 0 ] y Z X To prove that the bottom sum and integral in (5.0.13) converge absolutely it suﬃces to prove that the top sum and integral converge absolutely. Using Lemma 2.7 and Proposition 3.11 this reduces to the assertion that the integrals in (5.0.11) are convergent. Hence (5.0.14)

1/2 ∨ ∗ ∨ ∞ × ′ δP (β0 (t)m))|χs(tωP (m))Resz=s0 E(I, ιy(L(β0 (t)m)(f∞f ))χz )|dmd t ′der Z y∈P X(F )\Y (F ) 1 β0 converges, where the integral is over [Gm] × [A ]. Using (5.0.7) we see that (5.0.14) unfolds to (5.0.8). This completes the proof of the ﬁrst assertion of the lemma. ′ ′ Let n be the maximal order of the pole of E(I, f ′ ) at z = s as f ∈ S(X (A )) χz 0 P ∩Mβ0 F ∗ varies. Recall the continuity assertion for ιy contained in Proposition 3.5. Using the ﬁrst absolute convergence assertion of the lemma and Lemma 4.5 there are continuous linear functionals

∞ (5.0.15) Λi,y((·)f ): S(YP,P ′ (F∞)) −→ C

for 0 ≤ i ≤ n − 1 such that

1/2 ∨ ∗ ∗ ∨ ∗ δ (β (t)m)χ (tω (m))Res E (I, ι (L(β (t)m)f) ′ )dm P 0 s P z=s0 y 0 χz β ′ A 0 (AF ) y∈P (XF )\Y (F ) Z (5.0.16) n−1 −s0 i ∞ = χs(tωP (m))|t| (log |t|) χ(t)Λi,y(L(m)(f∞f )) dm β ′ A 0 (AF ) i=0 ! y∈P (XF )\Y (F ) Z X SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 43

× for all t ∈ AF . Here when F is a function ﬁeld we give S(YP,P ′ (F∞) the discrete topology. Varying f ∞ and using Lemma 3.10 we deduce that

∞ (5.0.17) |χs(ωP (m)Λi,y(L(m)(f∞f ))|dm β ′ A 0 (AF ) y∈P (XF )\Y (F ) Z

is convergent for each i. Using this fact we see that for Re(sβ0 ) suﬃciently large the integral − over t ∈ [Gm] of (5.0.16) is (5.0.18) n−1 −s0 i × ∞ χs(tωP (m))|t| (log |t|) χ(t)d tΛi,y(L(m)(f∞f )) dm; β − ′ A 0 (AF ) [Gm] i=0 ! y∈P (XF )\Y (F ) Z Z X in other words, it is permissible to bring the integral over t inside the other integral and the sum. The integrals over t may be evaluated in an elementary manner, yielding the remaining assertions of the lemma.

Proof of Theorem 1.1. Assume for the moment that f is KM -ﬁnite and that Re(sβ0 ) is large. Then

1/2 −1 EYP (fχs )= δP (m)χs(ωP (m)) f(m x)dm [M ab] Z x∈XYP (F ) 1/2 ∨ ∨ −1 × = δP (β0 (t)m)χs(tωP (m)) f((β0 (t)m) x)dmd t + β [Gm] ×[A 0 ] Z x∈XYP (F ) 1/2 ∨ ∨ −1 × + δP (β0 (t)m)χs(tωP (m)) f((β0 (t)m) x)dmd t 1 β [Gm] ×[A 0 ] Z x∈XYP (F ) 1/2 ∨ ∨ −1 × + δP (β0 (t)m)χs(tωP (m)) f((β0 (t)m) x)dmd t. − β [Gm] ×[A 0 ] Z x∈XYP (F ) × 1 Here d t is the measure on [Gm]. The subgroup [Gm] < [Gm] has nonzero measure with respect to d×t if and only if F is a function ﬁeld. By Lemma 3.4 one has

(5.0.19) FP |P ∗ ◦ L(m)= δP ∗∩M β0 (m)L(m) ◦FP |P ∗.

1/2 1/2 Moreover δP (m)δP ∗∩M (m)= δP ∗ (m). With this in mind we apply the Poisson summation 1 formula of Theorem 5.2 to 2 the second integral and the third integral above to see that

EYP (fχs ) is the sum of (5.0.20) and (5.0.21) below:

1/2 ∨ ∨ −1 × (5.0.20) δP (β0 (t)m)χs(tωP (m)) f((β0 (t)m) x)dmd t + β [Gm] ×[A 0 ] Z x∈XYP (F ) 1 1/2 ∨ ∨ −1 × + δP (β0 (t)m)χs(tωP (m)) f((β0 (t)m) x)dmd t 1 β 2 [Gm] ×[A 0 ] Z x∈XYP (F ) 44 YOUNGJU CHOIE AND JAYCE R. GETZ

1 1/2 ∨ ∨ −1 × + δP ∗ (β0 (t)m)χs(tωP (m)) FP |P ∗(f)((β0 (t)m) x)dmd t 1 β 2 [Gm] ×[A 0 ] Z x∈YXP ∗ (F ) 1/2 ∨ ∨ −1 × + δP ∗ (β0 (t)m)χs(tωP (m)) FP |P ∗(f)((β0 (t)m) x)dmd t − β [Gm] ×[A 0 ] Z x∈YXP ∗ (F ) and (5.0.21)

1 w(s0) 1/2 ∨ ∗ ∨ ′ χs(tωP (m)) δP (β0 (t)m)Resz=s0 E(I, ιy(L(β0 (t)m)f)χz ) 1 β 2 [Gm] ×[A 0 ] κF ′ Z s0∈C y,χ Re(Xs0)≥0 X

1/2 ∨ ∗ ∗ ∨ ∗ × − δ ∗ (β (t)m)Resz=s E (I, ι (L(β (t)m)FP |P ∗ (f)) ′ ) dmd t P 0 0 y 0 χ−z ′ ! Xy,χ w(s0) 1/2 ∨ ∗ ∨ ′ + χs(tωP (m)) δP (β0 (t)m)Resz=s0 E(I, ιy(L(β0 (t)m)f)χz ) − β [Gm] ×[A 0 ] κF ′ Z s0∈C y,χ Re(Xs0)≥0 X

1/2 ∨ ∗ ∗ ∨ ∗ × − δ ∗ (β (t)m)Resz=s E (I, ι (L(β (t)m)FP |P ∗ (f)) ′ ) dmd t P 0 0 y 0 χ−z ′ ! Xy,χ

′der ′ \× × where the sums over y are over P (F )\Y (F ) and the sums over χ are over AGm F \AF . We recall that the sums over s0 have ﬁnite support in a set depending only on the KM type

of f by Conjecture 1.3 in the number ﬁeld case and the fact that E(I, fχs ) is rational in the sense of [MW95, IV.1.5] in the function ﬁeld case [MW95, Proposition IV.1.12]. ∨ ∨ −1 Since ωP ∗ (β0 (t)) = ωP (β0 (t)) Lemma 5.3 implies that the lower two integrals in (5.0.20)

converge for Re(sβ0 ) small enough. Thus they converge for all s. Since we already know that the upper two converge absolutely for all s we deduce that (5.0.20) is a holomorphic function of s. Now consider (5.0.21). We have the functional equation

∗ ∗ ∗ (5.0.22) E(I, ι (L(m)f) )= δ ∗ (m)E (I, ι (L(m)F ∗ (f)) ) y χβ0s P ∩Mβ0 y P |P χβ0s by Theorem 4.2 and (5.0.19). Using Lemma 5.4 and (5.0.22) to justify switching sums and

integrals, we deduce that for Re(sβ0 ) suﬃciently large (5.0.21) is equal to (5.0.23) 1 w(s ) 0 χ (tω (m)) δ1/2(tω (m))Res E(I, ι∗(L(β∨(t)m)f) ) s P P P z=s0 y 0 χβ0z 1 β 2 κF [Gm] ×A 0 (AF ) s0∈C y Z Re(Xs0)≥0 X

1/2 ∗ ∗ ∨ ∗ × ∗ ∗ − δP (tωP (m))Ress=s0 E (I, ιy(L(β0 (t)m)FP |P (f))(χβ )−z ) dmd t 0 ! SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 45

w(s ) + 0 χ (tω (m)) δ1/2(β∨(t)m)Res E(I, ι∗(L(β∨(t)m)f) ) s P P 0 z=s0 y 0 χβ0z − β κF [Gm] ×A 0 (AF ) s0∈C y Z Re(Xs0)≥0 X

1/2 ∨ ∗ ∗ ∨ ∗ × ∗ ∗ − δP (β0 (t)m)Resz=s0 E (I, ιy(L(β0 (t)m)FP |P (f))(χβ )−z ) dmd t. 0 ! where the sums on y are now over P ′(F )\G(F ). Here we are using (5.0.7) to unfold the integral. By Lemma 5.4 and (5.0.22) the expression (5.0.23) is holomorphic for Re(s) large, and admits a meromorphic continuation to the plane. Thus under the assumption that f that is KM ﬁnite, we have proved the meromorphic continuation statement in the theorem. ∗ ∗ By a symmetric argument we deduce that the sum of (5.0.20)and(5.0.21) is E (F ∗ (f) ). YP ∗ P |P χs This proves the functional equation

∗ ∗ E (f )= E (F ∗(f) ). YP χs YP ∗ P |P χs

Thus far we have assumed that f is KM -finite. To remove this assumption we note that for \× × ′ any f ∈ S(XP (AF )) and any χ ∈ AGm F \AF there is a KM -finite f ∈ S(XP (AF )) such that ′ ′ ∗ ′ ∗ ∗ ∗ fχs = fχs . Moreover, using Lemma 3.4, we can choose f so that FP |P (f)χs = FP |P (f )χs . This allows us to remove the KM -finiteness assumption.

The proof of Theorem 1.1 shows that the poles of EYP (fχs ) are controlled by the poles of ′ ′ E(I, f ) for f ∈ S(XP ∩Mβ (AF )). In particular it is easy to deduce the following corollary χβ0s 0 from the proof:

∆ ′ Corollary 5.5. Under the hypotheses of Theorem 1.1, for ﬁxed (sβ) ∈ C P the order of

the pole of EYP (fχs ) at sβ0 = s0 is no greater than the maximum of the orders of the pole of ′ ′ E(I, f ) at z = s0 as f ranges over S(XP ∩Mβ (AF )). χβ0z 0

6. On the poles of degenerate Eisenstein series We assume for this section that F is a number ﬁeld. Our goal here is to verify conjectures

1.3 and 1.4 in some cases. Without loss of generality we take G = Mβ0 , thus P ≤ G is now a maximal parabolic subgroup containing our ﬁxed Borel B and P ∗ is the opposite parabolic. Let Q be the unique maximal parabolic subgroup containing B that is conjugate to P ∗.

Let K ≤ G(AF ) be a maximal compact subgroup and let

C+ := {z ∈ C : Re(z) > 0}.

Lemma 6.1. To prove Conjecture 1.3 it suffices to show that for each character χ ∈ \× × χs χs AGm F \AF there is a finite set Υ(χ) ⊂ C+ such that if E(g, ΦP ) or E(g, ΦQ ) has a pole at χs χs s = s0 with Re(s0) > 0 for any holomorphic K-finite section ΦP ∈ IP (χs) or ΦQ ∈ IQ(χs) then s ∈ Υ(χ). 46 YOUNGJU CHOIE AND JAYCE R. GETZ

Proof. By the observations in the proof of Lemma 3.1 if aP |P (χs) has a pole at s = s0 for

Re(s0) ≥ 0 then s0 =0. Moreover, the order of the pole is bounded by an integer depending

only on P and G. Thus for any f ∈ S(XP (AF )) Theorem 4.2 implies that the only possible

pole of E(g, fχs ) on the line Re(s) = 0 is at s = 0 and its order is bounded in a sense depending only on P and G.

Now consider poles of E(g, fχs ) for Re(s) < 0. Using notation from the proof of Theorem 4.2 and arguing as in that proof we have

∗ ∗ ∗ ∗ ∗ E(g, fχs )= E (g, FP |P (f)χs )= EQ(g, j(FP |P (f)χs )). ∗ ∗ Thus the order of the pole of E(g, fχs ) at s0 is equal to the order of the pole of EQ(g, j(FP |P (f)χs )) −1 at s0. Since FP |P ∗(f) ∈ S(XP ∗ (AF )) and aP ∗|P ∗ ((χs) ) is holomorphic for Re(s) < 0 by ∗ ∗ ∗ ∗ Lemma 4.1 we have that FP |P (f)χs ∈ IP (χs) is a holomorphic section for Re(s) < 0. This ∗ −1 implies j(FP |P (f)χs ) is a holomorphic section of IQ((χ )−s) for Re(s) < 0. The lemma follows. The proof of the following lemma is analogous:

Lemma 6.2. To prove Conjecture 1.4 it suffices to show that there is a finite set Υ ⊂ C+ χs χs and an integer n> 0 such that if E(g, ΦP ) or E(g, ΦQ ) has a pole at s = s0 with Re(s0) > 0 \× × χs χs for any χ ∈ AGm F \AF and holomorphic K-finite sections ΦP ∈ IP (χs) or ΦQ ∈ IQ(χs) n then s0 ∈ Υ and χ =1.

Theorem 6.3. Suppose that G = Sp2n and that P is the Siegel parabolic, that is, the parabolic subgroup with Levi subgroup isomorphic to GLn. Then Conjecture 1.4 is true.

Proof. We use the results of [Ike92]. We point out that aP |P (χs)= aI2n (χs) in the notation of loc. cit. by [GHL21, §4.3]. Thus the theorem follows from Lemma 6.2, the fact that

aP |P (χs) is holomorphic and nonzero for Re(s) > 0 by Lemma 4.1, and the Corollary of [Ike92, Proposition 1.6].

Theorem 6.4. Suppose that G = SLn. Then Conjecture 1.3 is true. Proof. This can be derived using Lemma 6.1 and the same arguments as those proving [HM15, Theorem 1.4]. The proof comes down to two statements explained in [HM15, §6]. The first is the statement that a certain quotient of completed L-functions depending on χ has only finitely many poles for Re(s) > 0. The second is that a certain normalized intertwining operator has only finitely many poles. These are proven exactly as in loc. cit. The required

facts on induced representations of GL2 may be found in [JL70]. Much of the work towards proving Conjecture 1.4 for arbitrary parabolic subgroups of symplectic groups is contained in [Han18], and much of the work towards proving Conjecture 1.4 for general linear groups is contained in [HM15]. The additional steps necessary seem to require careful analysis of the reducibility points of local principal series representations at the archimedean places. SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 47

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Department of Mathematics, Pohang University of Science and Technology, Pohang, South Korea 37673 Email address: [email protected]

Department of Mathematics, Duke University, Durham, NC 27708 Email address: [email protected]