A Relative Trace Formula for Pgl(2) in the Local Setting

Paciﬁc Journal of Mathematics

A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING

BROOKE FEIGON

Volume 260 No. 2 December 2012 PACIFIC JOURNAL OF MATHEMATICS Vol. 260, No. 2, 2012 dx.doi.org/10.2140/pjm.2012.260.395

A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING

BROOKE FEIGON

In memory of Jonathan Rogawski

We develop the local Kuznetsov trace formula on a unitary group in two variables for an unramified quadratic extension of local, non-Archimedean fields E/F and compare it to a local relative trace formula on PGL(2, E). To define the local distributions for the relative trace formula, we define a regularized local period integral and prove that it is a PGL(2, F)-invariant linear functional. By comparison of the two local trace formulas, we get an equality between a local PGL(2, F)-period and local Whittaker functionals.

1. Introduction 395 2. Notation 403 3. The local Kuznetsov trace formula for U(2) 405 4. The local relative trace formula and periods for PGL(2) 413 5. Comparison of local trace formulas and applications 427 Acknowledgments 430 References 431

1. Introduction

Base change is an important type of functoriality which is useful in the study of automorphic forms by relating automorphic representations on different groups. Hervé Jacquet shed light on a new technique for attacking certain cases of Robert Lang- lands’ important functoriality conjectures by comparing the relative and Kuznetsov trace formulas in the global setting. Jacquet’s comparison of trace formulas leads to global identities that characterize the image of the base change map associating automorphic representations of a unitary group for a quadratic extension of number ﬁelds E/F to automorphic representations of GL(2, ށE ) in terms of distinguished representations. While Jacquet’s global identities factor, they do not give unique local identities.

The author was supported by NSF grant DMS-1201446. MSC2010: 11F70, 22E50. Keywords: relative trace formula.

395 396 BROOKE FEIGON

This paper uses techniques of James Arthur to define and develop a local Kuznetsov trace formula on U(2) and a local relative trace formula on GL(2). Both local trace formulas are expanded geometrically in terms of orbital integrals and spectrally in terms of local Bessel distributions and local relative Bessel distributions. The latter involve regularized local period integrals. We then carry out Jacquet’s comparison in the local setting by relating these two local trace formulas for matching functions. This comparison yields identities between local Bessel distributions for automorphic representations on U(2) and local relative Bessel distributions for automorphic representations on GL(2). Before we describe more precisely the local relative trace formula developed in this paper, let us recall the relative trace formula for GL(2). Take E/F to be 0 a quadratic extension of number fields and ށF to be the adeles of F. Let ψ be ∼ a character on F\ށF = N(F)\N(ށF ) where N is the upper triangular unipotent 0 matrices of GL(2). Let ψ = ψ ◦ trE/F . A cuspidal automorphic representation π of GL(2, ށE ) with central character trivial on GL(2, ށF ) is distinguished by GL(2, ށF ) if there exists a φ ∈ Vπ , the vector space associated to π, such that the period integral, P(φ), is nonzero: Z P(φ) := φ(h) dh 6= 0. GL(2,F)Z(ށF )\ GL(2,ށF ) Where π 0 is a cuspidal automorphic representation of the quasisplit unitary group 0 U(2, ށF ) and φ ∈ Vπ 0 , let Z Z W(φ0) = φ0(n)ψ0(n) dn and W(φ) = φ(n)ψ(n) dn. N(F)\N(ށF ) N(E)\N(ށE ) We define the Bessel distribution as

0 0 X 0 0 0 0 0 0 Bπ 0 ( f ) := W (π ( f )φi )W (φi ), i and the relative Bessel distribution as X Bπ ( f ) := P(π( f )φ j )W(φ j ), j where the summations are over an orthonormal basis of Vπ 0 and Vπ respectively. Flicker[1991], following related work of Jacquet and Lai[1985] and Ye[1989], 0 0 showed that for “matching functions” f on U(2, ށF ) and f on GL(2, ށE ), if π maps to π under the unstable base change, then

X 0 0 0 0 0 0 X (1-1) W (π ( f )φi )W (φi ) = P(π( f )φ j )W(φ j ). i j In particular, this equality characterizes the image of the unstable base change lift associating every automorphic representation of U(2, ށF ) to an automorphic A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 397 representation of GL(2, ށE ) in terms of GL(2, ށF ) distinguished representations. The equality above is proved via the relative trace formula[Jacquet 2005], which tells us that for f and f 0 matching functions we have Z 0 − 0 1 K f (n1, n2)ψ (n1 n2) dn1 dn2 (N(F)\N(ށ ))2 F Z Z = K f (h, n)ψ(n) dn dh GL(2,F)Z(ށF )\ GL(2,ށF ) N(E)\N(ށE ) where X −1 K f (x, y) = f (x δy). δ∈Z(E)\ GL(2,E) 0 0 The distributions Bπ 0 ( f ) and Bπ ( f ) occur in the spectral expansions of the respec- tive trace formulas. In a different direction, Arthur[1989; 1991] developed a local version of the classical Arthur–Selberg trace formula. Let G be a connected reductive algebraic group over a local field F of characteristic zero. Diagonally embed G(F) into G(F) × G(F). Then L2(G(F)) is isomorphic to L2(G(F)\G(F) × G(F)) by 7→ 7→ −1 φ ((y1, y2) φ(y1 y2)). ∈ 2 = −1 For φ L (G(F)), let (ρ(g1, g2)φ)(x) φ(g1 xg2). The right regular representation of G(F) × G(F) on L2(G(F)\G(F) × G(F)) is equivalent to ρ of G(F) × G(F) on L2(G(F)). Thus to develop the local trace formula we look at ∞ ρ( f ) where f = f1 ⊗ f2 ∈ Cc (G(F) × G(F)). Then Z Z −1 (ρ( f )φ)(x) = f1(g) f2(y)φ(g xy) dg dy G(F) G(F) is an integral operator on L2(G(F)) with kernel Z −1 K f (x, y) = f1(g) f2(x gy) dg. G(F) The local trace formula develops an explicit formula for the regularized trace of ρ( f ). The main result of this paper is that, when evaluated with matching functions, the two local trace formulas described in Theorems 1.3 and 1.4 below, that is the local Kuznetsov trace formula and the local relative trace formula, are equal. Thus there is an equality between their local distributions on the spectral sides. This equality is stated in Theorem 1.1. This is the natural local counterpart to the global comparison from (1-1). In order to develop the local relative trace formula stated in Theorem 1.4, we have to define a local regularized period integral, prove it is a GL(2, F) × GL(2, F)-invariant linear functional and relate it to the truncated 398 BROOKE FEIGON period integral that initially appears in the relative trace formula. We state these properties about the local regularized period integral in Proposition 1.2. To describe our results more precisely we need to introduce some further notation. Let E/F now denote an unramified extension of local non-Archimedean fields of characteristic 0. Let ᏻF (respectively ᏻE ) denote the ring of integers in F (respectively E). Let H = GL(2)/F, G = ResE/F H and let 0 1 0 1 G0 = U(2, F) = g ∈ G : t g g = . −1 0 −1 0 Let N 0 and N be the upper triangular unipotent matrices of G0 and G, respectively, and let M0 and M be the diagonal subgroup of G0 and G, respectively. Let Z and Z 0 denote the center of G and G0, respectively. For any subgroup X of G let 0 eX = Z ∩ X\X and let X H = X ∩ H. Let ψ be an additive character on F with 0 ∞ conductor ᏻF and let ψ(x) = ψ ◦ trE/F . Let f = f1 ⊗ f2 ∈ Cc (Ge(F) × Ge(F)) 0 = 0 ⊗ 0 ∈ ∞ 0 × 0 and f f1 f2 Cc (Ge (F) Ge (F)). We define the local Kuznetsov trace formula as the equality between the geometric expansion (in terms of orbital integrals) and spectral expansion (in terms of representations) of Z 0 −1 lim K f 0 (n , n )ψ (n n )u(n , t)u(n , t) dn dn →∞ 1 2 1 2 1 2 1 2 t (N 0×N 0)(F) and the local relative trace formula as the equality between the expansions of Z Z lim K f (h, n)ψ(n)u(h, t)u(n, t) dn dh. t→∞ He(F) N(F) In this local setting Z Z −1 0 0 −1 K f (x, y) = f1(g) f2(x gy) dg, K f 0 (x, y) = f1(g) f2(x gy) dg Ge(F) Ge0(F) and u(n, t) and u(h, t) are truncation parameters that are needed due to convergence issues. They are defined analogously to Arthur’s truncation[1991, Section 3]. We use the following ideas in this paper to rewrite these local trace formulas in terms of orbital integrals and representations: • methods of Arthur[1991] from the local trace formula, • methods of Flicker[1991], Jacquet[2005] and Ye[1989] from the relative trace formula, • Harish-Chandra’s Plancherel formula[Harish-Chandra 1984; Waldspurger 2003], • Jacquet, Lapid and Rogawski’s methods for regularizing period integrals [Jacquet et al. 1999; Jacquet ≥ 2012]. A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 399

The power of the two trace formulas lies in the comparison. For “matching functions”, the geometric expansions of the two local relative trace formulas are equal. By comparing the spectral expansions in these two trace formulas, we get an analogue of (1-1), giving the following identity between local Bessel distributions for functions on U(2) and local relative Bessel distributions for functions on GL(2, E), and therefore local periods and local Whittaker functionals:

Theorem 1.1. If σ is a supercuspidal representation on Ge(F) that is the unstable 0 0 base change lift of the supercuspidal representation σ of Ge (F), and 0 0 0 ∞ 0 0 ∞ f = f1 ⊗ f2 ∈ Cc (Ge (F) × Ge (F)) and f = f1 ⊗ f2 ∈ Cc (Ge(F) × Ge(F)) are matching functions, then

0 X 0 0 0 0 0 0∨ 0 0 (1-2) d(σ ) Wσ 0 (σ ( f2)S σ ( f1 ))Wσ 0 (S ) S0∈Ꮾ(σ 0) X ∨ = d(σ ) Pσ (σ ( f2)Sσ ( f1 ))Wσ (S), S∈Ꮾ(σ ) where d(σ ) is the formal degree of σ , Ꮾ(σ ) is an orthonormal basis of the Hilbert space of Hilbert–Schmidt operators on Vσ , Z 0 0 0 0 0 −1 Wσ 0 (S ) = tr(σ (n)S )ψ (n ) dn, N 0(F) Z −1 Wσ (S) = tr(σ (n)S)ψ(n ) dn, N(F) Z Pσ (S) = tr(σ (h)S) dh. He(F) 0 0 The Bessel and relative Bessel distributions Bπ 0 ( f ) and Bπ ( f ) factor into 0 0 local (relative) Bessel distributions B 0 ( f ) and Bπ ( fv), but it is not clear how to πv v v normalize the local distributions. The distributions on the left and right-hand side of (1-2) are each the product of two local distributions and (1-2) can be restated as

0 0 0 0 0 d(σ )Bσ 0 ( f2)Bσ 0∗ ( f1) = d(σ )Bσ ( f2)Bσ ∗ ( f1).

We note that the local period integral Pσ (S) is not a convergent integral if σ is not a discrete series representation. To develop the local relative trace formula we have to define a local regularized period integral. Let K = G(ᏻF ) and let P = NM. α 0 λHM (m) For λ ∈ ރ and m = 0 β let e = |α/β|E where | · |E denotes the normalized valuation on E. For a principal series representation π of Ge and u, v ∈ π we define the matrix coefficient fu,v(g) = hπ(g)u, vi. Asymptotically on M, fu,v will equal a finite sum of functions of the form eλHM (m). We define the regularized period 400 BROOKE FEIGON integral as: Z ∗ Z fu,v(h) dh := fu,v(h)u(h, t) dh He(F) He(F) Z Z ] + DP (m) fu,v(k1mk2)(1 − u(m, t)) dm dk1 dk2 + H KeH ×KeH MeH (F) where Z ] eλHM (m)(1 − u(m, t)) dm + MeH (F) is the meromorphic continuation at ν = 0 of Z e(ν+λ)HM (m)(1 − u(m, t)) dm, + MeH (F) which is absolutely convergent for Re(ν) 0. We prove that the regularized period integral is an H(F)× H(F)-invariant linear functional, and we relate it to the truncated period integral that initially appears in the local relative trace formula as follows. By abuse of notation we identify a × a 0 = −1 character χ of Me(F) with a character χ of E by letting χ 0 b χ(a)χ (b). λ(H (m)) For λ ∈ ރ we let χλ(m) = χ(m)e M . We let IP (χλ) be the parabolically induced normalized representation acting on the Hilbert space ᏴP (χ). Then for S ∈ ᏮP (χ), = tr(IP (χλ, k1gk2)S) E P (g, 9S, λ)k1,k2 , where E P (g, 9, λ) is the Eisenstein integral and

P λHM (m) −λHM (m) (C E P )(m, ψ, λ) = (cP|P (1, λ)ψ)(m)e + (cP|P (w, λ)ψ)(m)e .

−1 We ﬁx a uniformizer $ in F (and E) and q = |$|F . Proposition 1.2. Fix a character χ of E× such that χ($ ) = 1. Then for t 0, Z Z ∗ tr(IP (χλ, h)S)u(h, t) dh = tr(IP (χλ, h)S) dh He(F) He(F) q2λ(t+1) Z −δ(χ)(1 + q−1) c(1, λ)9 (1) dk dk −q2λ S k1,k2 1 2 1 KeH ×KeH q−2λ(t+1) Z + c(w, λ)9 (1) dk dk , −q−2λ S k1,k2 1 2 1 KeH ×KeH where δ(χ) = 1 if χ| × = 1 and δ(χ) = 0 if χ| × 6= 1. ᏻF ᏻF Denote the action of the nontrivial element in Gal(E/F) on x ∈ E by x¯. Denote × × 1 × by NE/F the norm map from E to F . Let E = {x ∈ E : NE/F (x) = 1}. Let η −1 = 0 1 denote an element in G(F) such that η η 1 0 . A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 401

We deﬁne

0 0 X 0 0 0 0 X 0 = [ ] 0 = [ ] D χ ( f ) W 0 (Sλ f )W 0 (S ) and Dχλ ( f ) Pχλ (Sλ f )Wχλ (S), λ χλ χλ 0 0 S ∈ᏮP (χ ) S∈ᏮP (χ) where Z 0 0 0 0 0 −1 W 0 (S ) = lim tr(IP0 (χ , n)S )ψ (n )u(n, t) dn, χ →∞ λ λ t N 0(F) Z −1 W (S) = lim tr(IP (χ , n)S)ψ(n )u(n, t) dn, χλ →∞ λ t N(F) Z ∗ = Pχλ (S) tr(IP (χλ, h)S) dh, He(F) ∨ Sλ[ f ] = IP (χλ, f2)SIP (χλ, f1 ).

0 We let 52(Ge (F)) be a set of equivalence classes of irreducible, tempered square 0 0 integrable representations of Ge (F). We identify unitary characters on Me (F) × 1 0 with characters on E that are trivial on E . We let {52(Me (F))} be a set of 0 0 0 representatives of unitary characters χ on Me (F) such that χ ($ ) = 1. We let 0 µ(χλ) be Harish-Chandra’s µ-function. We take the analogous deﬁnitions for Ge(F). Theorem 1.3 (local Kuznetsov trace formula). For any

0 0 0 ∞ 0 0 f = f1 ⊗ f2 ∈ Cc (Ge (F) × Ge (F)), we have Z Z 0 −1 lim K f 0 (n , n )ψ (n n )u(n , t)u(n , t) dn dn →∞ 1 2 1 2 1 2 1 2 t N 0(F) N 0(F) Z 0 0 0 0 ¯ 0 × = O f1, ψ , a O f2, ψ , a |a|E d a a∈E×/E1 Z πi/log q X 0 0 0 1 X 0 0 0 0 = d(σ )D 0 ( f ) + d(χ ) µ(χ )D 0 ( f ) dλ, σ 2 λ χλ 0 0 0 0 0 σ ∈52(Ge (F)) χ ∈{52(Me (F))} where Z Z 0 0 0 = 0 −1 0 1 a 0 0 −1 O ( fi , ψ , a) fi n1 −1 0 0 a−1 n2 ψ (n1 n2) dn1 dn2. N 0(F) N 0(F) Theorem 1.4 (local relative trace formula). For any

∞ f = f1 ⊗ f2 ∈ Cc (Ge(F) × Ge(F)), we have 402 BROOKE FEIGON Z Z lim K f (h, n)ψ(n)u(h, t)u(n, t) dn dh t→∞ He(F) N(F) Z ¯ × = O f1, ψ, a O f2, ψ, a |a|E d a a∈E×/E1 = X + 1 X d(σ )Dσ ( f ) 2 Deχ ( f ) σ∈52(Ge(F)) χ∈{52(Me(F))} 2 χ 6=1,χ|F× =1 Z πi/log q + 1 X 2 d(χ) µ(χλ)Dχλ ( f ) dλ 0 χ∈{52(Me(F))} | = χ E1 1 where Z Z −1 a 0 O( fi , ψ, a) = fi h η n ψ(n) dn dh. He(F) N(F) 0 1 The representations that occur on the right-hand side of Theorem 1.4 are exactly 0 the representations that are in the image of the unstable base change lift on Ge (F). The additional discrete term Deχ ( f ) corresponds to the representations that lift from 0 discrete series on Ge (F) to principal series on Ge(F). In addition to the spectral comparison, these local trace formulas also have applications on the geometric side. If we deﬁne the inner product of two functions × 1 g1, g2 on E /E by Z × hg1, g2i = g1(a)g2(a)|a|E d a, a∈E×/E1 then:

Proposition 1.5 (orthogonality relations). For f1 and f2 matrix coefficients of the 0 0 supercuspidal representations σ1 and σ2 of Ge(F) and f1 and f2 matrix coefficients 0 0 0 of the supercuspidal representations σ1 and σ2 of Ge (F), 0 0 0 0 0 0−1 0 0 O ( f1, ψ , · ), O ( f2, ψ , · ) 6= 0 ⇐⇒ σ1 ∼ σ2, −1 O( f1, ψ, · ), O( f2, ψ , · ) 6= 0 ⇐⇒ σ1 ∼ σ2. The rest of this paper is organized as follows. In Section 2 we define notation and give normalizations of measures. In Section 3 we develop the local Kuznetsov trace formula. For the geometric expansion we rewrite our trace formula in terms of orbital integrals corresponding to the N 0\G0/N 0 double cosets. The orbital integrals 0 0 for f1 and f2 initially depend on the truncation and are intertwined. It is only through the multiplication of the two orbital integrals, integration over the space of double cosets, and the nontriviality of the character ψ0, that we are able to 0 0 untangle the orbital integral for f1 from the orbital integral for f2. For the spectral A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 403 expansion we apply Harish-Chandra’s Plancherel formula to rewrite the local kernel in terms of representations. We are left with truncated integrals over the unipotent subgroup of matrix coefficients against the character ψ0. By the smoothness of the matrix coefficients and the appearance of the character, we show these distributions stabilize for t large. In Section 4 we develop the local relative trace formula of H\G/N. In the spectral expansion we have truncated integrals of matrix coefficients over H that do not converge without the truncation. We define the regularized period integral Pχλ (S). We use the asymptotics of matrix coefficients of tempered representations to prove the truncated integral is a polynomial exponential function in the truncation param- eter t. We define the regularized integral as the constant term of this polynomial, and prove that this is an H × H invariant linear functional and the relevant term in the local relative trace formula. In Section 5 we compare our two local trace formulas. There is a bijection between the “admissible” N 0\G0/N 0 cosets and the “admissible” H\G/N cosets and both of these sets can be parametrized by E×/E1. This bijection allows us to compare the geometric sides. By work of Ye and Flicker, we know that for any f 0 there is an f such that the orbital integrals are equal for corresponding cosets. Thus, by their geometric expansions, our local trace formulas are equal for matching functions. This gives an equality of the spectral expansions and of local distributions. This paper would not have come into being had it not been for my teacher and advisor, Jonathan Rogawski. These thoughts originated as my PhD thesis under his direction, and his ideas, support, and guidance were critical to its completion. I am fortunate and will be forever grateful to have had him as a mentor. He could explain complicated math in a clear and simple way that aimed at the heart of the problem. He served, and continues to serve, as the role model of the inquisitive, patient, and approachable mathematician.

2. Notation

Let F be a non-Archimedean local ﬁeld of characteristic 0 and odd residual characteristic q. Let E be an unramiﬁed quadratic extension of F. Let ᏻF and ᏻE denote the rings of integers in F and E, respectively. Let $ denote a uniformizer in the maximal ideal of ᏻF . Thus $ is also a uniformizer in E. Let v( · ) denote the valuation on F, extended to E. Let | · |F and | · |E denote the normalized valuations × 2 on FE, respectively. Thus for a ∈ F , |a|E = |a|F . Denote the action of the nontrivial element in Gal(E/F) on x ∈ E by x¯. Denote by NE/F the norm map × × 1 × from E to F . Let E = {a ∈ E : NE/F (a) = 1}. 404 BROOKE FEIGON

Let H = GL(2)/F and let G = ResE/F H, the restriction of scalars of GL(2) from E to F. Thus G(F) = GL(2, E). Let

0 1 0 1 G0 = U(2, F) = g ∈ G : t g g = . −1 0 −1 0

We note that by deﬁning the quasisplit unitary group in this way, SL(2, F) ⊂ G0(F). Let N 0 and N be the upper triangular unipotent matrices of G0 and G, respectively. Let M0 and M be the diagonal subgroups of G0 and G, respectively. That is,

a 0 a 0 M0(F) = : a ∈ E× and M(F) = : a, b ∈ E× . 0 a−1 0 b

= 1 n = a 0 = Occasionally by abuse of notation we let n 0 1 and a 0 a¯−1 . Let P NM 0 0 0 0 0 0 and P = N M . Let K = G(ᏻF ) and K = G (ᏻF ). Let Z and Z denote the 0 centers of G and G , respectively. For any subgroup X of G let eX = Z ∩ X\X and X H = X ∩ H. By abuse of notation we identify a character χ of Me(F) with a × a 0 = −1 character χ of E by letting χ 0 b χ(a)χ (b). 0 Let ψ be an additive character on F with conductor ᏻF . Let ψ be the additive character on E defined by ψ(x) = ψ0(x + x). By abuse of notation we will also denote by ψ and ψ0 the corresponding characters on N(F) and N 0(F), respectively. = ⊗ ∈ ∞ × 0 = 0 ⊗ 0 ∈ ∞ 0 × 0 Let f f1 f2 Cc (Ge(F) Ge(F)) and f f1 f2 Cc (Ge (F) Ge (F)). For a function f on G, let f ∨(g) = f (g−1). To define the local Kuznetsov trace formula and local relative trace formula we first multiply our function by the characteristic function of a large compact subset of Ge(F) via Arthur’s local truncation[1991, §3], and then take the limit of the integral of the truncated function. For g ∈ G(F), t ∈ ޚ+, let 1 if g = zk 1 0 k , for some k , k ∈ K , z ∈ Z(F), 0 ≤ v(α) ≤ t, u(g, t) = 1 0 α 2 1 2 0 otherwise.

We note that u( · , t) is well-deﬁned on Ge(F) and 1 x 1 if x ∈ $ [−t/2]ᏻ , u , t = E 0 1 0 otherwise, where [x] is the integral part of x. If X is a closed subgroup of Ge(F) with the subgroup topology, supp(u( · , t))∩ X is a compact set. We normalize the Haar measure dx on F so that vol(ᏻF ) = 1. We deﬁne the multiplicative measure d×x on F× as

× = 1 1 d x −1 dx. 1−q |x|F A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 405

× Thus vol(ᏻF ) = 1. We let N(F) and M(F) have the measures induced by dx and d×x. We normalize the Haar measure dk on K so that vol(K ) = 1. We deﬁne the measure dg on G(F) by Z Z Z Z f (g) dg = f (mnk) dk dn dm. G(F) M(F) N(F) K 0 0 We deﬁne dg on G (F) similarly. We normalize Haar measure on Ke by taking vol(Ke) = 1. We let d×a be the unique Haar measure on E×/E1 such that

× 1 vol(ᏻ /E1) = . E 1+q−1

3. The local Kuznetsov trace formula for U(2)

In this section we develop a local Kuznetsov trace formula for the quasisplit unitary group in two variables. We expand this local Kuznetsov trace formula geometrically 0 0 in terms of separate orbital integrals for f1 and f2. Then we use Harish-Chandra’s Plancherel formula to rewrite this expression spectrally in terms of representations. We deﬁne the local Kuznetsov trace formula for 0 0 0 ∞ 0 0 f = f1 ⊗ f2 ∈ Cc (Ge (F) × Ge (F)) as the equality between the geometric and spectral expansions of Z −1 lim K f 0 (n , n )ψ(n n )u(n , t)u(n , t) dn dn →∞ 1 2 1 2 1 2 1 2 t (N 0×N 0)(F) where Z 0 0 −1 0 = K f (n1, n2) f1(g) f2(n1 gn2) dg. Ge0(F) We will show that for a ﬁxed f 0 this limit stabilizes, that is, there exists a T such that for all t0 ≥ T , Z 0 − 0 0 0 1 K f (n1, n2)ψ (n1 n2)u(n1, t )u(n2, t ) dn1 dn2 (N 0×N 0)(F) Z −1 = lim K f 0 (n , n )ψ(n n )u(n , t)u(n , t) dn dn . →∞ 1 2 1 2 1 2 1 2 t (N 0×N 0)(F) 3A. The geometric expansion. In this subsection we rewrite Z Z 0 −1 lim K f 0 (n , n )ψ (n n )u(n , t)u(n , t) dn dn →∞ 1 2 1 2 1 2 1 2 t N 0(F) N 0(F) 0 as an integral over admissible cosets of a product of an orbital integral for f1 and 0 an orbital integral for f2. 406 BROOKE FEIGON

= 0 1 ∈ × = a 0 3A1. Integration formula. Let w −1 0 . For a E , let βa 0 a−1 and = a 0 0 = 0 t 0 0 γa w 0 a−1 . By the Bruhat decomposition, G P P wP . Thus a is in a set of a is in a set of β : S γ : a representatives for E×/E1 a representatives for E×/E1

0 0 0 is a set of representatives for the double cosets of N (F)\Ge (F)/N (F). For g ∈ G0(F) let 0 0 Cg(N (F) × N (F)) = { ∈ 0 × 0 : −1 = ∈ 0 } (n1, n2) N (F) N (F) n1 gn2 zg for some z Z (F) . 0 Deﬁnition 3.1. An element g ∈ Ge (F) and its corresponding orbit are called admissible if the map 0 × 0 → : 7→ 0 −1 Cg(N (F) N (F)) ރ (n1, n2) ψ (n1 n2) is trivial. By a simple calculation we see that 1 x 1 x C (N 0(F) × N 0(F)) = , aa : x ∈ F , βa 0 1 0 1 0 × 0 = Cγa (N (F) N (F)) 1.

× 1 Thus the orbits represented by {β1} ∪ {γa : a ∈ E /E } are admissible. We use the following integration formula to rewrite K f 0 (n1, n2) as an integral over the admissible cosets. Unlike in the global case the trivial admissible coset, β1, will not contribute to the trace formula. 0 For any F ∈ Cc(Ge (F)), Z Z Z = −1 | | × (3-1) F(g) dg F(n1 γan2) dn1 dn2 a E d a. Ge0(F) E×/E1 (N 0×N 0)(F) 3A2. Separating the orbital integrals. Let Z Z t 0 −1 = 0 K ( f ) K f (n1, n2)ψ(n1 n2)u(n1, t)u(n2, t) dn1 dn2. N 0(F) N 0(F) t 0 0 · Clearly K ( f ) is absolutely convergent because f1 and u( , t) have compact 0 0 support on Ge (F) and N (F) respectively. By changing the order of integration and using (3-1), we see that K t ( f 0) equals Z Z Z 0 ˆ−1 ˆ 0 −1 ˆ−1 ˆ f1(n1 γan2) f2(n1 n1 γan2n2) E×/E1 (N 0×N 0)(F) (N 0×N 0)(F) × 0 −1 ˆ ˆ | | × ψ (n1 n2)u(n1, t)u(n2, t) dn1 dn2 dn1 dn2 a E d a. A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 407

This integral is absolutely convergent because the map

0 × 1 0 0 N (F) × E /E × N (F) → Ge (F) deﬁned by 7→ −1 (n1, a, n2) n1 γan2 0 is injective and f1 has compact support. By a change of variables we have Z t 0 t 0 × K ( f ) = K (γa, f )|a|E d a, E×/E1 where Z Z t 0 = 0 ˆ−1 ˆ 0 −1 0 −1 ˆ ˆ−1 K (γa, f ) f1(n1 γan2) f2(n1 γan2)ψ (n1 n1n2 n2) (N 0×N 0)(F) (N 0×N 0)(F) × ˆ−1 ˆ−1 ˆ ˆ u(n1 n1, t)u(n2 n2, t) dn1 dn2 dn1 dn2. To complete the geometric expansion of the local Kuznetsov trace formula we rewrite K t ( f 0) for t 0 as an integral of two separate orbital integrals. We begin by examining the dependence of the integrand on the truncation. 0 0 ∈ 0 Lemma 3.2. Let f1, f2 Cc(Ge (F)). For each t0 > 0 there exists a T > 0 such that for all t ≥ T ,

0 −1 0 −1 −1 −1 f1(x1 γ y1) f2(x2 γ y2)u(x1 x2, t)u(y1 y2, t0) = 0 −1 0 −1 −1 f1(x1 γ y1) f2(x2 γ y2)u(y1 y2, t0) 0 for all x1, x2, y1, y2, γ ∈ Ge (F). Proof. Let

0 0 0 1 = supp( f1), 2 = supp( f2), 3 = supp(u( · , t0)) ∩ Ge (F). 0 0 −1 0 −1 −1 6= These sets are all compact on Ge (F). If f1(x1 γ y1) f2(x2 γ y2)u(y1 y2, t0) 0, then the following conditions must hold:

• −1 ∈ −1 −1 x1 1 y1 γ . • ∈ −1 x2 γ y22 . • −1 ∈ y1 y2 3. 0 −1 0 −1 −1 6= −1 ∈ −1 Thus if f1(x1 γ y1) f2(x2 γ y2)u(y1 y2, t0) 0, then x1 x2 132 . Because −1 ⊆ this is a compact set, there exists a T > 0 such that 132 supp(u(g, T )). The lemma now follows. 408 BROOKE FEIGON

Now we use this lemma, along with the character ψ0, to separate the two orbital integrals. By abuse of notation, in the proof of the following lemma we let

$ n 0 a 0 $ n = and a = . 0 $ −n 0 a−1

0 = 0 ⊗ 0 ∈ ∞ 0 × 0 Lemma 3.3. For f f1 f2 Cc (Ge (F) Ge (F)), there exists a T such that for all t ≥ T and n ∈ ޚ,

Z Z t 0 × 0 0 0 0 0 ¯ 0 × (3-2) K (γa, f ) d a = O ( f1, ψ , a) O ( f2, ψ , a) d a, n × 1 n × 1 a∈$ ᏻE /E a∈$ ᏻE /E where Z Z 0 0 0 = 0 −1 0 −1 O ( f , ψ , a) f (n1 γan2)ψ (n1 n2) dn1 dn2. N 0(F) N 0(F)

Proof. We show that there is a hidden truncation on the right-hand side of (3-2) that comes from the fact that the two orbital integrals are simultaneously evaluated at 0 0 0 the same γa. Let K1 be an open compact subgroup of Ge (F) such that f1 and f2 are bi-K1-invariant. There exists a positive constant c such that

a 0 ∈ K for all a ∈ (1 + $ cᏻ )E1. 0 a−1 1 E

By deﬁnition

Z t 0 × K (γa, f ) d a n × 1 a∈$ ᏻE /E Z Z Z = 0 ˆ−1 n ˆ 0 ˆ ˆ−1 0 −1 n f1(n1 w$ an2)ψ (n1n2 ) f2(n1 w$ an2) × 1 0 0 0 0 a∈ᏻE /E (N ×N )(F) (N ×N )(F) × 0 −1 ˆ−1 ˆ−1 ˆ ˆ × ψ (n1 n2)u(n1 n1, t)u(n2 n2, t) dn2 dn1 dn2 dn1 d a Z Z = X 0 ˆ−1 n ˆ 0 ˆ ˆ−1 f1(n1 w$ ηan2)ψ (n1n2 ) ∈ + c 1 1 0× 0 × c 1 a (1 $ ᏻE )E /E (N N )(F) η∈ᏻE /(1+$ ᏻE )E Z × 0 −1 n 0 −1 ˆ−1 f2(n1 w$ ηan2)ψ (n1 n2)u(n1 n1, t) (N 0×N 0)(F) × ˆ−1 ˆ ˆ × u(n2 n2, t) dn2 dn1 dn2 dn1 d a. A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 409

By a change of variables and the fact that f 0 is locally constant the right-hand side of this equation is equal to Z X 0 ˆ−1 n ˆ 0 ˆ f1(n1 w$ ηn2)ψ (n1) 0× 0 × c 1 (N N )(F) η∈ᏻE /(1+$ ᏻE )E Z × 0 −1 n 0 −1 ˆ−1 ˆ f2(n1 w$ ηn2)ψ (n1 )u(n1 n1, t) dn1 dn1 (N 0×N 0)(F) Z × 0 −1 ˆ−1 −1 ˆ−1 × ˆ ψ (a n2 n2a)u(a n2 n2a, t) d a dn2 dn2. a∈(1+$ cᏻ )E1/E1 We can rewrite the inner integralE as Z ˆ−1 0 − ˆ ¯ −1 × u(n2 n2, t) ψ ((n2 n2)(aa) ) d a c 1 1 a∈(1+$ ᏻE )E /E Z = ˆ−1 0 − ˆ × u(n2 n2, t) ψ (b(n2 n2)) d b c b∈1+$ ᏻF Z = ˆ−1 1 0 − ˆ 0 − ˆ u(n2 n2, t) − ψ (n2 n2) ψ (b(n2 n2)) db − 1 c 1 q b∈$ ᏻF c − − vol($ ᏻ ) 0 − = u(nˆ 1n , t)u(nˆ 1n , 2c) F ψ (nˆ 1n ). 2 2 2 2 1−q−1 2 2 Thus for t ≥ 2c, Z Z Z t 0 × = 0 ˆ−1 ˆ 0 ˆ ˆ−1 K (γa, f ) d a f1(n1 γan2)ψ (n1n2 ) n × 1 n × 1 0 0 a∈$ ᏻE /E a∈$ ᏻE /E (N ×N )(F) Z × 0 −1 0 −1 ˆ−1 ˆ−1 ˆ ˆ × f2(n1 γan2)ψ (n1 n2)u(n1 n1, t)u(n2 n2, 2c) dn2 dn1 dn2dn1d a. (N 0×N 0)(F) By Lemma 3.2 there exists a T > 0 such that for all t ≥ max{T, 2c}, Z t 0 × K (γa, f ) d a n × 1 a∈$ ᏻE /E Z Z = 0 ˆ−1 ˆ 0 ˆ ˆ−1 ˆ f1(n1 γan2)ψ (n1n2 ) dn1 n × 1 0 0 a∈$ ᏻE /E (N ×N )(F) Z × 0 −1 0 −1 ˆ−1 ˆ × f2(n1 γan2)ψ (n1 n2)u(n2 n2, 2c) dn2 dn1 dn2 d a (N 0×N 0)(F) Z Z = 0 ˆ−1 ˆ 0 ˆ ˆ−1 ˆ ˆ f1(n1 γan2)ψ (n1n2 ) dn2 dn1 n × 1 0 0 a∈$ ᏻE /E (N ×N )(F) Z × 0 −1 0 −1 × f2(n1 γan2)ψ (n1 n2) dn2 dn1 d a. (N 0×N 0)(F) We have shown that the truncated local Kuznetsov trace formula stabilizes. 410 BROOKE FEIGON

0 = 0 ⊗ 0 ∈ ∞ 0 × 0 Proposition 3.4. For any f f1 f2 Cc (Ge (F) Ge (F)) and t 0, Z Z 0 − 0 1 K f (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t) dn1 dn2 N 0(F) N 0(F) Z 0 0 0 0 0 ¯ 0 × = O ( f1, ψ , a)O ( f2, ψ , a)|a|E d a. a∈E×/E1 3B. The spectral expansion. Now we derive a spectral expansion for the local Kuznetsov trace formula, Z Z 0 −1 lim K f 0 (n , n )ψ (n n )u(n , t)u(n , t) dn dn . →∞ 1 2 1 2 1 2 1 2 t N 0(F) N 0(F)

Our main tool is the Plancherel formula for p-adic groups, which was first stated, with an outlined proof, by Harish-Chandra[1984]. Silberger[1996] later filled in an important proof of one of the steps in the theorem. More recently Waldspurger [2003] provided a complete proof. As in[Arthur 1991, §2], we begin by rewriting K f 0 (x, y) using the Plancherel formula. First we introduce some additional notation. For an irreducible representa- 0 tion (σ, Vσ ) of G (F) let Ꮾ(σ ) be the Hilbert space of Hilbert–Schmidt operators on Vσ . The inner product on Ꮾ(σ ) is defined as

hS, S0i := tr(SS0∗) for S, S0 ∈ Ꮾ(σ ), where tr(SS0∗) = P hSS0∗u , u i and this sum converges o.n.b.Vσ i i absolutely and does not depend on the basis. For a discrete series representation σ of a group G let d(σ ) be the formal degree of σ . 0 Let 52(Ge (F)) be a set of representatives for the equivalence classes of irre- 0 0 ducible, tempered square integrable representations of Ge (F) and let {52(Me (F))} 0 be a set of representatives of unitary characters χ on Me (F) such that χ($ ) = 1. For 0 λ(H 0 (m)) 0 a character χ of M (F) and λ∈ރ, let χλ(m)=χ(m)e P . For χ ∈{52(Me (F))}, G0 0 = 0 IP0 (χλ) IP (χλ) is the normalized induced representation of Ge (F) acting on 0 a Hilbert space ᏴP0 (χ) of vector-valued functions on K . Let ᏮP0 (χ) be a fixed K 0-finite orthonormal basis of the Hilbert space of Hilbert–Schmidt operators on ᏴP0 (χ). Let m(σ ) be the Plancherel density. We normalize our measures following [Arthur 1991, §1]. The Plancherel density satisfies m(χλ) = d(χ)µ(χλ), where µ(χλ) is Harish-Chandra’s µ-function. For a fixed x ∈ G0(F), let Z 0 0 h(v) = f1(xu) f2(uvx) du. Ge0(F) A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 411

∞ 0 −1 Then h ∈ Cc (Ge (F)) and K f 0 (x, y) = h(yx ), so by the Plancherel formula, X −1 K f 0 (x, y) = d(σ ) tr(σ (R(yx )h)) 0 σ∈52(Ge (F)) Z πi 1 X log q −1 + 0 2 tr(IP (χ, R(yx )h))m(χλ) dλ. 0 ∈{ 0 } − χ 52(Me (F)) 0∨ 0 ∗ 0 1 = 0 0 0 0 Because IP (χλ, R(yx )h) IP (χλ, f1 )IP (χλ, x)IP (χλ, f2)(IP (χλ, y)) , we have −1 tr(IP0 (χλ, R(yx )h)) X 0∨ 0 ∗ ∗ = (IP0 (χλ, f1 )IP0 (χλ, x)IP0 (χλ, f2), S )(IP0 (χλ, y), S ) S∈ᏮP0 (χ) X 0∨ 0 = tr(IP0 (χλ, f1 )IP0 (χλ, x)IP0 (χλ, f2)S)tr(IP0 (χλ, y)S) S∈ᏮP0 (χ) X 0 = tr(IP0 (χλ, x)Sλ[ f ])tr(IP0 (χλ, y)S),

S∈ᏮP0 (χ) 0 0 0∨ [ ] = 0 0 where Sλ f IP (χλ, f2)SIP (χλ, f1 ). 0 ∞ 0 0 For f ∈ Cc (Ge (F)), π an admissible representation, π( f ) has finite rank. Thus K the sum over S is a finite sum of an orthonormal basis of operators on ᏴP (χ) 0 for some open compact K0. Putting everything together we have Z Z 0 −1 K 0⊗ 0 (n , n )ψ (n n )u(n , t)u(n , t) dn dn f1 f2 1 2 1 2 1 2 1 2 N 0(F) N 0(F) Z X X 0 0∨ 0 −1 = d(σ ) tr(σ (n)(σ ( f2)Sσ ( f1 )))ψ (n )u(n, t) dn 0 0 ∈ N (F) σ∈52(Ge (F)) S Ꮾ(σ ) Z × tr(σ (n)S)ψ0(n−1)u(n, t) dn N 0(F) Z πi Z 1 X log q X 0 0 −1 + d(χ)× tr(IP0 (χλ, n)Sλ[ f ])ψ (n )u(n, t) dn 2 0 0 0 ∈ 0 N (F) χ∈{52(Me (F))} S ᏮP (χ) Z 0 −1 × tr(IP0 (χλ, n)S)ψ (n )u(n, t) dn µ(χλ) dλ. N 0(F) To finish the spectral expansion we show that the above unipotent integrals stabilize. We first note that in the discrete series case the above integrals are absolutely convergent without any truncation for reasons similar to those in Section 4B1. 0 Lemma 3.5 (spectral stabilization). For any complex-valued function φ on Ge (F) that is biinvariant under a fixed open compact subgroup, there exists a positive 412 BROOKE FEIGON integer c such that for all t ≥ c, Z Z φ(n)ψ0(n)u(n, t) dn = φ(n)ψ0(n)u(n, c) dn. N 0(F) N 0(F) This c only depends on the open compact subgroup under which φ is biinvariant.

0 Proof. Let K1 be an open compact subgroup of Ge (F) under which φ is biinvariant. 0 K1 must contain a neighborhood of the identity, so there exists a positive integer c such that a 0 0 ∈ K for all a ∈ (1 + $ c ᏻ )E1. 0 a−1 1 E

We show that for m > c0,

Z 1 x φ ψ0(x) dx = 0. −m × 0 1 $ ᏻF We note that a 0 1 x a−1 0 1 aax = . 0 a−1 0 1 0 a 0 1

0 c0 Thus for x ∈ 1 + $ ᏻF ,

1 x0x 1 x φ = φ . 0 1 0 1

Hence Z 1 x φ ψ0(x) dx −m × 0 1 $ ᏻF X Z 1 αx = φ ψ0(αx) dx −m + c0 0 1 × c0 $ (1 $ ᏻF ) α∈ᏻF /(1+$ ᏻF ) X 1 $ −mα Z = φ ψ0($ −mα) ψ0(x) dx. 0 1 c0−m × c0 $ ᏻF α∈ᏻF /(1+$ ᏻF )

The last line equals 0 for m > c0. Thus for t > 2c0, Z Z 0 0 0 φ(n)ψ (n)u(n, t) dn = φ(n)ψ (n)u(n, 2c ) dn. N 0(F) N 0(F) We have now proved the following. A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 413

0 = 0 ⊗ 0 ∈ ∞ 0 × 0 Proposition 3.6. For any f f1 f2 Cc (Ge (F) Ge (F)), Z Z 0 −1 lim K f 0 (n , n )ψ (n n )u(n , t)u(n , t) dn dn →∞ 1 2 1 2 1 2 1 2 t N 0(F) N 0(F) Z πi X 0 0 0 1 X 0 log q 0 0 0 = d(σ )D 0 ( f ) + d(χ ) D 0 ( f )µ(χ ) dλ, σ 2 χλ λ 0 0 0 0 0 where σ ∈52(Ge (F)) χ ∈{52(Me (F))} 0 0 = X 0 0 0 0 0∨ 0 Dσ 0 ( f ) Wσ 0 (σ ( f2)Sσ ( f1 ))Wσ 0 (S), S∈Ꮾ(σ 0) Z 0 0 0 −1 Wσ 0 (S) = tr(σ (n)S)ψ (n ) dn, N 0(F) 0 0 X 0 0 0 0 0∨ 0 D 0 ( f ) = W 0 (IP0 (χλ, f2)SIP0 (χλ, f1 ))W 0 (S), χλ χλ χλ 0 S∈ᏮP0 (χ ) Z 0 0 0 −1 W 0 (S) = lim tr(IP0 (χ , n)S)ψ (n )u(n, t) dn. χ →∞ λ λ t N 0(F) We note that Theorem 1.3 now follows from the results of Propositions 3.4 and 3.6.

4. The local relative trace formula and periods for PGL(2)

In this section we define a local relative trace formula for PGL(2). We expand this local relative trace formula geometrically in terms of separate orbital integrals of f1 and f2. Then we use Harish-Chandra’s Plancherel formula to rewrite this expression spectrally in terms of representations. We define a regularized period integral, show that it is an H × H-invariant linear functional and that it is the term that appears in the spectral expansion of the local relative trace formula. ∞ We define the local relative trace formula for f = f1 ⊗ f2 ∈ Cc (Ge(F) × Ge(F)) as the equality between the geometric and spectral expansions of Z Z lim K f (h, n)ψ(n)u(h, t)u(n, t) dn dh t→∞ He(F) N(F) where Z −1 K f (h, n) = f1(g) f2(h gn) dg. Ge(F) As we did with the local Kuznetsov trace formula, we show that for a fixed f this limit stabilizes.

4A. The geometric expansion. We will rewrite Z Z lim K f (h, n)ψ(n)u(h, t)u(n, t) dn dh t→∞ He(F) N(F) 414 BROOKE FEIGON as an integral over admissible cosets of a product of an orbital integral for f1 and an orbital integral f2. 4A1. Integration formula. As pointed out in[Jacquet et al. 1999, §VI.13], by [Springer 1985], G(F) = H(F)P(F) t H(F)ηP(F), where η is any element in G(F) such that η −1η = 0 1 . Let η = η a 0 and γ = 1 0√ , where √ 1 0 a 0 1 α 0 α+ τ E = F( τ). Then 1 0 ∪ { : ∈ } ∪ { : × 1} 0 1 γα α F ηa a is in a set of representatives for E /E is a set of representatives for the double cosets of He(F)\Ge(F)/N(F). For g ∈ G(F), let

−1 Cg(He(F) × N(F)) = {(h, n) ∈ He(F) × N(F) : h gn = zg for some z ∈ Z(F)}.

Deﬁnition 4.1. An element g ∈ Ge(F) and its corresponding orbit is called admissible if the map Cg(He(F) × N(F)) → ރ : (h, n) 7→ ψ(n) is trivial. By a short calculation we see that √ 1 y 1 y(α + τ) C (H(F) × N(F)) = , : y ∈ F , γα e 0 1 0 1 × = Cηa (He(F) N(F)) 1.

× 1 Thus the orbits represented by {ηa : a ∈ E /E } ∪ {γ0} are admissible. We have the following integration formula. For any F ∈ Cc(Ge(F)), Z Z Z −1 × (4-1) F(g) dg = F(h ηan) dn dh|a|E d a. Ge(F) E×/E1 He(F)×N(F) 4A2. Separating the orbital integrals. Let Z Z t R ( f ) = K f (h, n)ψ(n)u(h, t)u(n, t) dn dh. He(F) N(F) t R ( f ) is absolutely convergent because f1(g), u(h, t) and u(n, t) have compact support on Ge(F), He(F) and N(F) respectively. By changing the order of integration and applying (4-1) we see that Rt ( f ) equals Z Z Z −1 −1 −1 f1(h1 ηan1) f2(h2 h1 ηan1n2) E×/E1 He(F)×N(F) He(F)×N(F) × × ψ(n2)u(h2, t)u(n2, t) dn2 dh2 dn1 dh1|a|E d a. By a change of variables we have Z t t × R ( f ) = R (ηa, f )|a|E d a, E×/E1 A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 415 where Z Z t = −1 −1 −1 R (ηa, f ) f1(h1 ηan1) f2(h2 ηan2)ψ(n1 n2) He(F)×N(F) He(F)×N(F) × −1 −1 u(h1 h2, t)u(n1 n2, t) dn2 dh2 dn1 dh1. To complete the geometric expansion of the local relative trace formula we rewrite Rt ( f ) for t 0 as an integral of a product of two separate orbital integrals that are not truncated. We omit the proof the lemma below as it is very similar to the proof of Lemma 3.3.

∞ Lemma 4.2. For f ∈Cc (Ge(F)×Ge(F)), there exists a T > 0 such that for all t ≥ T and n ∈ ޚ, Z Z t × × R (ηa, f ) d a = O( f1, ψ, a)O( f2, ψ, a) d a n × 1 n × 1 $ ᏻE /E $ ᏻE /E where Z Z −1 O( f, ψ, a) = f (h ηan)ψ(n) dn dh. He(F) N(F) We have proved the following proposition.

∞ Proposition 4.3. For any f = f1 ⊗ f2 ∈ Cc (Ge(F) × Ge(F)), Z Z lim K (h, n)ψ(n)u(h, t)u(n, t) dn dh t→∞ He(F) N(F) Z × = O( f1, ψ, a) O( f2, ψ, a)|a|E d a. a∈E×/E1 Here, as in the local Kuznetsov trace formula, we have actually shown that the limit of the truncated local relative trace formula stabilizes.

4B. The spectral expansion and period integrals. We want to develop a spectral expansion for the local relative trace formula, Z Z lim K f (h, n)ψ(n)u(h, t)u(n, t) dn dh. t→∞ He(F) N(F) As in the previous section, we expand the kernel via the Plancherel formula: Z Z (4-2) K (h, n)ψ(n)u(h, t)u(n, t) dn dh He(F) N(F) πi X X Z log q = d(σ )Dt ( f ) + 1 d(χ) µ(χ )Dt ( f ) dλ σ 2 λ χλ 0 σ ∈52(Ge(F)) χ∈{52(Me(F))} 416 BROOKE FEIGON where

t X t ∨ t Dσ ( f ) = Pσ (σ ( f2)Sσ ( f1 ))Wσ (S), S∈Ꮾ(σ ) X Dt ( f ) = Pt (I (χ , f )SI (χ , f ∨))W t (S), χλ IP (χλ) P λ 2 P λ 1 IP (χλ) S∈ᏮP (χ) Z t −1 Wπ (S) = tr(π(n)S)ψ(n )u(n, t) dn, N(F) Z t Pπ (S) = tr(π(h)S)u(h, t) dh. He(F) By Lemma 3.5, there exists a positive integer c, such that for t > c, Z t −1 Wπ (S) = tr(π(n)S)ψ(n )u(n, c) dn. N(F) Thus as in the previous section, we define Z −1 Wσ (S) = tr(σ (n)S)ψ(n ) dn, N(F) Z −1 W (S) = lim tr(IP (χ , n)S)ψ(n )u(n, t) dn. χλ →∞ λ t N(F) To finish the spectral expansion of the local relative trace formula we need to define the regularized integral Z ∗ tr(IP (χλ, h)S) dh He(F) because tr(IP (χλ, −)S) is not integrable over He(F). Many of the techniques in this section are inspired by the work of Jacquet, Lapid and Rogawski in[Jacquet et al. 1999]. In that paper they define a regularized period integral for an automorphic form φ on G(ށ) integrated over H where G is a reductive group over a number field F and H is the fixed point set of an involution of G. They focus on the case G = ResE/F H where E/F is a quadratic extension and they obtain explicit results for G = GL(n, E), H = GL(n, F). α 0 λHM (m) λ For λ ∈ ރ and m = 0 β ∈ M(F), let e = |α/β|E . If g = m(g)n(g)k(g), λH (g) λH (m(g)) m(g) ∈ M(F), n(g) ∈ N(F), k(g) ∈ K , we let e P = e M . Let δP (m) = λH HM (m) MH ∈ e . We give analogous definitions for e and δPH so that for m MH (F), λH (m) 2λH (m) e M = e MH . + We recall the Cartan decomposition H(F) = K H MH (F)K H , where + α 0 α M (F) = ∈ M(F) : v ≤ 0 . H 0 β β A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 417

Then for any absolutely integrable function f Z Z Z Z f (h) dh = DP (m) f (k1mk2) dm dk2 dk1, + H He(F) KeH KeH MeH (F) where α 0 |α/β|F (1 + |$|F ) v(α/β) ≤ 0, DP = H 0 β 0 v(α/β) > 0. To define the regularized integral, we begin by defining a regularized integral on + MH (F). We note that 1 0 0 ≤ v(α) ≤ t, 1 − u , t = α 1 v(α) > t. For Re ν < − Re λ, ∞ Z (t+1)2(ν+λ) (ν+λ)HM (m) X 2n(ν+λ) q (4-3) e (1 − u(m, t)) dm = q = + . + 1−q2(ν λ) MeH (F) n=t+1 We write Z ] eλHM (m)(1 − u(m, t)) dm + MeH (F) to denote the meromorphic continuation at ν = 0 of (4-3). This is well-defined so long as λ 6= 0. Let r X λi HM (m) φ(k1mk2) = φi (k1, k2) fi (m)e , k1, k2 ∈ K H , (4-4) i=1 1 m = , n ≥ 0, f ∈ C (M(F)) $ n i c e

6= − 1 with λi 2 . We deﬁne for t 0, Z ] φ(h)(1 − u(h, t)) dh He(F) r X Z Z ] = λi HM (m) − φi (k1, k2) DPH (m)e (1 u(m, t)) dm × + i=1 KeH KeH MeH (F) r Z Z ] −1 X (λi +1/2)HM (m) = (1 + q ) φi (k1, k2) e (1 − u(m, t)) dm. × + i=1 KeH KeH MeH (F)

If φ is a matrix coefﬁcient of IP (χλ) where χ($ ) = 1 then by smoothness and the asymptotics of matrix coefﬁcients there exists a function C P φ of the form in 418 BROOKE FEIGON

∈ { − 1 − − 1 } (4-4) with λi λ 2 , λ 2 and for n 0, 1 1 C P φ k k = φ k k . 1 $ n 2 1 $ n 2 Note that the condition for the regularized integral to exist is now that λ 6= 0.

Definition 4.4. For any matrix coefficient φ of IP (χλ) such that χ($ )=1 and λ6=0, Z ∗ Z Z ] φ(h) dh := φ(h)u(h, t) dh + φ(h)(1 − u(h, t)) dh He(F) He(F) He(F) for t 0. One can check that this definition of the regularized integral is independent of t and agrees with the usual integral if we start with something that is integrable. Now we will prove that it is H-invariant and then we will explicitly relate the regularized period to the truncated period that occurs in the local trace formula. h Let φ 0 (x) = φ(xh0) for h0 ∈ He. Note that if φ is a matrix coefficient of π then φh0 is as well. × Lemma 4.5. Fix h0 ∈ H, λ 6= 0 and a character χ of E with χ($ ) = 1. Then for any matrix coefficient φ of IP (χλ) and t 0, Z Z ] h0 − DPH (m)φ (k1mk2)(1 u(k1mk2h0, t)) dm dk1 dk2 KeH ×KeH MeH (F) Z Z ] = − DPH (m)φ(k1mk2)(1 u(m, t)) dm dk1 dk2. KeH ×KeH MeH (F) + Proof. For g ∈ G(F) let ᏹ(g) ∈ M (F) be such that g = k1ᏹ(g)k2, k1, k2 ∈ K . For Re ν 0 and t 0, Z Z h0 DPH (m)φ (k1mk2) KeH ×KeH MeH (F) ν(HM (ᏹ(k1mk2h0))) × e (1 − u(k1mk2h0, t)) dm dk1 dk2 Z ν(HM (ᏹ(hh0))) = φ(hh0)e (1 − u(hh0, t)) dh He(F) Z = φ(h)eν(HM (ᏹ(h)))(1 − u(h, t)) dh He(F) Z Z = ν(HM (m)) − DPH (m)φ(k1mk2)e (1 u(m, t)) dm dk1 dk2 KeH ×KeH MeH (F) by the invariance of Haar measure, since both sides are absolutely convergent. For t 0, if h ∈ supp(1 − u( · , t)), then ᏹ(hh0) = ᏹ(h)ᏹ(k2h0). Thus both sides of the equation above have a meromorphic continuation whose value at ν = 0 gives the statement of the lemma. A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 419

Proposition 4.6 (H-invariance). Let φ be a matrix coefficient of IP (χλ), where χ($ ) = 1 and λ 6= 0, and let h0 ∈ H(F). Then Z ∗ Z ∗ φh0 (h) dh = φ(h) dh. He(F) He(F) Proof. By the definition of the regularized integrals, the statement of the proposition will follow once we prove the following equality: Z Z ] h0 DP (m)φ (k1mk2)(1 − u(m, t)) dm dk1 dk2 + H KeH ×KeH MeH (F) Z Z ] − DP (m)φ(k1mk2)(1 − u(m, t)) dm dk1 dk2 + H KeH ×KeH MeH (F) Z Z = φ(h)u(h, t) dh − φh0 (h)u(h, t) dh. He(F) He(F) First we note that by Lemma 4.5 Z Z ] h0 DP (m)φ (k1mk2)(1 − u(m, t)) dm dk1 dk2 + H KeH ×KeH MeH (F) Z Z ] − DP (m)φ(k1mk2)(1 − u(m, t)) dm dk1 dk2 + H KeH ×KeH MeH (F) Z Z ] −1 = DP (m)φ(k1mk2)(1 − u(mk2h , t))) dm dk1 dk2 + H 0 KeH ×KeH MeH (F) Z Z ] − DP (m)φ(k1mk2)(1 − u(m, t)) dm dk1 dk2. + H KeH ×KeH MeH (F) · −1 − · For fixed h0 and t sufficiently large, u( h0 , t) u( , t) has support contained in an annulus. From this fact one can easily check that the previous line is equal to the convergent integral Z Z −1 DP (m)φ(k1mk2)[u(m, t) − u(mk2h , t)] dm dk1 dk2. + H 0 KeH ×KeH MeH (F) Z = [ − −1 ] φ(h) u(h, t) u(hh0 , t) dh He(F) Z Z h = φ(h)u(h, t) dh − φ 0 (h)u(h, t) dh. He(F) He(F) h We note that Proposition 4.6 also holds if we replace φ 0 with φ(h0−) so our regularized integral is H × H invariant. Now we derive an explicit formula relating regularized periods to truncated periods for the matrix coefficients that appear in the trace formula. We begin by recalling 420 BROOKE FEIGON some definitions of Harish-Chandra’s. For σ an admissible, tempered representation of G, Ꮽσ (G) is the space of functions on G spanned by K-finite matrix coefficients of σ , Ꮽtemp(G) is the sum of Ꮽσ (G) over all admissible tempered representations of G and Ꮽ2(G) is the sum of Ꮽσ (G) over all unitary, square integrable representations. For τ a finite dimensional, unitary, two-sided representation of K ,

Ꮽσ (G, τ) = { f ∈ Ꮽσ (G) ⊗ Vτ : f (k1gk2) = τ(k1) f (g)τ(k2), g ∈ G, k1, k2 ∈ K }.

Then Ꮽtemp(G, τ) and Ꮽ2(G, τ) are deﬁned similarly. Let τM = τ|K ∩M . By[Harish-Chandra 1984, §3] for f ∈ Ꮽσ (G, τ) there exists P a unique function C f ∈ Ꮽ(M, τM ) such that

1 2 α 0 α 0 P α 0 lim δP f − (C f ) = 0. | α | →∞ β E 0 β 0 β 0 β

We call C P f the weak constant term of f . For two parabolics P1, P2 with Levi component M, let = { ∈ : = ∈ ∩ ∈ ∩ } VP1|P2 v V τ(n1)vτ(n2) v, n1 NP1 K, n2 NP2 K ∈ and let τP1|P2 be the subrepresentation of τM on VP1|P2 . For 9 Ꮽ2(M, τP|P ) and λ ∈ [0, πi/log q], the Eisenstein integral E P (g, 9, λ) ∈ Ꮽtemp(G, τ) is deﬁned as Z −1 (λ+1/2)(HM (kg)) E P (g, 9, λ) = τ(k) 9P (kg)e dk K where 9P extends 9 to G by 9P (nmk) = 9(m)τ(k) for n ∈ N, m ∈ M, k ∈ K . The weak constant term of the Eisenstein integral uniquely deﬁnes Harish- Chandra’s c-functions[1984, §6]. For each element w in the Weyl group W of

Ge, the c-function cP|P (w, λ) is a linear map from Ꮽ2(M, τP|P ) to Ꮽ2(M, τP|P ) such that

−1 ∗ µ(χλ) = cP|P (s, λ)χ cP|P (s, λ)χ .

For the rest of this section we let c(1, λ) = cP|P (1, λ)χ and c(w, λ) = cP|P (w, λ)χ . K We note that the S we consider are actually in ᏴP (χ) 0 for some open compact K K0. Harish-Chandra[1976, §7] gives an isomorphism S → 9S from End(ᏴP (χ) ) 2 onto Ꮽχ (M, (τ)P|P ) where Vτ is a particular subspace of L (K × K ) such that = tr(IP (χλ, k1gk2)S) E P (g, 9S, λ)k1,k2 . A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 421

We can now relate the regularized integral to what appears in the local relative trace formula. −1 Proposition 4.7. For χ = (χ, χ ) ∈ {52(Me(F)}, χ($ ) = 1, λ 6= 0, t 0, Z ∗ tr(IP (χλ, h)S) dh He(F) Z = tr(IP (χλ, h)S)u(h, t) dh He(F) q2λ(t+1) Z +δ(χ)(1 + q−1) c(1, λ)9 (1) dk dk −q2λ S k1,k2 1 2 1 KeH ×KeH q−2λ(t+1) Z + c(w, λ)9 (1) dk dk , −q−2λ S k1,k2 1 2 1 KeH ×KeH where δ(χ) = 1 if χ| × = 1 and δ(χ) = 0 if χ| × 6= 1. ᏻF ᏻF Proof. For S ∈ ᏮP (χ), 9S ∈ Ꮽχ (M, (τ0)P|P ) and

c(1, λ)9S, c(w, λ)9S ∈ Ꮽχ (M, (τ0)P|P ).

Therefore 9 = 9S can be written as a sum of matrix coefﬁcients of χ. Thus P C E P (m, 9, λ)k1,k2 = λHM (m) + −λHM (m) c(1, λ)9(m)k1,k2 e c(w, λ)9(m)k1,k2 e

= λHM (m) + −λHM (m) χ(m) (c(1, λ)9)(1)k1,k2 e (c(w, λ)9)(1)k1,k2 e where χ(m) ∈ ރ×. Hence for t 0, Z ν(HM (m)) DP (m) tr(IP (χλ, k1mk2)S)e (1 − u(m, t)) dm + H MeH (F) Z − 1 2 (λ+ν)(HM (m)) = DP (m)δ (m)(c(1, λ)9)(1)k ,k e χ(m)(1 − u(m, t)) dm + H P 1 2 MeH (F) Z − 1 2 (−λ+ν)(HM (m)) + DP (m)δ (m)(c(w, λ)9)(1)k ,k e χ(m)(1−u(m, t)) dm + H P 1 2 MeH (F) Z −1 (λ+ν)(HM (m)) = (1 + q )(c(1, λ)9)(1)k ,k e χ(m)(1 − u(m, t)) dm 1 2 + MeH (F) Z −1 (−λ+ν)(HM (m)) + (1 + q )(c(w, λ)9)(1)k ,k e χ(m)(1 − u(m, t)) dm 1 2 + MeH (F) Z ∞ −1 × X 2(λ+ν)n = (1 + q ) χ(α) d α (c(1, λ)9)(1)k ,k q × 1 2 ᏻF n=t+1 + 2(−λ+ν)n (c(w, λ)9)(1)k1,k2 q . 422 BROOKE FEIGON

R × Clearly × χ(α) d α = 0 unless χ| × = 1. If χ| × = 1, the previous line equals ᏻF ᏻF ᏻF q2(λ+ν)(t+1) q2(−λ+ν)(t+1) (1 + q−1) c(1, λ)9(1) + c(w, λ)9(1) . 1−q2(λ+ν) k1,k2 1−q2(−λ+ν) k1,k2 Therefore for t 0 Z ] DP (m) tr(IP (χλ, k1mk2)S)(1 − u(m, t)) dm + H MeH (F) q2λ(t+1) q−2λ(t+1) = δ(χ)(1 + q−1) c(1, λ)9 (1) + c(w, λ)9 (1) 1−q2λ S k1,k2 1−q−2λ S k1,k2 and the proposition now follows. Lemma 4.8. Let χ = (χ, χ −1) where χ is a character of E× such that χ($ ) = 1. Then

(1) If χ|F× 6= 1 and χ|E1 6= 1, then Z ∗ Z tr(IP (χλ, h)S) dh = tr(IP (χλ, h)u(h, t) dh = 0. He(F) He(F)

(2) If χ|F× 6= 1 and χ|E1 = 1, then for t 0, Z ∗ Z tr(IP (χλ, h)S) dh = tr(IP (χλ, h)S)u(h, t) dh. He(F) He(F) R ∗ (3) If χ| × = 1 and χ| 1 6= 1, then tr(I (χ , h)S) dh is 0 whenever deﬁned and F E He P λ Z Z = c(1, λ)9S(1)k1,k2 dk1 dk2 c(s, λ)9S(1)k1,k2 dk1 dk2 KeH ×KeH KeH ×KeH at λ = 0. 2 (4) If χ|F× = 1 and χ|E1 = 1, then χ = 1. In this case c(1, λ) and c(s, λ) have a simple pole at λ = 0 and so µ(χλ) has a zero of order two at λ = 0 and µ(χλ)c(1, λ) = µ(χλ)c(s, λ) = 0 at λ = 0. In all cases, Z ∗ µ(χλ) tr(IP (χλ, h)S) dh He(F) is holomorphic for all λ ∈ iޒ, S ∈ ᏮP (χ). Proof. In this proof we follow the techniques of[Jacquet ≥ 2012]. Case 2 is obvious from the above work. Case 1 is obvious from the above work and the H-invariance ∗ of R tr(I (χ , h)S) dh [Jacquet et al. 1999, Proposition 22]. He P λ A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 423

The vanishing of the regularized period for λ 6= 0 in case 3 also follows from H-invariance. Then by the previous proposition we know that for λ 6= 0,

Z tr(IP (χλ, h)S)u(h, t) dh He(F) q2λ(t+1) Z = −(1 + q−1) c(1, λ)9 (1) dk dk −q2λ S k1,k2 1 2 1 KeH ×KeH q−2λ(t+1) Z + c(w, λ)9 (1) dk dk . −q−2λ S k1,k2 1 2 1 KeH ×KeH

Both sides are holomorphic and the left-hand side is also deﬁned and holomorphic for λ = 0. As

q2λ(t+1) −1 q−2λ(t+1) 1 Res = = and Res = = , λ 0 1−q2λ 2 log q λ 0 1−q−2λ 2 log q we must have that

Z Z = c(1, 0)9(1)k1,k2 dk1 dk2 c(w, 0)9(1)k1,k2 dk1 dk2. KeH ×KeH KeH ×KeH

In case 4 the poles and zeros are well-known and can also be seen by explicit computations of the intertwining operators. We have that

Z µ(χλ) tr(IP (χλ, h)S)u(h, t) dh He(F) Z ∗ = µ(χλ) tr(IP (χλ, h)S) dh He(F) q2λ(t+1) Z −(1 + q−1)µ(χ ) c(1, λ)9 (1) dk dk λ −q2λ S k1,k2 1 2 1 KeH ×KeH q−2λ(t+1) Z + c(w, λ)9 (1) dk dk . −q−2λ S k1,k2 1 2 1 KeH ×KeH

The left-hand side is 0 at λ = 0 and the last two terms are holomorphic at λ = 0 so the ﬁrst term must be holomorphic at λ = 0. 424 BROOKE FEIGON

Let = X [ ] Dχλ ( f ) Pχλ (Sλ f )Wχλ (S), S∈ᏮP (χ) Z ∗ = Pχλ (S) tr(IP (χλ, h)S) dh, He(F) ∨ Sλ[ f ] = IP (χλ, f2)SIP (χλ, f1 ), Z = + −1 X Deχ ( f ) (1 q )µ(χ0) Wχ0 (S) c(1, 0)ψS0[ f ](1)k1,k2 dk1 dk2. K H ×K H S∈ᏮP (χ) e e We now relate the distributions above to the truncated distributions from (4-2).

Lemma 4.9. Let χ = (χ, χ −1) where χ is a character of E× such that χ($ ) = 1.

(1) If χ|F× 6= 1 and χ|E1 6= 1, then Z πi log q t lim µ(χλ)D ( f ) dλ = 0. →∞ χλ t 0

(2) If χ|F× 6= 1 and χ|E1 = 1, then Z πi Z πi log q t log q lim µ(χλ)D ( f ) dλ = µ(χλ)Dχ ( f ) dλ. →∞ χλ λ t 0 0

(3) If χ|F× = 1 and χ|E1 6= 1, then Z πi log q t lim µ(χλ)D ( f ) dλ = Dχ ( f ). →∞ χλ e t 0

(4) If χ|F× = 1 and χ|E1 = 1, then Z πi Z πi log q t log q lim µ(χλ)D ( f ) dλ = µ(χλ)Dχ ( f ) dλ. →∞ χλ λ t 0 0 Proof. First we note that

πi πi Z log q Z log q X µ(χ )Dt ( f ) dλ = µ(χ ) Pt (S [ f ])W (S) dλ λ χλ λ IP (χλ) λ χλ 0 0 S∈ᏮP (χ) πi Z log q Z X −1 = tr(IP (χλ, n)S)ψ(n )u(n, t) dn 0 N(F) S∈ᏮP (χ) Z × µ(χλ) tr(IP (χλ, h)Sλ[ f ])u(h, t) dh dλ. He(F) A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 425

Cases 1 and 2 now follow directly from Lemma 4.8. For the remaining cases we note that by Proposition 4.7 for t 0, Z tr(IP (χλ, h)Sλ[ f ])u(h, t) dh He(F) Z ∗ = tr(IP (χλ, h)Sλ[ f ]) dh He(F) −1 Z 1+q 2λ(t+ 1 ) +δ(χ) q 2 c(1, λ)9 [ ](1) dk dk qλ −q−λ Sλ f k1,k2 1 2 KeH ×KeH Z −2λ(t+ 1 ) − 2 q c(w, λ)9Sλ[ f ](1)k1,k2 dk1 dk2 . KeH ×KeH In case 3, by Lemma 4.8 the regularized period vanishes and we are left computing

πi Z log q (4-5) (1 + q−1) lim µ(χ )W (S) →∞ λ χλ t 0 q2λ(t+1/2) Z c(1, λ)9 [ ](1) dk dk qλ −q−λ Sλ f k1,k2 1 2 KeH ×KeH q−2λ(t+1/2) Z − c(w, λ)9 [ ](1) dk dk dλ. qλ −q−λ Sλ f k1,k2 1 2 KeH ×KeH Let

1+q−1 Z f (λ) = µ(χ )W (S) c(1, λ)9 [ ](1) dk dk 1 2 λ χλ Sλ f k1,k2 1 2 KeH ×KeH Z − c(w, λ)9Sλ[ f ](1)k1,k2 dk1 dk2 , KeH ×KeH 1+q−1 Z f (λ) = µ(χ )W (S) c(1, λ)9 [ ](1) dk dk 2 2 λ χλ Sλ f k1,k2 1 2 KeH ×KeH Z + c(w, λ)9Sλ[ f ](1)k1,k2 dk1 dk2 . KeH ×KeH Then (4-5) equals

πi Z 2λ(t+ 1 ) −2λ(t+ 1 ) log q f (λ)(q 2 +q 2 ) lim 1 t→∞ qλ −q−λ 0 πi Z 2λ(t+ 1 ) −2λ(t+ 1 ) log q f (λ)(q 2 −q 2 ) + lim 2 dλ. →∞ λ −λ t 0 q −q

By Lemma 4.8, f1(0) = 0. Hence by Fourier analysis the ﬁrst integral will vanish. The limit of the second integral will be f2(0), which, by the identity in case 3 of 426 BROOKE FEIGON

Lemma 4.8, equals Z + −1 (1 q )µ(χ0)Wχ0 (S) c(1, 0)9S0[ f ](1)k1,k2 dk1 dk2. KeH ×KeH

For case 4 by Lemma 4.8 when multiplied by µ(χλ)Wχλ (S), f1(λ) and f2(λ) are holomorphic functions of λ and vanish at λ = 0, thus by similar analysis as above the last two terms vanish in the limit and we are left with the statement of the lemma. 4B1. Discrete series representations. Because the matrix coefficient of a supercuspidal representation σ has compact support it is obvious that Z Z lim tr(σ (h)S)u(h, t) dh = tr(σ (h)S) dh. t→∞ He(F) He(F) Now we will prove that this is also true for Steinberg representations. Lemma 4.10. For σ = St(χ), χ 2 = 1, the matrix coefficients are absolutely convergent over He(F). Thus the limit Z lim tr(σ (h)S)u(h, t) dh t→∞ He(F) exists and equals Z tr(σ (h)S dh. He(F) Proof. By [Borel and Wallach 1980, XI.4.3; Casselman 1995, 4.2.3], a matrix a 0 coefficient for σ evaluated at 0 b is equal to a matrix coefficient for the Jacquet a functor σN , evaluated at the same value, for b E sufficiently small. The Jacquet functor of σ is δP . Thus outside some compact set, our original matrix coeffi- − cient will behave like δP on MH (F). When we integrate over He(F), using the − −1/2 K H MH (F)K H decomposition, we get a measure factor of δP . Thus outside a R | | × set of compact support our integral will look like |a| 0. Putting everything together we have proved the following. ∞ Proposition 4.11. For any f ∈ Cc (Ge(F) × Ge(F)), Z Z lim K f (h, n)ψ(n)u(h, t)u(n, t) dn dh t→∞ He(F) N(F) = X + 1 X d(σ )Dσ ( f ) 2 Deχ ( f ) σ∈52(Ge(F)) χ∈{52(Me(F))} 2 χ 6=1,χ|F× =1 πi Z log q + 1 X 2 d(χ) µ(χλ)Dχλ ( f ) dλ, 0 χ∈{52(Me(F))} | = χ E1 1 A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 427 where

= X [ ] Dχλ ( f ) Pχλ (Sλ f )Wχλ (S), S∈ᏮP (χ) Z ∗ = Pχλ (S) tr(IP (χλ, h)S) dh, He(F) Z = + −1 X Deχ ( f ) (1 q )µ(χ0) Wχ0 (S) c(1, 0)ψS0[ f ](1)k1,k2 dk1 dk2, K H ×K H S∈ᏮP (χ) e e X ∨ Dσ ( f ) = Pσ (σ ( f2)Sσ ( f1 ))Wσ (S), S∈Ꮾ(σ ) Z Pσ (S) = tr(σ (h)S) dh. He(F) This proposition combined with Proposition 4.3 proves Theorem 1.4.

5. Comparison of local trace formulas and applications

We now combine the results of the previous two sections to compare the two trace × formulas. Let ωE/F be the quadratic character of F associated to E/F and let ω denote its trivial extension to E×.

0 ∞ 0 ∞ Deﬁnition 5.1. We say that f ∈ Cc (Ge (F)) and f ∈ Cc (Ge(F)) are matching functions if O0( f 0, ψ0, a) = ω(a)O( f, ψ, a) for all a ∈ E×.

By work of Ye[1989] and Flicker[1991, Proposition 3], we know that for 0 ∞ 0 ∞ any f ∈ Cc (Ge (F)) there exists a matching f ∈ Cc (Ge(F)) and vice versa. In fact, by the Fundamental Lemma, for f 0 spherical, we know that f is the corresponding function from the base change map between their Hecke algebras. Thus by the geometric expansion of the trace formulas in Propositions 3.4 and 4.3 we have the following statement.

0 ∞ 0 ∞ Proposition 5.2. For fi ∈ Cc (Ge (F)) and fi ∈ Cc (Ge(F)) matching functions for i = 1, 2, Z Z 0 −1 lim K f 0⊗ f 0 (n1, n2)ψ (n n2)u(n1, t)u(n2, t) dn1 dn2 →∞ 1 2 1 t N 0(F) N 0(F) Z Z = lim K f ⊗ f (h, n)ψ(n)u(h, t)u(n, t) dn dh. t→∞ 1 2 He(F) N(F) Now we use the equality of the trace formulas to compare the spectral expansions. By Propositions 3.6, 4.11 and 5.2 we have the following result. 428 BROOKE FEIGON

0 Theorem 5.3. For fi and fi matching functions for i = 1, 2,

Z πi X 0 0 0 0 1 X 0 log q 0 0 0 0 d(σ )D 0 ( f ⊗ f )+ d(χ ) µ(χ )D 0 ( f ⊗ f ) dλ σ 1 2 2 λ χλ 1 2 0 0 0 0 0 σ ∈52(Ge (F)) χ ∈{52(Me (F))} = X ⊗ + 1 X ⊗ d(σ )Dσ ( f1 f2) 2 Deχ ( f1 f2) σ ∈52(Ge(F)) χ∈{52(Me(F))} 2 χ 6=1,χ|F× =1 πi Z log q + 1 X ⊗ 2 d(χ) µ(χλ)Dχλ ( f1 f2) dλ. 0 χ∈{52(Me(F))} | = χ E1 1

The unstable base change map associated to ω lifts principal series representations 0 of Ge to principal series representations IP (χ) of Ge such that χ|E1 = 1. It also lifts 0 certain square integrable representations of Ge to the principal series representations 2 of Ge deﬁned by IP (χω) such that χ 6= 1, χ|F× = 1. It lifts the remaining square 0 integrable representations of Ge to square integral representations of Ge [Rogawski 1990; Flicker 1982]. Thus we could rephrase the right-hand side of Theorem 5.3 in terms of summing over the representations of Ge that are the unstable base change 0 lifts of representations of Ge . The extra discrete term Weχ ( f ) corresponds exactly to 0 the representations that lift from the discrete series of Ge to the principal series of Ge. We also note that the only representations that appear on the right-hand side of Theorem 5.3 are those σ or IP (χλ) for which there is a matrix coefﬁcient such that the regularized integral over H is nonzero. This gives us a more explicit description of the nonvanishing H invariant linear functional that characterizes the image of the unstable base change map. We would like to relate our distributions to the local factors in the Bessel and relative Bessel distributions. Recall from the introduction that Jacquet’s global 0 relative trace formula tells us that for f on U(2, ށF ) and f on GL(2, ށE ) matching 0 functions, if a cuspidal representation π of U(2, ށF ) maps to π of GL(2, ށE ) under unstable base change, then

0 0 Bπ 0 ( f ) = Bπ ( f ) A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 429 where 0 0 X 0 0 0 0 0 0 Bπ 0 ( f ) = W (π ( f )φ )W (φ ), 0 φ ∈o.n.b.(Vπ0 ) X Bπ ( f ) = P(π( f )φ)W(φ),

φ∈o.n.b.(Vπ ) Z W 0(φ0) = φ0(n)ψ0(n) dn, 0 0 N (F)\N (ށF ) Z W(φ) = φ(n)ψ(n) dn, N(E)\N(ށE ) Z P(φ) = φ(h) dh 6= 0. GL(2,F)Z(ށF )\ GL(2,ށF )

0 0 0 0 While B 0 ( f ) and Bπ ( f ) factor into local Bessel distributions B 0 ( f ) and Bπ ( fv), π πv v v it is not clear how to normalize the local Bessel distributions. We can rewrite our local distributions as a product of two local Bessel (or local relative Bessel) distributions:

0 0 Lemma 5.4. (1) For σ an irreducible supercuspidal representation of Ge (F), 0 there exists a local Bessel distribution Bσ 0 , unique up to a constant of absolute value 1, such that

0 0 0 0 0 0 0 Dσ 0 ( f1 ⊗ f2) = Bσ 0 ( f2)Bσ 0∗ ( f1).

(2) For σ an irreducible supercuspidal representation of Ge(F), there exists a local relative Bessel distribution Bσ , unique up to a constant of absolute value 1, such that

Dσ ( f1 ⊗ f2) = Bσ ( f2)Bσ ∗ ( f1).

Proof. We recall that Z 0 0 X 0 0 0 0 0∗ 0 0 −1 Dσ 0 ( f ) = tr(σ (n1)σ ( f2)S σ ( f1))ψ (n1) dn1 N 0 F S0∈Ꮾ(σ 0) ( ) Z 0 0 0 −1 tr(σ (n2)S )ψ (n2) dn2. N 0(F)

0 ∗ ∗ Let V = Vσ 0 . As S is an endomorphism on V there exist v ∈ V, v ∈ V such that S0 = v ⊗ v∗. Then the linear functional on V ⊗ V ∗ that acts by Z v ⊗ v∗ 7→ tr(σ 0(n)v ⊗ v∗)ψ0(n)−1 dn N 0(F) 430 BROOKE FEIGON transforms under n on v and v∗ by ψ0. Thus it is a Whittaker functional on V ⊗ V ∗. By the uniqueness of Whittaker models, Z tr(σ 0(n)S0)ψ0(n)−1 dn = W 0(v)W 0(v∗). N 0(F) Thus 0 0 X 0 0 0 0 0 0∗ 0 ∗ 0 ∗ Dσ 0 ( f ) = W (σ ( f2)v)W (v)W (σ ( f1)v )W (v ) v⊗v∗ 0 0 0 0 = Bσ 0 ( f2)Bσ 0∗ ( f1). 0 0 We note that if we change Bσ 0 by a constant c, then Bσ 0∗ will change by c. The proof for the local relative Bessel distributions is similar, using the uniqueness of the H-invariant linear functional[Hakim 1991; Flicker 1991, Proposition 11]. We can also describe matching functions by an equality of all the Bessel distributions. 0 ∈ ∞ 0 0 0 ⊗ 0 = Lemma 5.5 (density). (1) If f1 Cc (Ge (F)) is such that Dσ 0 ( f1 f2) 0 0 0 0 for all irreducible tempered representations σ of Ge (F) and all f2, then 0 0 0−1 = ∈ × O ( f1, ψ , a) 0 for all a E . ∞ (2) If f1 ∈ Cc (Ge(F)) is such that Dσ ( f1 ⊗ f2) = 0 and Deσ ( f1 ⊗ f2) = 0 for all ir- −1 reducible tempered representations σ of Ge(F) and f2, then O( f1, ψ , a) = 0 for all a ∈ E×. 0 0 ⊗ 0 = 0 Proof. If Dσ 0 ( f1 f2) 0 for all σ , then by Theorem 1.3, Z 0 0 0−1 0 0 0 × |a|E O ( f1, ψ , a)O ( f2, ψ , a) d a = 0 a∈E×/E1 0 ∈ ∞ 0 0 0 −1 for all f2 Cc (Ge (F)). As O ( f1, ψ , a) is a locally constant function of a there 0 0 0−1 exists some open compact U such that O ( f1, ψ , a) is biinvariant under it. Then 0 0 0 0−1 by choosing f2 such that O ( f2, ψ , a) has support contained in U we see that 0 0 0−1 = O ( f1, ψ , a) 0. The second case follows from the ﬁrst one. Combining Theorem 5.3 with Lemma 5.4 and the global relative trace formula, we have the following result: Corollary 5.6. If σ is the supercuspidal representation of Ge(F) that is the unstable 0 0 0 base change lift of the supercuspidal representation σ on Ge (F), and fi and fi are matching functions for i = 1, 2, then 0 0 0 d(σ )Dσ 0 ( f1 ⊗ f2) = d(σ )Dσ ( f1 ⊗ f2). Proof. From the global comparison of relative trace formulas[Flicker 1991; Lapid 2006; Ye 1989] and a standard globalization argument we know there exists a 0 0 = 0 0 constant cσ such that Bσ 0 ( fi ) cσ Bσ ( fi ) for all matching fi , fi . Take f1 and f2 to A RELATIVE TRACE FORMULA FOR PGL(2) IN THE LOCAL SETTING 431

0 0 0 6= 6= 0 be matrix coefﬁcients of σ and σ such that Bσ 0 ( f1) 0 and Bσ ( f2) 0. Take f2 0 a matching function to f2 and f1 a matching function to f1. Then by Theorem 5.3, 0 0 0 d(σ )Dσ 0 ( f ) = d(σ )Dσ ( f ). In addition to the spectral comparison, these local trace formulas also have applications on the geometric side. If we deﬁne the inner product of two functions × 1 g1, g2 on E /E by Z × hg1, g2i = g1(a)g2(a)|a|E d a, a∈E×/E1 then:

Corollary 5.7 (orthogonality relations). For f1 and f2 matrix coefﬁcients of the supercuspidal representations σ1 and σ2 of Ge(F), −1 O( f1, ψ, · ), O( f2, ψ , · ) 6= 0 ⇐⇒ σ1 ∼ σ2. 0 0 0 0 For f1 and f2 matrix coefﬁcients of the supercuspidal representations σ1 and σ2 of 0 Ge (F), 0 0 0 0 0 0−1 0 0 O ( f1, ψ , · ), O ( f2, ψ , · ) 6= 0 ⇐⇒ σ1 ∼ σ2. Proof. This follows directly from the local Kuznetsov and local relative trace formulas.

Acknowledgments

The author thanks Erez Lapid for his helpful suggestions in an early stage of this project and Omer Offen for his useful comments. Finally, the author thanks the referee for helpful comments.

References

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Received July 19, 2012. Revised October 9, 2012.

BROOKE FEIGON DEPARTMENT OF MATHEMATICS THE CITY COLLEGEOF NEW YORK,CUNY NAC 8/133 NEW YORK, NY 10031 UNITED STATES [email protected] PACIFIC JOURNAL OF MATHEMATICS http://paciﬁcmath.org

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Volume 260 No. 2 December 2012

Special issue devoted to the memory of Jonathan Rogawski

In memoriam: Jonathan Rogawski 257 DON BLASIUS,DINAKAR RAMAKRISHNAN and V. S. VARADARAJAN p-adic Rankin L-series and rational points on CM elliptic curves 261 MASSIMO BERTOLINI,HENRI DARMON and KARTIK PRASANNA

The syntomic regulator for K4 of curves 305 AMNON BESSER and ROBDE JEU Unique functionals and representations of Hecke algebras 381 BENJAMIN BRUBAKER,DANIEL BUMP and SOLOMON FRIEDBERG A relative trace formula for PGL(2) in the local setting 395 BROOKE FEIGON 2012 On the degrees of matrix coefﬁcients of intertwining operators 433 TOBIAS FINIS,EREZ LAPID and WERNER MÜLLER Comparison of compact induction with parabolic induction 457

GUY HENNIART and MARIE-FRANCE VIGNERAS 2 No. 260, Vol. The functional equation and beyond endoscopy 497 P. EDWARD HERMAN A correction to Conducteur des Représentations du groupe linéaire 515 HERVÉ JACQUET Modular L-values of cubic level 527 ANDREW KNIGHTLY and CHARLES LI On occult period maps 565 STEPHEN KUDLA and MICHAEL RAPOPORT A prologue to “Functoriality and reciprocity”, part I 583 ROBERT LANGLANDS Truncation of Eisenstein series 665 EREZ LAPID and KEITH OUELLETTE Some comments on Weyl’s complete reducibility theorem 687 JONATHAN ROGAWSKI and V. S. VARADARAJAN On equality of arithmetic and analytic factors through local Langlands 695 correspondence FREYDOON SHAHIDI