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

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

BROOKE FEIGON

In memory of Jonathan Rogawski

Abstract. 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.

Contents 1. Introduction 1 2. Notation 7 3. The local Kuznetsov trace formula for U(2) 8 3.1. The geometric expansion 9 3.2. The spectral expansion 12 4. The local relative trace formula and periods for PGL(2) 14 4.1. The geometric expansion 15 4.2. The spectral expansion and period integrals 17 5. Comparison of local trace formulas and applications 25 6. Acknowledgments 28 References 28

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. HervéJacquet shed light on a new technique for attacking certain cases of Robert Langlands’ 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, AE) in terms of distinguished representations. While Jacquet’s global identities factor, they do not give unique local identities. 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 a quadratic extension of number fields and AF to be

Date: October 10, 2012. 2010 Mathematics Subject Classiﬁcation. 11F70, 22E50. The author was supported by NSF grant DMS-1201446. 1 0 ∼ the adeles of F . Let ψ be a character on F \AF = N(F )\N(AF ) 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, AE) is distinguished by GL(2, AF ) 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 )\ GL(2,AF ) 0 0 For π a cuspidal automorphic representation of the quasi-split unitary group U(2, AF ) and φ ∈ Vπ0 , let Z W (φ0) = φ0(n)ψ0(n)dn N(F )\N(AF ) and Z W (φ) = φ(n)ψ(n)dn. N(E)\N(AE ) We deﬁne the Bessel distribution as 0 0 X 0 0 0 0 0 0 Bπ0 (f ) := W (π (f )φi)W (φi) i and the relative Bessel distribution as X Bπ(f) := P (π(f)φj)W (φj) j

where the summations are over an orthonormal basis of Vπ0 and Vπ respectively. Flicker [Fli91], following 0 ∞ related work of Jacquet-Lai [JL85] and Ye [Ye89], showed that for “matching functions” f ∈ Cc (U(2, AF )) ∞ 0 and f ∈ Cc (GL(2, AE)), if π maps to π under the nonstandard base change, then X 0 0 0 0 0 0 X W (π (f )φi)W (φi) = P (π(f)φj)W (φj). i j In particular, this equality characterizes the image of the nonstandard base change lift associating every automorphic representation of U(2, AF ) to an automorphic representation of GL(2, AE) in terms of GL(2, AF ) distinguished representations. The equality above is proved via the relative trace formula [Jac05], which tells us that for f and f 0 matching functions: Z Z Z 0 −1 Kf 0 (n1, n2)ψ (n1 n2)dn1dn2 = Kf (h, n)ψ(n)dndh 2 (N(F )\N(AF )) GL(2,F )\ GL(2,AF ) N(E)\N(AE ) where X −1 Kf (x, y) = f(x δy). δ∈GL(2,E) 0 0 The distributions Bπ0 (f ) and Bπ(f) occur in the spectral expansions of the respective trace formulas. In a different direction, Arthur developed a local version of the classical Arthur-Selberg trace formula ([Art89] and [Art91]). 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 −1 φ 7→ ((y1, y2) 7→ 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)dgdy G(F ) G(F ) is an integral operator on L2(G(F )) with kernel Z −1 Kf (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). 2 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, i.e. 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 Jacquet’s global comparison. 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 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 this 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 OF (respectively OE) 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 (resp. N) be the upper triangular unipotent matrices of G0 (resp G) and let M 0 (resp. M) be the diagonal subgroup of G0 (respectively G). Let Z (respectively Z0) denote the center of G (respectively G0). 0 For any subgroup X of G let Xe = Z ∩ X\X and let XH = X ∩ H. Let ψ be an additive character on F 0 ∞ 0 0 0 with conductor OF and let ψ(x) = ψ ◦ trE/F . Let f = f1 ⊗ f2 ∈ Cc (Ge(F ) × Ge(F )) and f = f1 ⊗ f2 ∈ ∞ 0 0 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 Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t), dn1dn2 t→∞ (N 0×N 0)(F ) and the local relative trace formula as the equality between the expansions of Z Z lim Kf (h, n)ψ(n)u(h, t)u(n, t)dndh. t→∞ He(F ) N(F ) In this local setting Z −1 Kf (x, y) = f1(g)f2(x gy)dg, Ge(F ) Z 0 0 −1 Kf 0 (x, y) = f1(g)f2(x gy)dg Ge0(F ) and u(n, t) and u(h, t) are truncation parameters defined analogously to Arthur’s truncation [Art91] that are needed due to convergence issues. We use the following ideas in this paper to rewrite these local trace formulas in terms of orbital integrals and representations: • methods of Arthur from the local trace formula [Art91], • methods of Jacquet and Ye from the relative trace formula ([Jac05], [Ye89]), • Harish-Chandra’s Plancherel formula ([HC84], [Wal03]) • Jacquet-Lapid-Rogawski’s methods for regularizing period integrals [JLR99]. 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 Jacquet’s result, giving an 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 the nonstandard base change lift of the supercuspidal representation σ0, and f 0 and f are matching functions, then

0 X 0 0 0 0 0 0∨ 0 0 X ∨ d(σ ) Wσ0 (σ (f2)S σ (f1 ))Wσ0 (S ) = d(σ) Pσ(σ(f2)Sσ(f1 ))Wσ(S), S0∈B(σ0) S∈B(σ) 3 where d(σ) is the formal degree of σ, B(σ) is an orthonormal basis of the Hilbert space of Hilbert-Schmidt operators on Vσ, Z 0 0 0 0 0 −1 Wσ0 (S ) = tr(σ (n)S )ψ (n )dn, N 0(F )

Z −1 Wσ(S) = tr(σ(n)S)ψ(n )dn, N(F ) and Z Pσ(S) = tr(σ(h)S)dh. He(F )

The Bessel and relative Bessel distributions factor locally into Bπv (fv), but it is not clear how to normalize the local distributions. Each distribution above is the product of two local distributions and the equality above can be restated as 0 0 0 0 0 d(σ )Bσ0 (f2)Bσ0 (f1) = d(σ)Bσ∗ (f1)Bσ(f2) 0 where Bσ (Bσ) is a local (relative) Bessel distribution. We note that 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(OF ) and let P = NM. For a principal series representation π of Ge and u, v ∈ π we define the matrix coefficient λHM (m) fu,v(g) = hπ(g)u, vi. Asymptotically on M, fu,v will equal a finite sum of functions of the form e . We define the regularized period integral as: Z ∗ Pπ(fu,v) := fu,v(h)dh He(F ) Z := fu,v(h)u(h, t)dh He(F ) Z Z ]

+ DPH (m)fu,v(k1mk2)(1 − u(m, t))dmdk1dk2 + (KeH ×KeH )(F ) MfH (F ) where Z ] eλHM (m)(1 − u(m, t))dm + MfH (F ) is the meromorphic continuation at ν = 0 of Z e(ν+λ)HM (m)(1 − u(m, t))dm + MfH (F ) which is absolutely convergent for Re(ν) 0. We prove that with this deﬁnition the period integral is a GL(2,F )×GL(2,F )-invariant linear functional, and we related it to the truncated period integral that initially appears in the relative trace formula as follows. By abuse of notation we identify a character χ of Mf(F ) with a character χ of E× by letting a 0 χ = χ(a)χ−1(b). For λ ∈ we let χ (m) = χ(m)eλ(HP (m)). We let I (χ ) be the parabolically 0 b C λ P λ induced normalized representation acting on the Hilbert space HP (χ). Then for S ∈ BP (χ),

tr(IP (χλ, k1gk2)S) = EP (g, ΨS, λ)k1,k2 , where EP (g, Ψ, λ) is the Eisenstein integral and

P λHM (m) −λHM (m) (C EP )(m, ψ, λ) = (cP |P (1, λ)ψ)(m)e + (cP |P (w, λ)ψ)(m)e . We ﬁx a uniformizer $ in F (and E). 4 Proposition 1.2. Fix a character χ of E× such that χ($) = 1. Then for t 0, Z tr(IP (χλ, h)S)u(h, t)dh He(F ) Z ∗ = tr(IP (χλ, h)S)dh He(F ) q2λ(t+1) Z q−2λ(t+1) Z −δ(χ)(1 + q−1) c(1, λ)Ψ (1) dk dk + c(w, λ)Ψ (1) dk dk , 1 − q2λ S k1,k2 1 2 1 − q−2λ S k1,k2 1 2 KeH ×KeH KeH ×KeH

where δ(χ) = 1 if χ| × = 1 and δ(χ) = 0 if χ| × 6= 1. OF OF

Let | · |E denote the normalized valuation on E. Denote the action of the nontrivial element in Gal(E/F ) × × 1 × on x ∈ E byx ¯. Denote by NE/F the norm map from E to F . Let E = {x ∈ E : NE/F (x) = 1}. Let η 0 1 denote an element in G(F ) such that η−1η = . 1 0 We deﬁne 0 0 X 0 0 0 0 0 D χ0 (f ) = Wχ0 (Sλ[f ])W 0 (S ), λ λ χλ 0 0 S ∈BP (χ ) and X Dχλ (f) = Pχλ (Sλ[f])Wχλ (S)

S∈BP (χ) where Z 0 0 0 0 0 −1 Wχ0 (S ) = lim tr(IP 0 (χλ, n)S )ψ (n )u(n, t)dn, λ t→∞ N 0(F )

Z −1 Wχλ (S) = lim tr(IP (χλ, n)S)ψ(n )u(n, t)dn t→∞ N(F ) and

∨ Sλ[f] = d(χ)IP (χλ, f2)SIP (χλ, f1 ).

0 We let Π2(Ge (F )) be a set of equivalence classes of irreducible, tempered square integrable representations of Ge0(F ). We identify unitary characters on Mf0(F ) with characters on E× that are trivial on E1. We let 0 0 0 0 {Π2(Mf (F ))} be a set of representatives of unitary characters χ on Mf (F ) such that χ ($) = 1. We let 0 µ(χλ) be Harish-Chandra’s µ-function. We take the analogous deﬁnitions for Ge(F ).

0 0 0 ∞ 0 0 Theorem 1.3 (Local Kuznetsov trace formula). For any f = f1 ⊗ f2 ∈ Cc (Ge (F ) × Ge (F )), Z Z 0 −1 lim Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2 t→∞ N 0(F ) N 0(F ) Z 0 0 0 0 ¯0 × = O (f1, ψ , a) O f2, ψ , a |a|Ed a a∈E×/E1 πi Z log q X 0 0 0 1 X 0 0 0 0 = d(σ )Dσ0 (f ) + d(χ ) µ(χλ)Dχ0 (f )dλ 2 0 λ 0 0 0 0 σ ∈Π2(Ge (F )) χ ∈{Π2(Mf (F ))} where Z Z 0 0 0 0 −1 0 1 a 0 0 −1 O (f , ψ , a) = f n1 −1 n2 ψ (n1n2 )dn1dn2. N 0(F ) N 0(F ) −1 0 0 a 5 ∞ Theorem 1.4 (Local relative trace formula). For any f = f1 ⊗ f2 ∈ Cc (Ge(F ) × Ge(F )), Z Z lim Kf (h, n)ψ(n)u(h, t)u(n, t)dndh t→∞ He(F ) N(F ) Z ¯ × = O (f1, ψ, a) O f2, ψ, a |a|Ed a a∈E×/E1 X 1 X = d(σ)Dσ(f) + Deχ(f) 2 σ∈Π2(Ge(F )) χ∈{Π2(Mf(F ))} χ26=1,χ| =1 F × πi 1 X Z log q + d(χ) µ(χλ)Dχλ (f)dλ 2 0 χ∈{Π2(Mf(F ))} χ| =1 E1 where Z Z a 0 O(f, ψ, a) = f h−1η n ψ(n)dndh. 0 1 He(F ) N(F ) The representations that occur on the right hand side of Theorem 1.4 are exactly the representations that 0 are in the image of the nonstandard base change lift on G (F ). The extra discrete term Deχ(f) corresponds to the representations that lift from the discrete series on Ge0(F ) to the principal series on Ge(F ). In addition to the spectral comparison, these local trace formulas also have applications on the geometric × 1 side. If we deﬁne the inner product of two functions g1, g2 on E /E by Z × hg1, g2i = |a|Eg1(a)g2(a)d a, a∈E×/E1 then

0 0 Proposition 1.5 (Orthogonality Relations). For f1 (resp. f1) and f2 (resp. f2) matrix coeﬃcients of the 0 0 supercuspidal representations σ1 (resp. σ1)and σ2 (resp. σ2),

0 0 0 0 0 0−1 0 0 hO (f1, ψ , ·) ,O f2, ψ , · i= 6 0 iﬀ σ1 ∼ σ2 and −1 hO (f1, ψ, ·) ,O f2, ψ , · i= 6 0 iﬀ σ1 ∼ σ2. Using Macdonald’s formula, we can explicitly evaluate the local trace formulas for spherical functions, a special subset of smooth functions. In this case there is no discrete spectrum and the spectral sides each reduce to a single term.

0 ∞ 0 0 Example 1.6. For any bi-K invariant function f ∈ Cc (Ge (F ) × Ge (F )) and t 0, Z 0 −1 lim Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2 t→∞ (N 0(F ))2

πi 2 Z log q Z 0 ˆ 0 0 = m(λ)Tf (λ) φλ(n)ψ (n)u(n, t)dn dλ 0 N 0(F ) πi Z log q 2λ −2λ −2 −1+2λ −1−2λ = Tˆf 0 (λ)(1 − q )(1 − q )(1 + q − q − q )dλ. 0 ∞ For any bi-K invariant function f ∈ Cc (Ge(F ) × Ge(F )) and t 0, Z Z lim Kf (h, n)ψ(n)u(h, t)u(n, t)dndh t→∞ He(F ) N(F ) πi ! ! Z log q Z ∗ Z = m(λ)Tˆf (λ) φλ(h)dh φλ(n)ψ(n)u(n, t)dn dλ 0 He(F ) N(F ) 6 πi Z log q 2λ −2λ −2 −1+2λ −1−2λ = Tˆf (λ)(1 + q )(1 + q )(1 + q + q + q )dλ 0 where Z ∗ Z 1 + q−1 q2λ(t+1) q−2λ(t+1) φλ(h)dh = φλ(h)u(h, t)dh − −2 c(λ) 2λ + c(−λ) −2λ , He(F ) He(F ) 1 + q 1 − q 1 − q

m(λ) is the Plancherel measure, φλ is a spherical matrix coeﬃcient corresponding to the characteristic ∨ ˆ function of K, IP (χλ, f2)IP (χλ, f1 )φλ = T (λ)φλ and 1 − q−2−4λ c(λ) = . 1 − q−4λ

Thus when f is the image of f 0 under the base change map between their Hecke algebras, Tˆ0(λ) and Tˆ(λ) are Laurent polynomials in q2λ that are equal under the mapping q2λ → −q2λ. As expected, when f is the image of f 0 under the base change map between their Hecke algebras, the trace formulas are equal.

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 cosets. The orbital integrals for these cosets depend on the truncation and are intertwined. It is only through the multiplication of the two orbital integrals together, integration over the space of double cosets, and the non triviality of 0 0 0 the character ψ , that we are able to untangle the orbital integral for f1 from the orbital integral for f2. For the spectral expansions we apply Harish-Chandra’s Plancherel formula to rewrite the local kernel in terms of representations. We are left with integrals of matrix coefficients against the character over the unipotent subgroup. 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 Pσ(S). We use the asymptotics of matrix coefficients of tempered representations to prove the truncated integral is a polynomial exponential function in the truncation parameter t. We define the regularized integral as the constant term of this polynomial, and prove that this is the relevant term in the local relative trace formula and an H invariant linear functional. 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 parameterized by E×/E1. This allows us to compare the geometric sides. By work of Ye, 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. This tells us the spectral expansions are equal and we get the equality 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 OF (respectively OE) denote the ring of integers in F (respectively E). Let $ denote a uniformizer in the maximal ideal of OF . Thus $ is also a uniformizer in E. Let v(·) denote the valuation on F , extended to E. Let | · |F (respectively | · |E) denote the normalized valuation on × 2 F (respectively E). Thus for a ∈ F , |a|E = |a|F . Denote the action of the nontrivial element in Gal(E/F ) × × on x ∈ E byx ¯. Denote by NE/F the norm map from E to F . 7 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 quasi-split unitary group in this way SL(2,F ) ⊂ G0(F ). Let N 0 (resp. N) be the upper triangular unipotent matrices of G0 (resp. G). Let M 0 (respectively M) be the diagonal subgroup of G0 (respectively G). That is, a 0 M 0(F ) = : a ∈ E× 0 a−1 and a 0 M(F ) = : a, b ∈ E× . 0 b 1 n a 0 Occasionally by abuse of notation we let n = and a = . Let P = NM and P 0 = N 0M 0. 0 1 0a ¯−1 0 0 0 0 Let K = G(OF ) and K = G (OF ). Let Z (respectively Z ) denote the center of G (respectively G ). For 0 0 any subgroup X of G let Xe = Z ∩ X\X and XH = X ∩ H. By abuse of notation we identify a character χ a 0 of M(F ) with a character χ of E× by letting χ = χ(a)χ−1(b). f 0 b 0 Let ψ be an additive character on F with conductor OF . 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 0 ∞ 0 0 0 ∞ 0 0 N(F ) and N (F ) respectively. 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 insert Arthur’s local truncation [Art91, §3], and then take the limit of the integral of the truncated function. For Arthur’s truncation we multiply our function by the characteristic function of a large compact subset of Ge(F ). For g ∈ Ge(F ), t ∈ Z+, let  !  1 0 1 if g = zk1 k2, for some k1, k2 ∈ K, z ∈ Z(F ), 0 ≤ v(α) ≤ t u(g, t) = 0 α 0 otherwise. We note that ( −t 1 x 1 if x ∈ $[ 2 ]O 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(OF ) = 1. We define the multiplicative measure d x × × 1 1 × on F as d x = −1 dx. Thus vol(O ) = 1. We let N(F ) and M(F ) have the measures induced by 1−q |x|F F 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)dkdndm. G(F ) M(F ) N(F ) K We define dg0 on G0(F ) similarly. We normalize Haar measure on Ke by taking vol(Ke)=1. × × 1 × 1 1 We let d a be the unique Haar measure on E /E such that vol(OE /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 quasi-split unitary group in two variables. 0 We expand this local Kuznetsov trace formula geometrically in terms of separate orbital integrals for f1 and 0 f2. Then we use Harish-Chandra’s Plancherel formula to rewrite this expression spectrally in terms of representations. 8 0 0 0 ∞ 0 0 We deﬁne the local Kuznetsov trace formula for f = f1 ⊗ f2 ∈ Cc (Ge (F ) × Ge (F )) as the equality between the geometric and spectral expansions of Z −1 lim Kf 0 (n1, n2)ψ(n1 n2)u(n1, t)u(n2, t)dn1dn2 t→∞ (N 0×N 0)(F ) where Z 0 0 −1 Kf 0 (n1, n2) = f1(g)f2(n1 gn2)dg. Ge0(F ) We show that for a ﬁxed f 0 this limit stabilizes, that is, there exists a T such that for all t0 ≥ T , Z 0 −1 0 0 Kf 0 (n1, n2)ψ (n1 n2)u(n1, t )u(n2, t )dn1dn2 (N 0×N 0)(F ) Z −1 = lim Kf 0 (n1, n2)ψ(n1 n2)u(n1, t)u(n2, t)dn1dn2. t→∞ (N 0×N 0)(F )

3.1. The geometric expansion. We rewrite Z Z 0 −1 lim Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2 t→∞ N 0(F ) N 0(F )

0 0 as an integral over admissible cosets of a product of an orbital integral for f1 and an orbital integral for f2.

0 1 a 0 a 0 3.1.1. Integration formula. Let w = . For a ∈ E×, let β = and γ = w . Let −1 0 a 0 a−1 a 0 a−1 1 × 0 0 0 0 E = {a ∈ E : NE/F (a) = 1}. By the Bruhat decomposition, G = P t P wP . Thus

× 1 [ × 1 βa : a is in a set of representatives for E /E γa : a is in a set of representatives for E /E

is a set of representatives for the double cosets of N 0(F )\Ge0(F )/N 0(F ). For g ∈ Ge0 let

0 0 0 0 −1 0 Cg(N × N ) = {(n1, n2) ∈ N × N : n1 gn2 = zg for some z ∈ Z }.

Deﬁnition 3.1. An element g ∈ Ge0(F ) and its corresponding orbit is called admissible if the map

0 0 0 −1 Cg(N (F ) × N (F )) → C :(n1, n2) 7→ ψ (n1 n2) is trivial. An orbit which is not admissible is called inadmissible.

By a simple calculation we see that 1 x 1 x C (N 0(F ) × N 0(F )) = , aa : x ∈ F and βa 0 1 0 1 0 0 Cγa (N (F ) × N (F )) = 1.

× 1 1 Thus the orbits represented by {βa : a ∈ (E − E )/E } are inadmissible and the orbits represented by × 1 {β1} ∪ {γa : a ∈ E /E } are admissible. We use the following integration formula to rewrite Kf 0 (n1, n2) as an integral over only the admissible cosets. Unlike in the global case the trivial admissible coset, β1, will not contribute to the local Kuznetsov trace formula. 0 For any F ∈ Cc(Ge (F )), Z Z Z −1 × (3.1) F (g)dg = F (n1 γan2)dn1dn2|a|Ed a. Ge0(F ) E×/E1 (N 0×N 0)(F ) 9 3.1.2. Separating the orbital integrals. Let Z Z t 0 −1 K (f ) = Kf 0 (n1, n2)ψ(n1 n2)u(n1, t)u(n2, t)dn1dn2. N 0(F ) N 0(F )

t 0 0 0 0 Clearly K (f ) is absolutely convergent because f1 and u(·, t) have compact support on Ge (F ) and N (F ) respectively. By changing the order of integration and using (3.1) we see that Kt(f 0) equals Z Z Z 0 −1 0 −1 −1 f1(ˆn1 γanˆ2)f2(n1 nˆ1 γanˆ2n2) E×/E1 (N 0×N 0)(F ) (N 0×N 0)(F ) 0 −1 × ×ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2dnˆ1dnˆ2|a|Ed a.

We note that the above integral is absolutely convergent as the map from N 0(F ) × E×/E1 × N 0(F ) to Ge0(F ) −1 0 deﬁned by (n1, a, n2) 7→ n1 γan2 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|Ed a, E×/E1 where Z Z t 0 0 −1 0 −1 0 −1 −1 K (γa, f ) = f1(ˆn1 γanˆ2)f2(n1 γan2)ψ (n1 nˆ1nˆ2 n2) (N 0×N 0)(F ) (N 0×N 0)(F ) −1 −1 ×u(ˆn1 n1, t)u(ˆn2 n2, t)dn1dn2dnˆ1dnˆ2. To complete the geometric expansion of the local Kuznetsov trace formula we rewrite Kt(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 0 −1 0 −1 −1 f1(x1 γy1)f2(x2 γy2)u(x1 x2, t)u(y1 y2, t0) = f1(x1 γy1)f2(x2 γy2)u(y1 y2, t0) 0 for all x1, x2, y1, y2, γ ∈ Ge (F ).

0 0 0 Proof. Let Ω1 = supp(f1), Ω2 = supp(f2) and Ω3 = supp(u(·, t0)) ∩ Ge (F ). These sets are all compact on 0 0 −1 0 −1 −1 Ge (F ). If f1(x1 γy1)f2(x2 γy2)u(y1 y2, t0) 6= 0, then the following conditions must hold: −1 −1 −1 • x1 ∈ Ω1y1 γ ; −1 • x2 ∈ γy2Ω2 ; −1 • y1 y2 ∈ Ω3. 0 −1 0 −1 −1 −1 −1 Thus if f1(x1 γy1)f2(x2 γy2)u(y1 y2, t0) 6= 0, then x1 x2 ∈ Ω1Ω3Ω2 . Because this is a compact set, there −1 exists a T > 0 such that Ω1Ω3Ω2 ⊆ supp(u(g, T )). The lemma now follows.

Now we use this lemma, along with the character ψ, to separate the two orbital integrals. Let K1 be an 0 0 0 open compact subgroup of Ge (F ) such that f1 and f2 are bi-K1-invariant. There exists a positive constant c a 0 such that ∈ K for all a ∈ 1 + $cO . By abuse of notation, in the proof of the following lemma 0 a−1 1 E we let $n 0 a 0 $n = and a = . 0 $−n 0 a−1

0 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 ∈ Z , 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∈$ OE /E a∈$ OE /E where Z Z 0 0 0 0 −1 0 −1 O (f , ψ , a) = f (n1 γan2)ψ (n1 n2)dn1dn2. N 0(F ) N 0(F ) 10 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 the same γa. By deﬁnition

Z t 0 × K (γa, f )d a n × 1 a∈$ OE /E Z Z 0 −1 n 0 −1 = f1(ˆn1 w$ anˆ2)ψ (ˆn1nˆ2 ) × 1 0 0 a∈OE /E (N ×N )(F ) Z 0 −1 n −1 −1 −1 × × f2(n1 w$ an2)ψ(n1 n2)u(ˆn1 n1, t)u(ˆn2 n2, t)dn2dn1dnˆ2dnˆ1d a (N 0×N 0)(F ) Z Z X 0 −1 n 0 −1 = f1(ˆn1 w$ ηanˆ2)ψ (ˆn1nˆ2 ) c 1 1 0 0 × c 1 a∈(1+$ OE )E /E (N ×N )(F ) η∈OE /(1+$ OE )E Z 0 −1 n 0 −1 −1 −1 × × f2(n1 w$ ηan2)ψ (n1 n2)u(ˆn1 n1, t)u(ˆn2 n2, t)dn2dn1dnˆ2dnˆ1d a. (N 0×N 0)(F )

By a change of variables and the fact that f 0 is locally constant the above line equals

Z X 0 −1 n 0 f1(ˆn1 w$ ηnˆ2)ψ (ˆn1) 0 0 × c 1 (N ×N )(F ) η∈OE /(1+$ OE )E Z 0 −1 n 0 −1 −1 × f2(n1 w$ ηn2)ψ (n1 )u(ˆn1 n1, t)dn1dnˆ1 (N 0×N 0)(F ) Z 0 −1 −1 −1 −1 × × ψ (a nˆ2 n2a)u(a nˆ2 n2a, t)d adn2dnˆ2. c 1 1 a∈(1+$ OE )E /E

We can rewrite the inner integral as

Z −1 0 −1 × u(ˆn2 n2, t) ψ (n2 − nˆ2)(aa¯) d a c 1 1 a∈(1+$ OE )E /E Z −1 0 × =u(ˆn2 n2, t) ψ (b(n2 − nˆ2))d b c b∈1+$ OF Z −1 1 0 0 =u(ˆn2 n2, t) −1 ψ (n2 − nˆ2) ψ (b(n2 − nˆ2))db c 1 − q b∈$ OF q−c =u(ˆn−1n , t)u(ˆn−1n , 2c) ψ0(ˆn−1n ). 2 2 2 2 1 − q−1 2 2

Thus for t ≥ 2c,

Z t 0 × K (γa, f )d a n × 1 a∈$ OE /E Z Z 0 −1 0 −1 = f1(ˆn1 γanˆ2)ψ (ˆn1nˆ2 ) n × 1 0 0 a∈$ OE /E (N ×N )(F ) Z 0 −1 0 −1 −1 −1 × × f2(n1 γan2)ψ (n1 n2)u(ˆn1 n1, t)u(ˆn2 n2, 2c)dn2dn1dnˆ2dnˆ1d a. (N 0×N 0)(F ) 11 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∈$ OE /E Z Z 0 −1 0 −1 = f1 nˆ1 γanˆ2 ψ (ˆn1nˆ2 )dnˆ1 n × 1 0 0 a∈$ OE /E (N ×N )(F ) Z 0 −1 0 −1 −1 × × f2 n1 γan2 ψ (n1 n2)u(ˆn2 n2, 2c)dn2dn1dnˆ2d a (N 0×N 0)(F ) Z Z 0 −1 0 −1 = f1 nˆ1 γanˆ2 ψ (ˆn1nˆ2 )dnˆ2dnˆ1 n × 1 0 0 a∈$ OE /E (N ×N )(F ) Z 0 −1 0 −1 × × f2 n1 γan2 ψ (n1 n2)dn2dn1d a. (N 0×N 0)(F ) We have shown that the truncated local Kuznetsov trace formula stabilizes.

0 0 0 ∞ 0 0 Proposition 3.4. For any f = f1 ⊗ f2 ∈ Cc (Ge (F ) × Ge (F )) and t 0, Z Z 0 −1 Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2 N 0(F ) N 0(F ) Z 0 0 0 0 0 ¯0 × = O (f1, ψ , a) O f2, ψ , a |a|Ed a. a∈E×/E1 3.2. The spectral expansion. Now we derive a spectral expansion for the local Kuznetsov trace formula, Z Z 0 −1 lim Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2. 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 [HC84]. Silberger later filled in an important proof of one of the steps in the theorem [Sil96]. More recently Waldspurger provided a complete proof [Wal03]. As Arthur does in [Art91, §2] we begin by rewriting Kf 0 (x, y) using the Plancherel formula. First we 0 introduce some additional notation. For an irreducible representation (σ, Vσ) of G (F ) let B(σ) be the Hilbert space of Hilbert-Schmidt operators on Vσ. The inner product on B(σ) is defined as hS, S0i := tr(SS0∗) for S, S0 ∈ B(σ) where tr(SS0∗) = P hSS0∗u , u i and this sum converges absolutely and does not o.n.b.Vσ i i 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 Π2(Ge (F )) be a set of representatives for the equivalence classes of irreducible, tempered square 0 0 integrable representations of Ge (F ) and let {Π2(Mf (F ))} be a set of representatives of unitary characters χ 0 1 0 on Mf (F ) such that χ is a character on OE that is trivial on E . For a character χ of M (F ) and λ ∈ C, let 0 λ(HP 0 (m)) 0 G χλ(m) = χ(m)e . For χ ∈ Π2(M (F )), IP 0 (χ) = IP 0 (χ) is the normalized induced representation of 0 0 G(F ) which acts on a Hilbert space HP 0 (χ) of vector valued functions on K . Let BP 0 (χ) be a fixed K -finite orthonormal basis of the Hilbert space of Hilbert-Schmidt operators on HP 0 (χ). Let m(σ) be the Plancherel density. We normalize our measures following [Art91, §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 ) ∞ 0 −1 Then h ∈ Cc (Ge (F )) and Kf 0 (x, y) = h(yx ) so by the Plancherel formula, πi Z log q X −1 1 X −1 Kf 0 (x, y) = d(σ) tr(σ(R(yx )h)) + tr(IP 0 (χ, R(yx )h))m(χλ)dλ. 2 0 0 0 σ∈Π2(Ge (F )) χ∈{Π2(Mf (F ))} 12 Because −1 0∨ 0 ∗ IP 0 (χλ,R(yx )h) = IP 0 (χλ, f1 )IP 0 (χλ, x)IP 0 (χλ, f2)(IP 0 (χλ, y)) , we have

−1 X 0∨ 0 ∗ ∗ tr(IP 0 (χλ,R(yx )h)) = (IP 0 (χλ, f1 )IP 0 (χλ, x)IP 0 (χλ, f2),S )(IP 0 (χλ, y),S )

S∈BP 0 (χ) X 0∨ 0 = tr(IP 0 (χλ, f1 )IP 0 (χλ, x)IP 0 (χλ, f2)S)tr(IP 0 (χλ, y)S)

S∈BP 0 (χ) X 0 0 0 = tr(IP (χλ, x)Sχλ [f ])tr(IP (χλ, y)S),

S∈BP 0 (χ) 0 0 0∨ 0 0 where Sχλ [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 the sum over S is a K0 ﬁnite sum of an orthonormal basis of operators on HP (χ) 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 Z X X 0 0∨ 0 −1 0 −1 = d(σ) tr(σ(n)(σ(f2)Sσ(f1 )))ψ (n )u(n, t)dn tr(σ(n)S)ψ (n )u(n, t)dn 0 0 0 N (F ) N (F ) σ∈Π2(Ge (F )) S∈B(σ) 1 X + d(χ) 2 0 χ∈{Π2(Mf (F ))}

πi   Z log q Z Z X 0 0 −1 0 0 0 −1 ×  tr(IP (χλ, n)Sχλ [f ])ψ (n )u(n, t)dn tr(IP (χλ, n)S)ψ (n )u(n, t)dn µ(χλ)dλ. 0 N 0(F ) N 0(F ) S∈BP 0 (χ) To ﬁnish the spectral expansion we show that the above 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 4.2.1.

Lemma 3.5 (Spectral Stabilization). For any complex valued function φ on Ge0(F ) that is bi-invariant under an open compact subgroup, there exists a positive 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 ) Note: This c only depends on the open compact subgroup under which φ is bi-invariant.

0 Proof. Let K1 be an open compact subgroup of Ge (F ) under which φ is bi-invariant. K1 must con- 0 c0 tain a neighborhood of the identity, so there exists a positive integer c such that for a, b, c, d ∈ $ OF , 1 + a b ∈ K . We show that for m > c0, c 1 + d 1

Z 1 x φ ψ0(x)dx = 0. −m × 0 1 $ OF We note that x0 0 1 x x0−1 0 1 x0x = . 0 1 0 1 0 1 0 1

0 c0 Thus for x ∈ 1 + $ OF , 1 x0x 1 x φ = φ . 0 1 0 1 13 Hence Z 1 x φ ψ0(x)dx −m × 0 1 $ OF X Z 1 αx = φ ψ0(αx)dx −m c0 0 1 × c0 $ (1+$ OF ) α∈OF /(1+$ OF ) X 1 $−mα Z = φ ψ0($−mα) ψ0(x)dx. 0 1 c0−m × c0 $ OF α∈OF /(1+$ OF ) The last line equals 0 for m > c0. Thus for t > 2c0, Z φ(n)ψ0(n)u(n, t)dn N 0(F ) Z = φ(n)ψ0(n)u(n, 2c0)dn. N 0(F ) We have now proved the following.

0 0 0 ∞ 0 0 Proposition 3.6. For any f = f1 ⊗ f2 ∈ Cc (Ge (F ) × Ge (F )), Z Z 0 −1 lim Kf 0 (n1, n2)ψ (n1 n2)u(n1, t)u(n2, t)dn1dn2 t→∞ N 0(F ) N 0(F ) πi Z log q X 0 0 0 1 X 0 0 0 0 = d(σ )Dσ0 (f ) + d(χ ) Dχ0 (f )µ(χλ)dλ, 2 0 λ 0 0 0 0 σ ∈Π2(Ge (F )) χ ∈{Π2(Mf (F ))} where 0 0 X 0 0 0 0 0∨ 0 Dσ0 (f ) = Wσ0 (σ (f2)Sσ (f1 ))Wσ0 (S), S∈B(σ0) Z 0 0 0 −1 Wσ0 (S) = tr(σ (n)S)ψ (n )dn, N 0(F )

0 0 X 0 0 0 0 0∨ 0 Dχ0 (f ) = Wχ0 (IP 0 (χλ, f2)SIP 0 (χλ, f1 ))W 0 (S) λ λ χλ 0 S∈BP 0 (χ ) and Z 0 0 0 −1 Wχ0 (S) = lim tr (IP 0 (χλ, n)S) ψ (n )u(n, t)dn. 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-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 Kf (h, n)ψ(n)u(h, t)u(n, t)dndh t→∞ He(F ) N(F ) where Z −1 Kf (h, n) = f1(g)f2(h gn)dg. Ge(F ) 14 As with the local Kuznetsov trace formula, we show that for a fixed f this limit stabilizes.

4.1. The geometric expansion. We will rewrite Z Z lim Kf (h, n)ψ(n)u(h, t)u(n, t)dndh t→∞ He(F ) N(F )

as an integral over admissible cosets of a product of an orbital integral for f1 and an orbital integral f2.

4.1.1. Integration formula. As pointed out in [JLR99, §VI.13], by [Spr85], G(F ) = H(F )P (F ) t H(F )ηP (F ), 0 1 a 0 1 0 where η is any element in G(F ) such that η−1η = . Let η = η and γ = √ , 1 0 a 0 1 α 0 α + τ √ where E = F ( τ). Then 1 0 ∪ {γ : α ∈ F } ∪ η : a is in a set of representatives for E×/E1 0 1 α a is a set of representatives for the double cosets of He(F )\Ge(F )/N(F ). For g ∈ Ge, let −1 Cg(He × N) = {(h, n) ∈ He × N : h gn = zg for some z ∈ Z}.

Deﬁnition 4.1. An element g ∈ Ge(F ) and its corresponding orbit is called admissible if the map

Cg(He(F ) × N(F )) → C :(h, n) 7→ ψ(n) is trivial. An orbit which is not admissible is called inadmissible. By a short calculation we see that √ 1 y 1 y(α + τ) C (H(F ) × N(F )) = , : y ∈ F and γα e 0 1 0 1

Cηa (He(F ) × N(F )) = 1. × Thus the orbits represented by {γα : α ∈ F } ∪ {1} are inadmissible and the orbits represented by {ηa : a ∈ × 1 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)dndh|a|Ed a. Ge(F ) E×/E1 He(F )×N(F ) 4.1.2. Separating the orbital integrals. Let Z Z t R (f) = Kf (h, n)ψ(n)u(h, t)u(n, t)dndh. 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)dn2dh2dn1dh1|a|Ed a.

We note that the above integral is absolutely convergent as the map from He(F ) × E×/E1 × N(F ) to Ge(F ) −1 deﬁned by (h, a, n) 7→ h ηan is injective and f1 has compact support. By a change of variables we have Z t t × R (f) = R (ηa, f)|a|Ed a, E×/E1 15 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)dn2dh2dn1dh1.

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. Let K1 be an open compact subgroup of Ge(F ) such that f1 and f2 are bi-K1-invariant. There exists a a 0 0 c0 > 0 such that ∈ K for a ∈ 1 + $c O . By abuse of notation we let 0 1 1 E

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

× Lemma 4.2. For f ∈ Cc (Ge(F ) × Ge(F )), there exists a T > 0 such that for all t ≥ T , Z t × R (ηa, f)d a n × 1 $ OE /E Z 0 0 ¯ × = O (f1, ψ, a)O(f2, ψ, a)d a n × 1 $ OE /E for all n ∈ Z where Z Z −1 O(f, ψ, a) = f(h ηan)ψ(n)dndh. He(F ) N(F ) Proof. This proof is very similar to the proof of Lemma 3.3 so we will omit some details. Z t × R (ηa, f)d a n × 1 $ OE /E Z Z Z −1 n −1 = f1(h1 η$ an1)ψ(n1 ) × 1 a∈OE /E He(F ) N(F ) Z Z −1 n −1 −1 × × f2(h2 η$ an2)ψ(n2)u(h1 h2, t)u(n1 n2, t)dn2dh2dn1dh1d a He(F ) N(F ) Z Z Z X −1 n −1 = f1(h1 η$ βan1)ψ(n1 ) c0 1 × c0 a∈(1+$ OE )/E He(F ) N(F ) β∈OE /(1+$ OE ) Z Z −1 n −1 −1 × × f2(h2 η$ βan2)ψ(n2)u(h1 h2, t)u(n1 n2, t)dn2dh2dn1dh1d a. He(F ) N(F ) By a change of variables this equals Z Z Z Z X −1 n −1 n −1 f1(h1 η$ βn1) f2(h2 η$ βn2)u(h1 h2, t)dh2dh1 × c0 1 He(F ) N(F ) He(F ) N(F ) β∈OE /(1+$ OE )E Z −1 −1 −1 −1 × × ψ(a n1 n2a)u(a n1 n2a, t)d adn2dn1 c0 1 1 a∈(1+$ OE )E /E Z Z Z −1 −1 = f1(h1 ηan1)ψ(n1 ) n × 1 a∈$ OE /E He(F ) N(F ) Z Z −1 −1 −1 0 × × f2(h2 ηan2)ψ(n2)u(h1 h2, t)u(n1 n2, c )dn2dh2dn1dh1d a He(F ) N(F ) 16 for t ≥ c0. By Lemma 3.2 (whose proof is identical if we replace Ge0(F ) with Ge(F )), there exists a T > 0 such that for all t ≥ max{T, c0} Z t × R (ηa, f)d a n × 1 a∈$ OE /E Z Z Z −1 −1 = f1(h1 ηan1)ψ(n1 )dh1 n × 1 a∈$ OE /E He(F ) N(F ) Z Z −1 −1 0 × × f2(h2 ηan2)ψ(n2)u(n1 n2, c )dn2dh2dn1d a He(F ) N(F ) Z Z Z −1 −1 = f1(h1 ηan1)ψ(n1 )dn1dh1 n × 1 a∈$ OE /E He(F ) N(F ) Z Z −1 × × f2(h2 ηan2)ψ(n2)dn2dh2d a. 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)dndh t→∞ He(F ) N(F ) Z × = O (f1, ψ, a) O f2, ψ, a |a|Ed 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.

4.2. The spectral expansion and period integrals. We want to get a spectral expansion for the local relative trace formula, Z Z lim Kf (h, n)ψ(n)u(h, t)u(n, t)dndh. 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)dndh He(F ) N(F ) πi X 1 X Z log q = d(σ)Dt (f) + d(χ) µ(χ )Dt (f)dλ σ λ χλ 2 0 σ∈Π2(Ge(F )) χ∈{Π2(Mf(F ))} where t X t ∨ t Dσ(f) = Pσ(σ(f2)Sσ(f1 ))Wσ(S), S∈B(σ)

t X t ∨ t D (f) = P (IP (χλ, f2)SIP (χλ, f ))W (S), χλ IP (χλ) 1 IP (χλ) S∈BP (χ) Z t −1 Wπ(S) = tr(π(n)S)ψ(n )u(n, t)dn N(F ) and Z t Pπ(S) = tr(π(h)S)u(h, t)dh. He(F ) By Lemma 3.5, there exists a positive integer c1, such that for t > c1, Z t −1 Wπ(S) = tr(π(n)S)ψ(n )u(n, c1)dn. N(F ) 17 Thus as in the previous section, we define Z −1 Wπ(S) = lim tr(π(n)S)ψ(n )u(n, t)dn. 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-Rogawski in [JLR99]. In that paper they define a regularized period integral for an automorphic form φ on G(A) integrated over H where G is a reductive group over a number field F and H is the fixed point set of an involution θ 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 α Let δP = β , 0 β E α 0 log |α|E 0 HM = 0 β 0 log |β|E and α 0 1 α ρ H = log | | . P M 0 β 2 β E

λHM (m) α λ We note that for g ∈ H, HM (g) = 2HMH (g). For λ ∈ C, e = | β |E. α 0 We recall the Cartan decomposition H(F ) = K M +K where M + = : v( α ) ≤ 0 . Then H H H H 0 β β Z Z Z Z f(h)dh = DP (m)f(k1mk2)dmdk2dk1 + H He(F ) KeH KeH MgH where ( α α 0 | β |F (1 + |$|F ) v(α/β) ≤ 0 DP = H 0 β 0 v(α/β) > 0 for any absolutely integrable function f. We note that ( 1 0, 0 ≤ v(α) ≤ t 1 − u , t = α 1, v(α) > t

+ Let M(g) ∈ M be such that g = k1M(g)k2, k1, k2 ∈ K. For Re ν < −Re λ,

∞ Z X q(t+1)2(ν+λ) (4.3) e(ν+λ)HM (m)(1 − u(m, t))dm = q2n(ν+λ) = . + 1 − q2(ν+λ) MfH n=t+1 We write Z ] eλHM (m)(1 − u(m, t))dm + MfH to denote the meromorphic continuation at ν = 0 of (4.3). This is well deﬁned so long as λ 6= 0. If

r X 1 (4.4) φ(k mk ) = φ (k , k )f (m)eλiH(m), k , k ∈ K , m = , n ≥ 0, f ∈ C (M(F )) 1 2 i 1 2 i 1 2 H $n i c f i=1 18 where λi 6= −1/2 we deﬁne for t 0 Z ] r Z Z ] X λiHM (m) φ(h)(1 − u(h, t))dh = φi(k1, k2) DPH (m)e (1 − u(m, t))dm + He(F ) i=1 KeH ×KeH MfH 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 MfH

If φ is a matrix coeﬃcient of IP (χλ) where χ($) = 1 then by smoothness and the asymptotics of matrix P coeﬃcients there exists a function C φ of the form in (4.4) with λi ∈ {λ − 1/2, −λ − 1/2} and for n 0, 1 1 CP φ k k = φ k k . Note that the condition for the regularized integral to exist 1 $n 2 1 $n 2 is now that λ 6= 0.

Deﬁnition 4.4. For any matrix coeﬃcient φ of IP (χλ) where χ($) = 1 and λ 6= 0, Z ∗ f(h)dh He(F ) Z Z Z ]

:= f(h)u(h, t)dh + DPH (m)f(k1mk2)(1 − u(m, t))dmdk1dk2 + He(F ) KeH ×KeH MfH 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 He-invariant and then we will explicitly relate the regularized period to the truncated period that occurs in the local trace formula. h0 h0 Let φ (x) = φ(xh0) for h0 ∈ He. Note that if φ is a matrix coeﬃcient of π then φ is as well.

Lemma 4.5. For any matrix coeﬃcient φ of IP (χλ) where χ($) = 1 and λ 6= 0, h0 ∈ H and t 0, Z Z ] h0 DPH (m)φ (k1mk2)(1 − u(k1mk2h0, t))dmdk1dk2 KeH ×KeH MfH Z Z ]

= DPH (m)φ(k1mk2)(1 − u(m, t))dmdk1dk2 KeH ×KeH MfH Proof. For Re ν 0 and t 0, Z Z h0 ν(HM (M(k1mk2h0))) DPH (m)φ (k1mk2)e (1 − u(k1mk2h0, t))dmdk1dk2 KeH ×KeH MfH Z ν(HM (M(hh0))) = φ(hh0)e (1 − u(hh0, t))dh He(F ) Z = φ(h)eν(HM (M(h)))(1 − u(h, t))dh He(F ) Z Z ν(HM (m)) = DPH (m)φ(k1mk2)e (1 − u(m, t))dmdk1dk2 KeH ×KeH MfH by the invariance of Haar measure, since both sides are absolutely convergent. For t 0, if M(h) > t where h = k1M(h)k2, then M(hh0) = M(h)M(k2h0). Thus both sides of the equation above have a meromorphic continuation whose value at ν = 0 gives the statement of the lemma.

Proposition 4.6 (H-invariance). For φ a matrix coefficient of IP (χλ) where χ($) = 1 and λ 6= 0, and h0 ∈ H, Z ∗ Z ∗ φh0 (h)dh = φ(h)dh. He(F ) He(F ) 19 Proof. By the definition of the regularized integrals, the statement of the proposition will follow once we prove the following equality Z Z ] Z Z ] h0 DPH (m)φ (k1mk2)(1 − u(m, t))dmdk1dk2 − DPH (m)φ(k1mk2)(1 − u(m, t))dmdk1dk2 + + KeH ×KeH MfH KeH ×KeH MfH 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 ] Z Z ] h0 DPH (m)φ (k1mk2)(1 − u(m, t))dmdk1dk2 − DPH (m)φ(k1mk2)(1 − u(m, t))dmdk1dk2 + + KeH ×KeH MfH KeH ×KeH MfH Z Z ] Z Z ] −1 = DPH (m)φ(k1mk2)(1 − u(mk2h0 , t)))dmdk1dk2 − DPH (m)φ(k1mk2)(1 − u(m, t))dmdk1dk2. + + KeH ×KeH MfH KeH ×KeH MfH −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 DPH (m)φ(k1mk2)[u(m, t) − u(mk2h0 , t)]dmdk1dk2. + KeH ×KeH MfH (F ) Z −1 = φ(h)[u(h, t) − u(hh0 , t)]dh He(F ) Z Z = φ(h)u(h, t)dh − φh0 (h)u(h, t)dh. He(F ) He(F ) To derive an explicit formula in terms of regularized periods for the truncated periods of matrix coefficients that appear in the trace formula we recall some definitions of Harish-Chandra’s. For σ an admissible, tempered representation of G, Aσ(G) is the space of functions on G spanned by K-finite matrix coefficients of σ, Atemp(G) is the sum of Aσ(G) over all admissible tempered representations of G and A2(G) is the sum of Aσ(G) over all unitary, square integrable representations. For τ a finite dimensional, unitary, two-sided representation of K,

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

Then Atemp(G, τ) and A2(G, τ) are deﬁned similarly. P Let τM = τ|K∩M . By [HC84, §3] for f ∈ Aσ(G, τ) there exists a unique function C f ∈ A(M, τM ) such that 1/2 α 0 α 0 P α 0 lim δP f − (C f) = 0. | α | →∞ 0 β 0 β 0 β β E We call CP 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}

πi and let τP1|P2 be the subrepresentation of τM on VP1|P2 . For Ψ ∈ A2(M, τP |P ) and λ ∈ [0, log q ], the Eisenstein integral EP (g, Ψ, λ) ∈ Atemp(G, τ) is deﬁned as Z −1 (λ+ρP )(HP (kg)) EP (g, Ψ, λ) = τ(k) ΨP (kg)e dk K where ΨP extends Ψ to G by

ΨP (nmk) = Ψ(m)τ(k) for n ∈ N(E), m ∈ M, k ∈ K. 20 The weak constant term of the Eisenstein integral uniquely deﬁnes Harish-Chandra’s c-functions [HC84,

§6]. For each element w in the Weyl group W of Ge, the c-function cP |P (w, λ) is a linear map from A2(M, τP |P ) to A2(M, τP |P ) such that P (C EP )(m, Ψ, λ)

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. K0 We note that the S we consider are actually in HP (χ) for some open compact K0. Harish-Chandra K [HC76, §7] gives an isomorphism S → ΨS from End(HP (χ) ) onto Aχ(M, (τ)P |P ) where Vτ is a particular subspace of L2(K × K) such that

tr(IP (χλ, k1gk2)S) = EP (g, ΨS, λ)k1,k2 . We can now relate the deﬁnition of the regularized integral to what appears in the local relative trace formula. −1 Proposition 4.7. For χ = (χ, χ ) ∈ {Π2(Mf(F )}, 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 q−2λ(t+1) Z δ(χ)(1 + q−1) c(1, λ)Ψ (1) dk dk + c(w, λ)Ψ (1) dk dk , 1 − q2λ S k1,k2 1 2 1 − q−2λ S k1,k2 1 2 KeH ×KeH KeH ×KeH

where δ(χ) = 1 if χ| × = 1 and δ(χ) = 0 if χ| × 6= 1. OF OF

Proof. For S ∈ BP (χ), ΨS ∈ Aχ(M, (τΓ)P |P ) and c(1, λ)ΨS, c(w, λ)ΨS ∈ Aχ(M, (τΓ)P |P ). Therefore Ψ = ΨS can be written as a sum of matrix coeﬃcients of χ. Thus P (C EP )(m, Ψ, λ)k1,k2

λHM (m) −λHM (m) =(c(1, λ)Ψ)(m)k1,k2 e + (c(w, λ)Ψ)(m)k1,k2 e

λHM (m) −λHM (m) =χ(m)[(c(1, λ)Ψ)(1)k1,k2 e + (c(w, λ)Ψ)(1)k1,k2 e ] where χ(m) ∈ C×. Hence Z −1/2 (λ+ν)(HM (m)) DP (m)δ (m)(c(1, λ)Ψ)(1)k ,k e χ(m)(1 − u(m, t))dm + H P 1 2 MgH Z −1/2 (−λ+ν)(HM (m)) + DP (m)δ (m)(c(w, λ)Ψ)(1)k ,k e χ(m)(1 − u(m, t))dm + H P 1 2 MgH Z −1 (λ+ν)(HM (m)) =(1 + q )(c(1, λ)Ψ)(1)k ,k e χ(m)(1 − u(m, t))dm 1 2 + MgH Z −1 (−λ+ν)(HM (m)) +(1 + q )(c(w, λ)Ψ)(1)k ,k e χ(m)(1 − u(m, t))dm 1 2 + MgH Z ∞ −1 × X 2(λ+ν)n 2(−λ+ν)n =(1 + q ) χ(α)d α [(c(1, λ)Ψ)(1)k1,k2 q + (c(w, λ)Ψ)(1)k1,k2 q ]. × OF n=t+1 R × Clearly × χ(α)d α = 0 unless χ| × = 1. If χ| × = 1, the previous line equals OF OF OF q2(λ+ν)(t+1) q2(−λ+ν)(t+1) δ(χ)(1 + q−1) c(1, λ)Ψ(1) + c(w, λ)Ψ(1) . 1 − q2(λ+ν) k1,k2 1 − q2(−λ+ν) k1,k2 21 Hence Z ] −1/2 P DPH (m)δP (m)C EP (m, ΨS, λ)k1,k2 (1 − u(m, t))dm + MfH (F ) q2λ(t+1) q−2λ(t+1) =δ(χ)(1 + q−1) c(1, λ)Ψ (1) + c(w, λ)Ψ (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). 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)Ψ(1)k1,k2 dk1dk2 = c(s, λ)Ψ(1)k1,k2 dk1dk2 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 ∈ iR, S ∈ BP (χ). Proof. 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. He P λ The vanishing of the regularized period 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 q−2λ(t+1) Z = − (1 + q−1) c(1, λ)Ψ (1) dk dk + c(w, λ)Ψ (1) dk dk . 1 − q2λ S k1,k2 1 2 1 − q−2λ S k1,k2 1 2 KeH ×KeH KeH ×KeH Both sides are holomorphic and the left hand side is also deﬁned and holomorphic for λ = 0. As q2λ(t+1) −1 Res = λ=0 1 − q2λ 2 log q and q−2λ(t+1) 1 Res = λ=0 1 − q−2λ 2 log q we must have that Z Z

c(1, 0)Ψ(1)k1,k2 dk1dk2 = c(w, 0)Ψ(1)k1,k2 dk1dk2. KeH ×KeH KeH ×KeH 22 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 q−2λ(t+1) Z − (1 + q−1)µ(χ ) c(1, λ)Ψ (1) dk dk + c(w, λ)Ψ (1) dk dk . λ 1 − q2λ S k1,k2 1 2 1 − q−2λ S k1,k2 1 2 KeH ×KeH 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.

Let X Dχλ (f) = Pχλ (Sχλ [f])Wχλ (S), 0 S ∈BP (χ) Z ∗

Pχλ (S) = tr(IP (χλ, h)S)u(h, t)dh He(F ) and Z −1 X Deχ(f) = (1 + q )µ(χ0) c(1, 0)ψS0[f](1)k1,k2 dk1dk2 Wχ0 (S). KeH ×KeH S∈BP (χ) We now relate the distributions above to the truncated distributions from (4.2).

Lemma 4.9. Let χ = (χ, χ−1).

(1) If χ|F × 6= 1 and χ|E1 6= 1, then

πi Z log q lim µ(χ )Dt (f)dλ = 0. λ χλ t→∞ 0

(2) If χ|F × 6= 1 and χ|E1 = 1, then

πi πi Z log q Z log q lim µ(χ )Dt (f)dλ = µ(χ )D (f)dλ. λ χλ λ χλ t→∞ 0 0

(3) If χ|F × = 1 and χ|E1 6= 1, then

πi Z log q lim µ(χ )Dt (f)dλ = D (f). λ χλ eχ t→∞ 0

(4) If χ|F × = 1 and χ|E1 = 1, then

πi πi Z log q Z log q lim µ(χ )Dt (f)dλ = µ(χ )D (f)dλ. λ χλ λ χλ t→∞ 0 0 Proof. First we note that

πi πi Z log q Z log q X µ(χ )Dt (f)dλ = µ(χ ) P t (S[f])W (S)dλ λ χλ λ χλ χλ 0 0 S∈BP (χ) πi X Z log q Z Z = tr(IP (χλ, n)S)ψ(n)u(n, t)dnµ(χλ) tr(IP (χ, h)S)u(h, t)dh. 0 N(F ) He(F ) S∈BP (χ)

23 By Proposition 4.7 for t 0, (4.5) Z tr(IP (χλ, h)S)u(h, t)dh He(F ) Z ∗ = tr(IP (χλ, h)S)dh+ He(F ) q2λ(t+1/2) Z −q−2λ(t+1/2) Z δ(χ)(1 + q−1) c(1, λ)Ψ(1) dk dk + c(w, λ)Ψ(1) dk dk . qλ − q−λ k1,k2 1 2 qλ − q−λ k1,k2 1 2 KeH ×KeH KeH ×KeH Cases 1 and 2 now follow directly from Lemma 4.8. In case 3, by Lemma 4.8 the regularized period vanishes and we are left computing

πi Z log q 2λ(t+1) Z −1 q (4.6) (1 + q ) lim µ(χλ)Wχ (S)( c(1, λ)Ψ(1)k ,k dk1dk2 t→∞ λ 1 − q2λ 1 2 0 KeH ×KeH q−2λ(t+1) Z + c(w, λ)Ψ(1) dk dk )dλ. 1 − q−2λ k1,k2 1 2 KeH ×KeH Let 1 + q−1 Z Z f (λ) = µ(χ )W (S) c(1, λ)Ψ(1) dk dk − c(w, λ)Ψ(1) dk dk 1 2 λ χλ k1,k2 1 2 k1,k2 1 2 KeH ×KeH KeH ×KeH and ! 1 + q−1 Z Z f (λ) = µ(χ )W (S) c(1, λ)Ψ(1) dk dk + c(w, λ)Ψ(1) dk dk . 2 2 λ χλ k1,k2 1 2 k1,k2 1 2 KeH (F )×KeH (F ) KeH (F )×KeH (F ) Then (4.6) equals

πi 1 1 πi 1 1 Z log q 2λ(t+ ) −2λ(t+ ) Z log q 2λ(t+ ) −2λ(t+ ) f1(λ)(q 2 + q 2 ) f2(λ)(q 2 − q 2 ) lim λ −λ + lim λ −λ dλ. t→∞ 0 q − q 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 Lemma 4.8 equals Z −1 (1 + q )µ(χ0)Wχ0 (S) c(1, 0)Ψ(1)k1,k2 dk1dk2. 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. 4.2.1. Discrete series representations. Because the matrix coeﬃcient of a supercuspidal representation σ has compact support, it is obvious by the deﬁnition of the regularized integral 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), χ2 = 1, the matrix coefficients are absolutely convergent over He. 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 ) 24 a 0 Proof. By [BW80, XI.4.3] and [Cas, 4.2.3], a matrix coefficient for σ evaluated at is equal to a matrix 0 b a coefficient for the Jacquet functor σN , evaluated at the same value, for b E sufficiently small. The Jacquet functor of σ is δP . Thus outside some ball of compact support, our original matrix coefficient will behave − − like δP on MH . When we integrate over He(F ), using the KH MH KH decomposition, we get a measure factor −1/2 of δP . Thus outside a ball of compact support our integral will look like Z × |a|F d a |a| 0. Putting everything together we have proved the following.

∞ Proposition 4.11. For any f ∈ Cc (Ge(F ) × Ge(F )), Z Z lim Kf (h, n)ψ(n)u(h, t)u(n, t)dndh t→∞ He(F ) N(F ) πi X 1 X 1 X Z log q = d(σ)Dσ(f) + Deχ(f) + µ(χλ)Dχλ (f)dλ, 2 2 0 σ∈Π2(Ge(F )) χ∈{Π2(Mf(F ))} χ∈{Π2(Mf(F ))} χ26=1,χ| =1 χ| 1 =1 F × E where X Dχλ (f) = Pχλ (Sλ[f])Wχλ (S),

S∈BP (χ) Z ∗

Pχλ (S) = tr(IP (χλ, h)S)dh, He(F ) "Z # −1 X Deχ(f) = (1 + q )µ(χ0) c(1, 0)ψS0[f](1)k1,k2 dk1dk2 Wχ0 (S), KeH (F )×KeH (F ) S∈BP (χ) X ∨ Dσ(f) = Pσ(σ(f2)Sσ(f1 ))Wσ(S), S∈B(σ) and Z Pσ(S) = tr(σ(h)S)dh. He(F ) The result of this proposition, combined with Proposition 4.3, proves Theorem 1.4.

5. Comparison of local trace formulas and applications We now put together the results of the last two sections to compare the two trace formulas.

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) = O(f, ψ, a) for all a ∈ E×.

0 ∞ 0 By work of Flicker [Fli91] and Ye [Ye89], we know that for any f ∈ Cc (Ge (F )) there exists a matching ∞ 0 f ∈ Cc (Ge(F )) and vice versa. In fact, by the Fundamental Lemma, for f 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.

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 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 t→∞ N 0(F ) N 0(F ) Z Z = lim Kf ⊗f (h, n)ψ(n)u(h, t)u(n, t)dndh. t→∞ 1 2 He(F ) N(F ) 25 Now we use the equality of the trace formulas to compare the spectral expansions. By Propositions 3.6, 4.11 and 5.2,

0 Theorem 5.3. For fi and fi matching functions for i = 1, 2, πi Z log q X 0 0 0 0 1 X 0 0 0 0 0 d(σ )Dσ0 (f1 ⊗ f2) + d(χ ) µ(χλ)Dχ0 (f1 ⊗ f2)dλ 2 0 λ 0 0 0 0 σ ∈Π2(Ge (F )) χ ∈{Π2(Mf (F ))} X 1 X = d(σ)Dσ(f1 ⊗ f2) + Deχ(f1 ⊗ f2) 2 σ∈Π2(Ge(F )) χ∈{Π2(Mf(F ))} χ26=1,χ| =1 F × πi 1 X Z log q + d(χ) µ(χλ)Dχλ (f1 ⊗ f2)dλ. 2 0 χ∈{Π2(Mf(F ))} χ| =1 E1

The nonstandard base change map lifts principal series representations of Ge0 to principal series repre- 0 sentations IP (χ) of Ge such that χ|E1 = 1. It also lifts certain square integrable representations of Ge to 2 the principal series representations of Ge deﬁned by IP (χ) such that χ 6= 1, χ|F × = 1. It lifts the remain- ing square integrable representations of Ge0 to square integral representations of Ge [Rog90]. Thus we could rephrase the right hand side of Theorem 5.3 in terms of summing over the representations of Ge that are the 0 nonstandard base change lifts of representations of Ge . The extra discrete term Wfχ(f) corresponds exactly to the representations that lift from the discrete series of Ge0 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 non-zero. This gives us a more explicit description of the nonvanishing H invariant linear functional that characterizes the image of the nonstandard base change map. We would like to relate our distributions to the local factors in the Bessel and relative Bessel distributions. 0 ∞ Recall from the introduction that Jacquet’s global relative trace formula tells us that for f ∈ Cc (U(2, AF )) ∞ 0 and f ∈ Cc (GL(2, AE)) matching functions, if a cuspidal representation π of U(2, AF ) maps to π of GL(2, AE) under nonstandard base change, then 0 0 Bπ0 (f ) = Bπ(f) where 0 0 X 0 0 0 0 0 0 Bπ0 (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 (AF ) Z W (φ) = φ(n)ψ(n)dn, N(E)\N(AE ) and Z P (φ) = φ(h)dh 6= 0. GL(2,F )\ GL(2,AF ) 0 0 0 0 While B 0 (f ) and Bπ(f) factor into local Bessel distributions B 0 (f ) and Bπ (fv), it is not clear how to π πv v v normalize the local Bessel distributions. We can rewrite our local distributions as a product of two local Bessel (or local relative Bessel) distributions:

Lemma 5.4. (1) For σ0 an irreducible supercuspidal representation of Ge0, there exists a local Bessel 0 distribution Bσ0 , that is unique up to a constant of absolute value 1, such that 0 0 0 0 0 0 0 Dσ0 (f1 ⊗ f2) = Bσ0 (f2)Bσ0∗ (f1). 26 (2) For σ an irreducible supercuspidal representation of Ge, there exists a local relative Bessel distribution Bσ, that is unique up to a constant of absolute value 1, such that

Dσ(f1 ⊗ f2) = Bσ(f2)Bσ∗ (f1). Proof. We recall that Z Z 0 0 X 0 0 0 0 0∗ 0 0 −1 0 0 0 −1 Dσ0 (f ) = tr(σ (n1)σ (f2)S σ (f1))ψ (n1) dn1 tr(σ (n2)S )ψ (n2) dn2. 0 0 S0∈B(σ0) N (F ) N (F ) 0 ∗ ∗ 0 ∗ Let V = Vσ0 . As S in an endomorphism on V there exist v ∈ V, v ∈ V such that S = v ⊗ v . Then the linear functional on V ⊗ V ∗ that acts by Z v ⊗ v∗ 7→ tr(σ0(n)v ⊗ v∗)ψ0(n)−1dn N 0(F ) 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(σ0(n)S0)ψ0(n)−1dn = W 0(v)W 0(v∗). N 0(F ) Thus 0 0 X 0 0 0 0 0 0∗ 0 ∗ 0 ∗ Dσ0 (f ) = W (σ (f2)v)W (v)W (σ (f1)v )W (v ) v⊗v∗ 0 0 0 0 =Bσ0 (f2)Bσ0∗ (f1). 0 0 We note that if we change Bσ0 by a constant c, then Bσ0∗ will change by c. The proof for the local relative Bessel distributions is similar, using the uniqueness of the H-invariant linear functional. These results allow us to describe matching functions by an equality of all the Bessel distributions. 0 ∞ 0 0 0 Lemma 5.5 (Density). (1) If f1 ∈ Cc (Ge (F )) is such that Bσ0 (f1) = 0 for all irreducible tempered 0 0 0 0 0−1 × representations σ of Ge , then O (f1, ψ , a) = 0 for all a ∈ E . ∞ (2) If f1 ∈ Cc (Ge(F )) is such that Bσ(f1) = 0 for all irreducible tempered representations σ of Ge, then −1 × O(f1, ψ , a) = 0 for all a ∈ E . 0 0 0 Proof. If Bσ0 (f1) = 0 for all σ , then by Theorem 1.3 and Lemma 5.4, Z 0 0 0−1 0 0 0 × |a|O (f1, ψ , a)O (f2, ψ , a)d a = 0 a∈E×/E1 0 ∞ 0 0 0 −1 for all f2 ∈ Cc (Ge ). As O (f1, ψ , a) is a locally constant function of a there exists some open compact U 0 0 0−1 0 0 0 0−1 such that O (f1, ψ , a) is bi-invariant under it. Then by choosing f2 such that O (f2, ψ , a) has support 0 0 0−1 contained in U we see that 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: 0 0 Corollary 5.6. If σ is the nonstandard base change lift of the supercuspidal representation σ , and fi and fi are matching functions for i = 1, 2, then 0 0 0 d(σ )Dσ0 (f1 ⊗ f2) = d(σ)Dσ(f1 ⊗ f2). Proof. From the global comparison of relative trace formulas [Fli91, Lap06, Ye89], a standard globalization 0 0 argument and Lemma 5.4, we know there exists a constant cσ such that Dσ0 (f ) = cσDσ(f) for all matching 0 0 0 0 0 0 f, f . Take f1 and f2 to be matrix coefficients of σ such that Bσ0 (fi ) 6= 0. Take f1 to be a matching 0 0 0 0 0 function of f1. Then Bπ(f1) = cBπ0 (f1) = 0 unless π ∼ σ . Let P rσ(f1) denote the projection of f1 to σ. Then Bπ(P rσ(f1)) = 0 if π 6∼ σ and Bσ(P rσ(f1)) = Bσ(f1). Thus by the density lemma above, f1 0 and P rσ(f1) have identical orbital integrals. As f1 is a matching function for f1 then P rσ(f1) must be a 0 0 matching function for f1 as well. Take f2 a matching function to f2. Then by Theorem 5.3, 0 0 0 d(σ )Dσ0 (f ) = d(σ)Dσ(f) 27 d(σ) so c = d(σ0) . In addition to the spectral comparison, these local trace formulas also have applications on the geometric × 1 side. If we define the inner product of two functions g1, g2 on E /E by Z × hg1, g2i = |a|Eg1(a)g2(a)d a, a∈E×/E1 then 0 0 Corollary 5.7 (Orthogonality Relations). For f1 (resp. f1) and f2 (resp. f2) matrix coefficients of the 0 0 supercuspidal representations σ1 (resp. σ1) and σ2 (resp. σ2), 0 0 0 0 0 0−1 0 0 hO (f1, ψ , ·) ,O f2, ψ , · i= 6 0 iff σ1 ∼ σ2 and −1 hO (f1, ψ, ·) ,O f2, ψ , · i= 6 0 iff σ1 ∼ σ2. Proof. This follows directly from the local Kuznetsov and local relative trace formulas. 6. 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.

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Department of Mathematics, The City College of New York, CUNY, NAC 8/133, New York, NY 10031, USA E-mail address: [email protected]