Plancherel Distribution of Satake Parameters of Maass Cusp Forms on $ GL 3$

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Af n h νf orbital integrals, f p q p q“ ÿ whereas the Kuznetsov trace formula is

Af n Af m p q p qh ν δ main term Kloosterman sums, L 1, Ad f f m“n f p q“ ` ÿ p q where is summing over the cuspidal spectrum of GL2 and h is a test function on the spectral pa- f ´1 rametersř νf . The weight removal technique of Luo in [Luo99] and [Luo01] is to remove L 1, Ad f p q from the Kuznetsov trace formula. Luo proves the distribution of L 1, Ad f for f being Maass p q forms on GL2 and the Phillips-Sarnak theorem on the Weyl’s law, both from the Kuznetsov trace formula on GL2. Such a method is also used in [LW11] to study the equidistribution of Hecke eigenvalues on GL2. The weight removal technique of Luo depends on a precise formula of Shimura for L s, Ad f . Shimura has the famous formula for the adjoint L-function from GL2 from [Shi75] p q 8 2 Af n L s, Ad f ζ 2s p q ns p q“ p q n“1 ÿ for a Hecke-Maass cusp form (or a modular form) f for SL2 Z with Hecke eigenvalues Af n . The p q p q GLN -analogue of this formula was not clear. Thanks to Joseph Hundley, we have the following formula for GL3

AF n1,n2 L s,F, Ad ζ 2s ζ 3s p q p q“ p q p q ns n1,n2 rn1,n2s|n|n1n2 3 n1{n2ÿPpQˆq ÿ for a Maass cusp form F for SL3 Z with Fourier coefficients AF , . In the previous formula, ˆ 3 p q p˚ ˚q n1 n2 Q means n1 n2 is the cube of a rational number. The formula above can be traced { P p q { back to the Rankin-Selberg integral of Ginzburg in [Gin91], and independently the representation- theoretic works of Kostant, Lusztig and Hesselink. A formula for GLN can be found in Theorem 2.2 but it is less explicit. It will turn out that the adjoint L-function has connection with the Kazhdan-Lusztig polynomial. Unlike some other applications of the Kuznetsov trace formula, the weight L 1, F, Ad ´1 is cru- p q cial for the equidistribution problems of the Satake parameters. With the weight L 1, F, Ad ´1, it p q is proved in [BBR14] and [Zho14] that the Satake parameter of F at a fixed prime p is equidistributed with respect to the Sato-Tate measure, as F varies over Maass cusp forms on GL3. In this paper, we are going to prove that without the weight, the equidistribution is with respect to another measure, the p-adic Plancherel measure µp (see (5)). There is a lot of literature on the equidistribution problems on GL2, both with and without weight (see [Zho14, Section 1]). In some sense, our weight removal technique is not to throw away the weight but to absorb it. The Kuznetsov trace formula on GL3 used in this paper is a significant improvement over those in [Blo13] and [GK13]. It is comparable to the convexity-breaking paper [BB15b], although the latter is for a different purpose. The crucial improvements in that paper are the integral formulae 2 [BB15b, Section 5], following the harmonic analysis of the integral transforms in [But16]. Careful study of these integral formulae yields a sharp cut-off point on the geometric side the Kuznetsov formula. In this paper, we provide even stronger integral representations (24) and (25), easily giving the sharp upper bounds needed for the current work; these integral representations hold largely independent of the particular family of test functions used here. We expect that such formulae will lead to asymptotic expansions and thereby solutions to previously intractable problems such as removing the maximal Eisenstein contribution (see the discussion after Proposition 3 of [BB15a]). Our method also relies on the functional equation and the analytic properties of L s,F, Ad . p q For a general automorphic representation π of GL3 AK and a number field K, it is proved by p q Ginzburg that L s,π, Ad has functional equation and analytic continuation (except finitely many p q possible but unlikely poles) in [Gin91]. For a Maass cusp form F for SL3 Z , with the recent p q work [Hun16] of Hundley, we are able to prove that L s,F, Ad and its completed L-function are p q holomorphic. Our work on the Kuznetsov trace formula could have been saved by a good bound toward the Lindelöf hypothesis of L s,F, Ad on average over the spectrum (see Lemma 3.2). Indeed, the p q Lindelöf hypothesis is known on average for a few families of L-functions, such as that in [Luo99]. However, it is not available here. We compensate that with our improvement in the Kuznetsov trace formula.

1.1 Equidistribution with and without weight Lp1,F, Adq´1

The equidistribution of Hecke eigenvalues of automorphic forms on GL2 has been studied extensively. Let f be a cuspidal automorphic form on GL2. For a majority of f, the Sato-Tate conjecture predicts Af p is equidistributed with respect to the Sato-Tate measure, as p varies over all primes. p q Big progress has been made by Harris, Taylor, et al. for modular forms. From a spectral perspective, it is proved by Sarnak in [Sar87], Conrey-Duke-Farmer in [CDF97], Serre in [Ser97] that Af p is equidistributed with respect to the p-adic Plancherel measure, as f varies over a family. p q By the Kuznetsov-Bruggeman trace formula, Bruggeman proves that if each Af p is given a weight ´1 p q L 1, Ad f , then the equidistribution of Af p is changed to the Sato-Tate measure again (see p q p q [Bru78]). Later works on the equidistribution problems on GL2 are so numerous that we do not include any here. On GL3, Bruggeman’s analogue is proved in [BBR14], [Zho13], [Zho14], by using the Kuznetsov trace formula on GL3 of [But13], [Blo13] and [GK13].

1.2 Hecke-Maass cusp forms on GLN

The dual group of PGLN Qp is SLN C . The standard maximal torus of SLN C is p q p q p q N ˚ T diag α1, , αN : αi C for all i, αi 1 SLN C “ # t ¨ ¨ ¨ u P i“1 “ + Ă p q ź The group SUN is the standard maximal compact subgroup of SLN C . The standard maximal p q torus of SUN is

N ˚ T0 diag α1, , αN : αi C and αi 1 for all i, αi 1 SLN C “ # t ¨ ¨ ¨ u P | |“ i“1 “ + Ă p q ź The Weyl group W ( SN ) acts on T and T0 by permutating the diagonal entries. – 3 Let F be a Hecke-Maass cusp form for SLN Z with Fourier coeﬃcients AF m1, ,mN´1 C N´1 p q p1q pNpq ¨ ¨ ¨ q P for m1, ,mN 1 N , as deﬁned in [Gol06]. Let diag α p , , α p T W be the p ¨ ¨ ¨ ´ q P t F p q ¨ ¨ ¨ F p qu P { Satake parameter of F at a prime p. We have the Shintani formula

m1 mN´1 p1q pNq AF p , ,p Sm1, ,m 1 α p , , α p , p ¨ ¨ ¨ q“ ¨¨¨ N´ F p q ¨ ¨ ¨ F p q ´ ¯ where Sm1, ,m 1 x1, ,xN is the Schur polynomial (see [Gol06, p.233]). The L-factor of F at ¨¨¨ N´ p ¨ ¨ ¨ q a prime p is given by ´1 N αpiq p L s, F : 1 F p q p ps p q “ i“1 ˜ ´ ¸ ź and the standard L-function of F has the Euler product

8 AF 1, , 1,n L s, F : p ¨ ¨ ¨ q ns p q “ n“1 ÿ Lp s, F . “ p is a prime p q ź The generalized Ramanujan-Petersson conjecture predicts

p1q pNq diag α p , , α p T0 W, t F p q ¨ ¨ ¨ F p qu P { namely, αpiq p 1 for all i. F p q “ ˇ ˇ ˇ ˇ 1.3 Hecke-Maassˇ ˇ cusp form for SL3pZq and a family of test functions 2 Let ν1,ν2 C and deﬁne ν3 : ν1 ν2. It will be convenient to also use coordinates µ p q P 3 “ ´ ´ “ µ1,µ2,µ3 C , µ1 µ2 µ3 0 given by p q P ` ` “

µ1 ν1 ν3, µ2 ν2 ν1, µ3 ν3 ν2. (1) “ ´ “ ´ “ ´ ˚ 3 ˚ 3 Deﬁne a µ1,µ2,µ3 R µ1 µ2 µ3 0 and aC µ1,µ2,µ3 C µ1 µ2 µ3 0 . “ tp q P | ` ˚ ` “˚ u “ tp q P | ` ` “ u The Weyl group W ( S3) acts on a and aC by permutation of coordinates. Identify bijectively 2 – ˚ 2 ˚ ν ν1,ν2 C with µ aC as in (1) and we have C aC. “ p q P P – 2 ˚ Let F be a Hecke-Maass cusp form for SL3 Z of type νF ν1 F ,ν2 F C (also aC). Let ˚ p q “ p p q p qq P P µ1 F ,µ2 F ,µ3 F a W be the Langlands parameter of F , satisfying (1), with the properties p p q p q p qq P C{

µ1 F µ2 F µ3 F 0 p q` p q` p q“ and µ1 F , µ2 F , µ3 F µ1 F , µ2 F , µ3 F . (2) t´ p q ´ p q ´ p qu “ t p q p q p qu The functional equation for the standard L-function L s, F is p q 3 3 L s, F ΓR s µj F L 1 s, F˜ ΓR 1 s µj F . p q j“1 p ´ p qq “ p ´ q j“1 p ´ ` p qq ź ź Let Ω ia˚ be a compact Weyl-group invariant subset disjoint from the Weyl chamber walls Ď µ a˚ w µ µ for some w W and w 1 and T 1 a large parameter. We utilize the test t P C| p q “ P ‰ u ą 4 function of [BB15a, Section 5.2], which approximates the characteristic function on T Ω: Let ν0 Ω. P For ν a˚ , we put ψ ν exp 3 ν2 ν2 ν2 and P C p q“ 1 ` 2 ` 3 ` ` ˘˘3 2 1 2 νj 1 2n P ν : p q ´ 9 p ` q p q “ T 2 0ďnďA j“1 ź ź for some large, ﬁxed constant A to compensate poles of the spectral measure in a large tube. Now we choose 2 2 w ν Tν0 hT ν : P ν ψ p q´ (3) p q “ p q T 1´ε wPW ´ ÿ ´ ¯¯ for some very small 0 ε 1 2. Then T ε such functions give a majorant of the characteristic ă ă { function of T Ω.

Theorem 1.1. Let µp be the p-adic Plancherel measure supported on T0 W , deﬁned in (5). Let { hT for T 1 be the family of test functions on the spectral parameters of Hecke-Maass cusp forms ą on GL3, deﬁned in (3). For any continuous function φ on T W , we have the limit { p1q p2q p3q φ diag αF p , αF p , αF p hT νF F t p q p q p qu p q lim ´ ¯ φ dµp, T Ñ8 ř hT νF “ T0{W F p q ż ř where is a summation over all Hecke-Maass cusp forms for SL3 Z and νF is the spectral pa- F p q rameterř of F .

2 Representation theory and adjoint L-function

2.1 Kazhdan-Lusztig polynomial

Let G SLN C be a semi-simple algebraic group over C with Lie algebra g slN C . Let Φ be “ p q ` “ p q 1 the root system and Φ the set of positive roots. Let W be its Weyl group. Let ρ : ` α “ 2 αPΦ be the half sum of positive roots. Let P` be the (positive) Weyl chamber of dominant weights. ř Let q be a symbol. For a weight β deﬁne the Kostant q partition by

npαq Pq β q . p q“ β“ npαqα ř αPΦ`ÿ,npαqě0 ř Deﬁne Lusztig’s q polynomial

β lengthpwq M q 1 Pq w λ ρ β ρ . λp q“ p´ q p p ` q ´ p ` qq wPW ÿ This is also called the Kostka-Foulkes polynomial or the Kazhdan-Lusztig polynomial (for root systems of A type). Such polynomials were studied extensively but we are only interested in M0 q λp q in this work.

5 2.2 Hilbert-Poincar´eseries

` Deﬁne Vλ as the ﬁnite-dimensional complex representation of G for the highest weight λ P . By P the Weyl character formula, we have

1 lengthpwqewpρ`λq V wPW . char λ ρ p´ q ´β p q“ e ` 1 e ř βPΦ p ´ q ś 2 Recall g slN C and G acts on g as the adjoint representation of dimension N 1. The “ p q ´ symmetric algebra S g 8 n g becomes a graded representation of G. A fundamental paper p q“‘n“0 _ of Kostant [Kos63] proves S g I H where G-invariant part I is a free module (generated p q “8 bn by known degrees) and H n“0H is the graded module of harmonic polynomials. Deﬁne 8 “n ‘n F V dim Homg V,H q , as in [Kir92, §2]. It is proved in [Hes80] p q“ n“0 p q ř 8 n n 0 dim Homg Vλ,H q Mλ q . n“0 p q “ p q ÿ Hence we have 8 n n 0 char H q M q char Vλ . p q “ λp q p q n“0 λ P` ÿ Pÿ Combining it with S g I H, we have for S g 8 n g p q“ b p q“‘n“0 _

8 N ´1 n n l 0 char g q 1 q M q char Vλ . (4) p_ q “ ´ λp q p q n“0 ˜l“2 ¸ λ P` ÿ ź ´ ¯ Pÿ 2.3 Plancherel measure

Let p be a prime number. Let ds be the normalized Haar measure on T0. The Sato-Tate measure is defined on T0 W as { 1 dµ 1 eβ s ds 8 “ W ´ p q βPΦ | | ź ´ ¯ Let µp be the unramified Plancherel measure of SLN C , which is supported on T0 W , and it is p q { defined as W p´1 dµp p q dµ (5) “ 1 p´1eβ s 8 βPΦp ´ p qq ś with W q : qlengthpwq. p q “ wPW ÿ The formula for µp is due to Macdonald in [Mac71].

Proposition 2.1 ([Kat82, (3.4)]). For β P` we have P

0 ´1 char Vβ dµp Mβ p . T p q “ p q ż 0{W

6 2.4 Adjoint L-function on GLN ` N´1 ` Let the Weyl chamber P be parameterized by Zě0 . Let λ1 be the highest weight in P for the standard inclusion SLN C ã GLN C (the first minuscule representation). For i 2, ,N 1, p q Ñ ` p q i “ ¨ ¨ ¨ ´ let λi be the highest weight in P for the exterior power representation Vλ1 . Define a bijective N´1 ` ^ map ℵ : Zě0 P by Ñ N´1 lN´1, , l1 liλi. p ¨ ¨ ¨ q ÞÑ i“1 ÿ Let F be a Hecke-Maass cusp form for SLN Z . Define p q L s, F F˜ L s,F, Ad : p ˆ q p q “ ζ s p q as the adjoint L-function of F . It is a Dirichlet series with Euler product of degree N 2 1. The ´ functional equation and holomorphy of the adjoint L-function is studied in [Shi75] for N 2, in “ [Gin91] for N 3, in [BG98] for N 4, and in [GH08] for N 5. “ “ “ Theorem 2.2. Let F be a Hecke-Maass cusp form for SLN Z . We have the local L-function at p p q for Re s 1 p qą N 8 8 0 ´s lN´1 l1 Lp s,F, Ad ζp ls Mℵ p AF p , ,p p q“ p q ¨ ¨ ¨ ¨ plN´1,¨¨¨ ,l1qp q p ¨ ¨ ¨ q˛ ˜l“2 ¸ l1“0 l ´1“0 ź ÿ Nÿ ˝ ‚ l ´1 l1 Proof. The theorem follows from (4), with the Casselman-Shalika formula for AF p N , ,p . p ¨ ¨ ¨ q

Lemma 2.3. For the special case of N 3, we have “ l1`l2 i 0 q , if 3 l1 l2, Mℵ q i“maxtl1,l2u | ´ pl2,l1qp q“ $ &’ ř 0 , otherwise.

Proof. Let us assume l1 l2. The Weyl%’ group W has six elements. Recall the deﬁnition ě 0 lengthpwq Mℵpl2,l1q q 1 Pq w ℵ l2, l1 ρ ρ . p q“ wPW p´ q p p p q` q´ q ÿ 0 It is easy to see that 3 l1 l2 is necessary for M q to be nonzero. For w 1, we have | ´ ℵpl2,l1qp q “ i Pq ℵ l2, l1 q . p p qq “ 2l1`l2 3 ÿďiďl1`l2

For the Weyl group element w W which sends λ1 to λ1 and λ2 to λ1 λ2, we have P ´ lengthpwq i 1 Pq w ℵ l2, l1 ρ ρ q . p´ q p p p q` q´ q“´ 2l1`l2 3 ÿďiďl1´1

lengthpwq For all other w W , the Kostant q partition 1 Pq w ℵ l2, l1 ρ ρ is zero. P p´ q p p p q` q´ q 7 2.4.1 The case of N 3 “ Assume N 3 throughout this subsection. Let F be a Hecke-Maass cusp form for SL3 Z . The “ p q functional equation and analytic continuation of L s,F, Ad is studied by Ginzburg in [Gin91] using p q an integral representation involving Eisenstein series on the exceptional group G2 and by [GJ00] using a Siegel-Weil identity for G2. It is not yet generally known that L s,π, Ad is holomorphic p q for an automorphic representation π of GL3 AK and a number ﬁeld K. The functional equation p q for the adjoint L-function L s,F, Ad is p q Λ s,F, Ad Λ 1 s,F, Ad , (6) p q“ p ´ q where we have 2 Λ s,F, Ad L s,F, Ad ΓR s ΓR s µj F µi F . p q“ p q p q i‰j p ´ p q` p qq ź By the recent work of Hundley [Hun16], we are able to prove that Λ s,F, Ad is holomorphic on p q the whole complex plane.

Theorem 2.4. The adjoint L-function L s,F, Ad is holomorphic on the complex plane. The p q completed adjoint L-function Λ s,F, Ad is holomorphic on the complex plane. p q Proof. By [Hun16, Theorem 6.1], L s,F, Ad is holomorphic on Re s 1 2. If F satisﬁes the p q p q ě { generalized Ramanujan (Selberg) conjecture at the archimedean place, i.e.,

Im µi F 0 p p qq “ for i 1, 2, 3, then all the poles of the gamma factors are on Re s 0. Therefore, Λ s,F, Ad is “ p q “ p q holomorphic on Re s 1 2. If F does not satisfy the generalized Ramanujan conjecture at the p q ě { archimedean place, by (2), we have

µ1 F ,µ2 F ,µ3 F σ it, σ it, 2it , t p q p q p qu “ t ` ´ ` ´ u with σ, t R and 0 σ 5 14 (Kim-Sarnak bound in [Kim03, Appendix 2]). If Λ s,F, Ad has a P ď ď { p q pole in 1 2 Re s 1, it must be s 2σ with 1 4 σ 5 14. As we know from the classical { ď p q ă “ { ď ď { Rankin-Selberg theory, the completed Rankin-Selberg L-function Λ s, F F˜ is holomorphic in p ˆ q 0 Re s 1. Thus from the perspective of ă p qă Λ s, F F˜ Λ s,F, Ad p ˆ q, p q“ ζ s ΓR s p q p q we must have ζ 2σ 0. This is false and a contradiction is reached. In conclusion, Λ s,F, Ad is p q“ p q holomorphic on Re s 1 2 and by its functional equation it is holomorphic on the whole complex p qě { plane.

We are given by Joseph Hundley the following formula for the adjoint L-function on GL3. This formula appears in a diﬀerent form in his earlier work [Hun12, (7)]. It can be viewed as a special case of Theorem 2.2 for the local part Lp s,F, Ad . Hundley derived the formula from [Gin91], p q independently from [Hes80].

8 Theorem 2.5. For a Hecke-Maass cusp form F with Fourier coeﬃcients AF , . We have for p˚ ˚q Re s 1 p qą 8 m1`m2 m1 m2 AF p ,p Lp s,F, Ad ζp 2s ζp 3s p q (7) p q“ p q p q pjs m1,m2“0 j“maxpm1,m2q 3|mÿ1´m2 ÿ and L s,F, Ad Lp s,F, Ad . Moreover, we have for Re s 1 p q“ p is a prime p q p qą ś L s,F, Ad ζ 2s ζ 3s L# s,F, Ad (8) p q“ p q p q p q and AF n1,n2 L# s,F, Ad p q. p q“ ns n1,n2 rn1,n2s|n|n1n2 3 n1{n2ÿPpQˆq ÿ Proof. Obviously (7) implies (8). Lemma 2.3 and Theorem 2.2 imply (7).

3 The proof of the main theorem

In order to prove the main theorem (Theorem 1.1), by the Stone-Weierstrass theorem (also the Peter-Weyl theorem as in [Zho14, Theorem 7.1]), it is suﬃcient to prove the following one, with eﬀective error term.

Theorem 3.1. For integers l1, l2 0, we have ě 1 14´4ε l1`l2 A pl2 ,pl1 h ν M0 p´1 h ν spec ν dν O T 3 `ǫp 2 `ǫ F T F ℵpl2,l1q 64π5 T F p q p q“ p q˜ Repνq“0 p q p q ¸ ` ÿ ż ´ ¯ for ǫ 0. ą The value of L 1, F, Ad can be approximated by the Dirichlet series in Theorem 2.5. p q ˚ Lemma 3.2. Let F be a Maass cusp form with spectral parameters µ1 F ,µ2 F ,µ3 F a . p p q p q p qq P C‘ For 0 σ 1, we have for X 0 ă ă ą AF n1,n2 n L 1, F, Ad ζ 2 ζ 3 p qe IF X p q“ p q p q n ´X ` p q n1,n2 rn1,n2s|n|n1n2 3 ´ ¯ n1{n2ÿPpQˆq ÿ with 1 4 `ǫ 1 ´ IF X µi F µj F X 2 . p q! ˜i‰j | p q´ p q|¸ ź Proof. We begin with

ζ 2 ζ 3 L s 1, F, Ad s IF X p q p q p ` q Γ s X ds ´ p q“ 2πi ζ 2s 2 ζ 3s 3 p q żp´1{2q p ` q p ` q ζ 2 ζ 3 L s 1, F, Ad L 1, F, Ad p q p q p ` q Γ s Xsds “´ p q` 2πi ζ 2s 2 ζ 3s 3 p q żp`q p ` q p ` q AF n1,n2 n L 1, F, Ad ζ 2 ζ 3 p qe “´ p q` p q p q n ´X n1,n2 rn1,n2s|n|n1n2 3 ´ ¯ n1{n2ÿPpQˆq ÿ 9 by the Mellin inversion. Because Λ s,F, Ad is holomorphic from Theorem 2.4 and satisfies the p q functional equation (6), L s,F, Ad satisfies the convexity bound by the Phragmén-Lindelöf prin- p q ciple. We apply the convexity bound to L s 1, F, Ad on the vertical line Re s 1 1 2 and we p ` q p ` q“ { get the bound for IF X . p q The following theorem improves [Blo13, Theorem 5] and [GK13, Theorem 1.3]. It is an application of the Kuznetsov trace formula on GL3 and it will be proved in the remaining sections.

Theorem 3.3. Let P m1m2n1n2 0, then we have “ ‰

AF n1,n2 AF m1,m2 C : p q p qh ν ∆ O T P ǫ T P 1{2 T 3P 1{6 , L 1, F, Ad T F “ F p q“ ` p q ` ÿ p q ´ ´ ¯¯ where we have

1 3 3π ∆ δ h ν spec ν dν, spec ν : 3ν tan ν . n“m 64π5 T j 2 j “ Repνq“0 p q p q p q “ j“1 ż ź ˆ ´ ¯˙ Proof of Theorem 3.1. We start with the formal sum

l2 l1 AF p ,p hT νF F p q p q ÿ l2 l1 AF p ,p p qL 1, F, Ad h ν L 1, F, Ad T F “ F p q p q ÿ p q

l2 l1 AF p ,p AF n1,n2 n p q¨ζ 2 ζ 3 p qe IF X ˛hT νF “ L 1, F, Ad p q p q n ´X ` p q p q F p q n1,n2 rn1,n2s|n|n1n2 ÿ ˚ ÿ ˆ 3 ÿ ´ ¯ ‹ ˚ n1{n2PpQ q ‹ ˚ ‹ ζ 2 ζ 3 M E˝1 E2, ‚ “ p q p qp ` q` where the main term is by Theorem 3.3

1 pl 1 M l e 5 hT ν spec ν dν “ ¨ p ´X ˛ 64π Re ν 0 p q p q maxtl1,l2uďlďl1`l2 ˜ p q“ ¸ ÿ ˆ ˙ ż ˝ ‚ and the error terms are (E1 comes from Theorem 3.3)

1 n ǫ 1{2 3 1{6 l1 l2 E1 e O T P T P T P , with P n1n2p p “ n ´X p q ` “ n1,n2 rn1,n2s|n|n1n2 3 ´ ¯ ´ ´ ¯¯ n1{n2ÿPpQˆq ÿ 8 1 n e O T P ǫ T P 1{2 T 3P 1{6 , with P n2pl1`l2 “ n ´X p q ` “ n“1 ´ ¯ ´ ´ ¯¯ ÿ l1`l2 O T X2 ǫ T X T 3X1{3 p 2 `ǫ “ p q ` ´ ´ ¯ ¯

10 and

l2 l1 AF p ,p E2 p qIF X hT νF “ L 1, F, Ad p q p q F p q ÿ 1{2 1{2 l2 l1 2 2 AF p ,p IF X | p q| h ν | p q| h ν L 1, F, Ad T F L 1, F, Ad T F ď ˜ F p q¸ ˜ F p q¸ ÿ p q ÿ p q l `l 3 1 1 1 2 `ǫ `ǫ ´ hT ν spec ν dν p 2 T 2 X 2 , ! 64π5 p q p q ˜ żRepνq“0 ¸ because Lemma 3.2 and Theorem 3.3 imply that for δ 0 and for X T 3`δ, we have ą " 2 IF X 1 | p q| h ν h ν spec ν dν T 3`ǫX´1. L 1, F, Ad T F 64π5 T F p q! ˜ Repνq“0 p q p q ¸ ÿ p q ż Since we have

1 5´2ε 5´3ε hT ν spec ν dν CT O T for some constant C 0, 64π5 p q p q “ ` p q ą żRepνq“0 11´4ε l2 l1 balancing E1 and E2 by taking X T 3 , we get AF p ,p hT νF is equal to “ F p q p q ř 1 1 14´4ε l1`l2 3 `ǫ 2 `ǫ ζ 2 ζ 3 l 5 hT ν spec ν dν O T p . p q p q¨ p ˛ 64π Re ν 0 p q p q ` maxtl1,l2uďlďl1`l2 ˜ p q“ ¸ ÿ ż ´ ¯ ˝ ‚ Remark 3.4. The bound for

2 IF X 1 | p q| h ν h ν spec ν dν T 3`ǫX´1 L 1, F, Ad T F 64π5 T F p q! ˜ Repνq“0 p q p q ¸ ÿ p q ż comes from the individual convexity bound for L s,F, Ad in Lemma 3.2. This is far from p q what can be conjectured. Under the generalized Lindel¨of hypothesis, T 3`ǫ can be replaced with T ǫ and greater power saving can be achieved. The average of the Lindel¨of hypothesis or the subconvexity bound is not available for this family of L-functions L s,F, Ad : t p q F is a Hecke-Maass cusp form for SL3 Z . p qu 4 The Kuznetsov Formula

In this section we state the Kuznetsov formula for the particular test function hT described in Section 1.3 above. We will be brief and refer to [BB15b, Section 3] for more details and no- tation. In particular, we will not require the precise deﬁnition of the two Kloosterman sums S˜ n1,n2,m1; D1, D2 and S n1,n2,m1,m2; D1, D2 given in [BB15a, Section 5.1 and (1.1)], and p q p q we will treat the two Eisenstein series terms, which we denote Emax and Emin, occuring in [But16, Theorem 4] trivially.

11 2 3 For s s1,s2 C , µ C with µ1 µ2 µ3 0 deﬁne the meromorphic functions “ p q P P ` ` “ 1 3 G s,µ : Γ s1 µ Γ s2 µ , (9) Γ s s j j p q “ 1 2 j“1 p ´ q p ` q p ` q ź ´3s 3 1 3 1 π Γ s µj Γ 1 s µj G˜˘ s,µ : p 2 p ´ qq i p 2 p ` ´ qq , (10) π7{2 1 s µ 1 s µ p q “12288 ˜j“1 Γ 2 1 j ˘ j“1 Γ 2 2 j ¸ ź p p ´ ` qq ź p p ´ ` qq and the following trigonometric functions

3 `` 1 3 S s,µ : cos πνj , p q “ 24π2 2 j“1 ˆ ˙ ź 3 `´ 1 cos 2 πν2 sin π s1 µ1 sin π s2 µ2 sin π s2 µ3 S s,µ : 2 p q p 3p ´ qq3 p p ` qq p p ` qq, p q “´32π sin 2 πν1 sin 2 πν3 sin π s1 s2 3 p q p q p p ` qq ´` 1 cos 2 πν1 sin π s1 µ1 sin π s1 µ2 sin π s2 µ3 S s,µ : 2 p q p 3p ´ qq3 p p ´ qq p p ` qq, p q “´32π sin 2 πν2 sin 2 πν3 sin π s1 s2 3 p q p q p p ` qq ´´ 1 cos 2 πν3 sin π s1 µ2 sin π s2 µ2 S s,µ : 2 p q p 3p ´ qq3 p p ` qq. p q “ 32π sin πν2 sin πν1 p 2 q p 2 q 2 Then for y y1,y2 R 0 with sgn y1 α1, sgn y2 α2, let “ p q P p zt uq p q“ p q“ i8 i8 α1,α2 2 ´s1 2 ´s2 α1,α2 ds1 ds2 K y; µ 4π y1 4π y2 G s,µ S s,µ , (11) w6 p q“ | | | | p q p q 2πi 2 żí8 żí8 p q and for y R 0 with sgn y α, let P zt u p q“ i8 ´s ˜α ds Kw4 y; µ y G s,µ . (12) p q“ | | p q2πi żí8 The paths of integration must be chosen according to the Barnes convention as in [BB15b, Definition 1]. Then for n1,n2,m1,m2 N and hT as above we have P

C ∆ Σ4 Σ5 Σ6 Emax Emin, “ ` ` ` ´ ´ with C and ∆ as in Theorem 3.3,

S˜ αn2,m2,m1; D2, D1 αm1m2n2 Σ4 p´ qΦw4 , “ D1D2 D1D2 α“˘1 D2|D1 ˆ ˙ ÿ ÿ 2 m2D1“n1D2

S˜ αn1,m1,m2; D1, D2 αn1m1m2 Σ5 p qΦw5 , “ D1D2 D1D2 α“˘1 D1|D2 ˆ ˙ ÿ ÿ 2 m1D2“n2D1

S α2n2, α1n1,m1,m2; D1, D2 α2m1n2D2 α1m2n1D1 Σ6 p qΦw6 2 , 2 , (13) “ D1D2 ´ D ´ D α1,α2“˘1 D1,D2 1 2 ÿ ÿ ˆ ˙ 12 and

Φw4 y hT µ Kw4 y; µ spec µ dµ, p q“ p q p q p q żReµ“0

Φw5 y hT µ Kw4 y; µ spec µ dµ, (14) p q“ p q p´ ´ q p q żReµ“0 sgnpy1q,sgnpy2q Φw6 y1,y2 hT µ K y1,y2 ; µ spec µ dµ. p q“ p q w6 pp q q p q żReµ“0 As in [BB15a, Section 5.2], we may truncate the sums of Kloosterman sums at some high power 100 of T , say D1D2 T , and then replace hT , up to a negligible error, with a real-analytic, Weyl- ! group invariant function that is compactly supported in T Ω1, where Ω1 Ω is a slightly bigger Ě compact subset not intersecting the Weyl chamber walls, and satisﬁes

jpε´1q DjhT T (15) ! for every diﬀerential operator of order j. Theorem 3.3 will require strong bounds on the Φw functions, which we provide in the following proposition. Proposition 4.1.

´100 100 2 2 3`ǫ (a) For y T , T , α 1 , we have Φw6 αy T . P p q P t˘ u p q! ´100 100 3`ǫ 1{6 ´1{6 (b) For y T , T , α 1 , we have Φw4 αy T y y . P p q P t˘ u p q! ` ` ˘ We ﬁrst note that Kw6 has a much simpler sum-of-Mellin-Barnes integral expression than given above. For α 1 2, deﬁne P t˘ u d1`s1´µ d2`s2`µ 1`d3´s1´s2 3 i i ?π Γ Γ 2 Γ 2 Gα s,µ αd1 αd2 1 d1d2 2 , (16) 1 2 d3`s1`s2 p q“768 p´ q 1´`d1´s1`µi ¯ ´ 1`d2´s2´¯µi 2 Γ` 2 ˘ i“1 Γ Γ dPtÿ0,1u ź 2 2 ` ˘ ´ ¯ ´ ¯ using d3 d1 d2 mod 2 , d3 0, 1 . Then the long-element kernel function has the following ” ` p q P t u expression

1 Kα1,α2 y; w µ Ksym αy; µ , (17) 6 w6 p p qq “ w6 p q wPW ÿ where i8 i8 sym 2 ´s1 2 ´s2 α ds1 ds2 K αy; µ π y1 π y2 G s,µ , (18) w6 p q“ p q 2πi 2 żí8 żí8 ˇ ˇ ˇ ˇ p q and the unbounded portion of the s1,s2 integralsˇ ˇ mustˇ passˇ to the left of the zero line. This identity of functions may be verified by shifting the contours to the left and comparing power series expansions. Some care must be taken that

Re 2s1 s2 0, Re 2s2 s1 0, p ´ qă p ´ qă on the unbounded portions to maintain absolute convergence; this requires shifting the contours in stages. Then using (12) and the simpliﬁed (18), we can prove the following integral representations. 13 Lemma 4.2. The integral kernels above may be expressed as

´ν1 ´ν2 8 8 3{2 3{2 sym 1 d1 d2 y2z1 y1z2 Kw6 αy; µ η1η3α1 η2α2 (19) p q“6144π 2p q p q 0 0 y1 y2 dPt0,1u ż ż ˜ ¸ ˜ ¸ ÿ 3 ηPt˘1u d1`d2 d1`d2 d1`d2 sgn 1 η1z1 sgn 1 η2z2 sgn 1 η3z3 p ` q p ` q p ` q 2 d3 dz1dz2 K0 4π z4 1 Y0 4π z4 , π | | ´ p´ q | | z z ˆ ˙ 1 2 ´ a ¯ ´ a ¯ with 2 z2y1 y2 z3 2 , z4 1 η1?z1 1 η2?z2 1 η3?z3 , d3 d1 d2 2d1d2, (20) “ z1y2 “ p ` q p ` q p ` q ?z2 “ ` ´ and

y 2 3 z3 1 η1?z1 1 η2?z2 . (22) “ ?z1z2 p ` q p ` q The integral (21) converges in the Riemannian sense.

Here K0 and Y0 are the usual Bessel functions.

Proof of Proposition 4.1. For the compactly supported test function hT described above, deﬁne 8 8 ´3 1 iu1 iu2 hT v1, v2 3T hT iu v v spec iu du1 du2. (23) p q“´ 6144π p q 1 2 p q ż´8 ż´8 The support ofqhT allows us to remove the hyperbolic tangents from spec iu at a negligible cost, p q ε´1`ǫ and integration by parts in hT v1, v2 constrains v1 and v2 to a region vi 1 O T . We p q2´2ε “ ` have the trivial bound hT v1, v2 T . p q! 2 ` ˘ Now for y T ´100, T 100 q2, α 1 , Lemma 4.2 implies P p q q P t˘ u 8 8 3{2 3{2 3 d1 d2 y2z1 y1z2 Φw6 α1y1, α2y2 T η1η3α1 η2α2 hT , (24) p q“ 2p q p q 0 0 y1 y2 dPt0,1u ż ż ˜ ¸ ÿ 3 ηPt˘1u q d1`d2 d1`d2 d1`d2 sgn 1 η1z1 sgn 1 η2z2 sgn 1 η3z3 p ` q p ` q p ` q 2 d3 dz1dz2 K0 4π z4 1 Y0 4π z4 . π | | ´ p´ q | | z z ˆ ˙ 1 2 ´ a ¯ ´ a ¯ ´1 Then using the trivial bound log 3 z4 for the Bessel functions implies Φw6 α1y1, α2y2 ! ` | | p q! T 3`ǫ. ´ ¯ 14 For y T ´100, T 100 , α 1 , Lemma 4.2 again implies P p q P t˘ u T 3 8 8 z αy η η α d h 1 , z z η z Φw4 11{2 1 2 T ? 1 2 1 1? 1 (25) p q“´ 8π p´ q ?z2 1 η1?z1 | ` | dPt0,1u ż0 ż0 ˆ ˙ ÿ 2 ` ηPt˘1u q ˇ ˇ 1´2d ˇ dz dz ˇ d 6 d 3 1 2 sgn 1 η2z2 z3 sin π 2 2 z3 , p ` q | | ` | | z1z2 ´ a ¯ 3`ǫ 1{6 ´1{6 and trivially bounding the integral gives Φw4 αy T y y . p q! ` ` ˘ The next two sections are devoted to proving the integral representations of Lemma 4.2.

4.1 The long element weight function

Starting from (18), substitute s1 2s1 µ2, s2 2s2 µ2 so that ÞÑ ` ÞÑ ´ Ksym αy; µ w6 p q“ µ2 i8 i8 ?π d1 d2 d1d2 y2 2 ´2s1 2 ´2s2 α1 α2 1 π y1 π y2 768 2 p´ q y1 í8 í8p q p q dPtÿ0,1u ˆ ˙ ż ż 1`d3 d1 d2 d1 3 d2 3 Γ s1 s2 Γ s1 Γ s2 Γ s1 ν1 Γ s2 ν1 2 ´ ´ 2 ` 2 ` 2 ` ´ 2 2 ` ` 2 d3 1`d1 1`d2 1`d1 3 1`d2 3 Γ s1 s2 Γ s1 Γ s2 Γ s1 ν1 Γ s2 ν1 ` 2 ` ` ˘ `2 ´ ˘ ` 2 ´ ˘ `2 ´ ` 2 ˘ ` 2 ´ ´ 2˘ d1 3 d2 3 Γ s1 ν2 Γ s2 ν2 ds ds ` 2 ` ` 2˘ ` 2 ` ´˘ 2 ` 1 ˘ 2 .` ˘ ` ˘ 1`d1 3 1`d2 3 2 Γ s1 ν2 Γ s2 ν2 2πi `2 ´ ´ 2 ˘ ` 2 ´ ` 2˘ p q ` ˘ ` ˘ 1 Lemma 4.3. For Re s1 u , Re s2 u 0,Re s1 s2 , we have p ´ q p ` qą p ` qă 2 1`d3 d1 d2 Γ s1 s2 Γ s1 u Γ s2 u ?π 2 ´ ´ 2 ` ´ 2 ` ` (26) d3 1`d1 1`d2 Γ s1 s2 Γ s1 u Γ s2 u “ `2 ` ` ˘ 2` ´ ` ˘ ` 2 ´ ´ ˘ d1d2 8 1 ´2s1´2s2 dz p´` q ˘ηd1` sgn 1 ηz˘ d1`d2 1 η?z ˘ zú`s1 , 2 p ` q ` z η“˘1 ż0 ÿ ˇ ˇ i8 d1 d2 1 Γ s1 u Γ s2 u ˇ ˇ 2 ` ´ 2 ` ` zudu (27) 1`d1 1`d2 2πi í8 Γ s1 u Γ s2 u “ ż `2 ´ ` ˘ ` 2 ´ ´˘ d1d2 d3 1 ` s1 Γ 2 s˘1 ` s2 d1 ˘ d1`d2 ´2s1´2s2 p´ q z ` ` η sgn 1 ηz 1 η?z , 2?π 1`d3 p ` q ` Γ ` 2 s1 s˘2 η“˘1 ´ ´ ÿ ˇ ˇ ` ˘ ˇ ˇ Proof. By elementary substitutions, the right-hand side of (26) is given by the sum of beta functions

d1 d2 1 B 1 2s1 2s2, 2s1 2u 1 B 1 2s1 2s2, 2s2 2u B 2s1 2u, 2s2 2u . p´ q p ´ ´ ´ q`p´ q p ´ ´ ` q` p ´ ` q Then (26) follows by applying reﬂection to the gamma functions in the denominators and trigonom- etry, and (27) follows from Mellin inversion. Note that Mellin inversion produces an integral which converges in the Riemannian sense; the contour may then be deformed so the unbounded portion passes to the left of the zero line for absolute convergence.

15 Plugging (26) into the preceeding equation, and substituting s1 s1 s2, ÞÑ ´ Ksym αy; µ w6 p q“ µ2 i8 8 8 1 d1 d2 d1d2 y2 d1`d2 d1`d2 η1α1 η2α2 1 sgn 1 η1z1 sgn 1 η2z2 3072?π 2p q p q p´ q y1 ´i8 0 0 p ` q p ` q dPt0,1u ˆ ˙ ż ż ż ÿ 2 ηPt˘1u d3 i8 2 s2 d1 d2 Γ s1 z y Γ s1 s2 Γ s2 ds π2y ´2s1 2 ` 2 1 2 ` ´ 2 ` 2 1 1`d3 2 1`d1 1`d2 p q Γ s z1y Γ s s Γ s 2πi ` 2 ˘1 ż´i8 ˆ 2 ˙ `2 1 2˘ ` 2 ˘ 2 ´ 3 3 ´ ` ´ ´2s1 ´2s1 ´ 2 ν1`s1 ´ 2 ν2 dz1dz2 ds1 1 η1?z1 ` 1 η2?˘ z2 z1 z2` .˘ ` ˘ | ` | | ` | z1z2 2πi

Using z3 as in (20) and taking s2 0 in (27), we have Ñ ´ν1 ´ν2 8 8 3{2 3{2 sym 1 d1 d2 y2z1 y1z2 Kw6 αy; µ η1η3α1 η2α2 p q“6144π 2p q p q 0 0 y1 y2 dPt0,1u ż ż ˜ ¸ ˜ ¸ ÿ 3 ηPt˘1u d1`d2 d1`d2 d1`d2 sgn 1 η1z1 sgn 1 η2z2 sgn 1 η3z3 p ` q p ` q p ` q i8 d3 2 Γ 2 s1 ´2s1 ` 1 η3?z3 1`d3 2 | ` | ´i8 Γ ` s˘1 ż 2 ´ 4 2 ´s1 ` ´2s1 ˘ ´2s1 π y2 ds1 dz1dz2 1 η1?z1 1 η2?z2 . | ` | | ` | z 2πi z z ˆ 2 ˙ 1 2 Then (19) follows by applying the following lemma.

Lemma 4.4. For z 0, we have ą i8 d3 2 1 Γ 2 s1 ´s1 2 1{4 d3 1{4 ` z ds1 K0 4z 1 Y0 4z . 2πi 1`d3 2 “π p q´p´ q p q ´i8 Γ ` s˘1 ż 2 ´

Proof. The d3 0 case is given` in [GR15,˘ 6.422.16] or [PBM90, 8.4.23.15], but the d3 1 case “ “ seems to be missing from the literature. So from [GR15, 6.561.15 and 6.561.16], we have

8 2 d u´1 u d πu 2 2K0 2x 1 πY0 2x x dx Γ 1 1 cos . 0 p q´p´ q p q “ 2 ` p´ q 2 ż ´ ¯ ´ ¯ ´ ¯ By the half-angle and duplication formulae, the right-hand side is

2 Γ d`u{2 π u 2 2 2 , 2 ´1`d´u{¯2 Γ 2 ´ ¯ and the claim follows by Mellin inversion.

16 4.2 The w4 weight function

µ1 In (12), we write µ3 µ1 µ2 and apply (26) with u , giving “´ ´ “ 2

2s`µ2 d`s´µ2 i8 Γ Γ 1 d 3 ´s 2 2 Kw4 αy; µ 13 4 η1α π y p q“2 3π p´ q ´i8 1´´2s´µ2 ¯ ´ 1`d´s`¯µ2 dPt0,1u ż Γ 2 Γ 2 η1Pt˘ÿ 1u ˇ ˇ ˇ ˇ ´ ¯ ´ ¯ 8 ´µ1`s ´2s´µ2 2 dz1 ds 1 η1?z1 z , | ` | 1 z 2πi ż0 1 and again with u µ2 giving “ 2 8 8 µ1 µ2 1 d d ´µ2 ´ 2 ´ 2 Kw4 αy; µ η1η2α sgn 1 η2z2 1 η1?z1 z z p q“214 3π9{2 p´ q p ` q | ` | 1 2 dPt0,1u 0 0 2 ż ż ηPt˘ÿ1u

i8 d`3s s s Γ 2 ´2s ´3s 3 ´s 2 2 ds dz1 dz2 1`d´3s 1 η1?z1 1 η2?z2 π y z1 z2 . Γ | ` | | ` | 2πi z1z2 ż´i8 ` 2 ˘ ˇ ˇ ˇ ˇ We have [GR15, 6.422.9], ` ˘

i8 Γ u s ´s ds p´ ` qx Ju 2?x , Γ 1 s 2πi “ p q ż´i8 p ´ q which holds for Re u 0, but extends to all u by analytic continuation (deforming the contour p q ą for convergence). Using the known values of J˘1{2 (see [GR15, 8.464]),

i8 Γ d 1 s ds ´ 2 ` x´s π´1{2x´1{4 sin πd 2?x , Γ 1 s 2πi “ 2 ` ż´i8 ` ˘ p ´ q ` ˘ and (21) follows.

5 Proof of Theorem 3.3

By [BB15b, Lemma 9] and the argument of [BB15a, Section 5.2], the long element term sums ´4 ǫ over D1D2 P T T P , and we apply the proof of [Blo13, Proposition 3] and the bound 3`ǫ ! p q T for Φw6 from Proposition 4.1a. Similarly, by [BB15b, Lemma 8], the w4 term sums over ´3 ǫ 3`ǫ 1{6 ´1{6 D1D2 P T T P , and we use the bound T y y for Φw4 from Proposition 4.1b, in ! p q 3´ǫ ` combination with y T , again from [BB15b, Lemma 8], and Larsen’s bound on the w4 Kloost- " ` ˘ erman sum [BFG88, Appendix]; the treatment of the w5 term is identical. As in the appendix to [Blo13], the contribution of the Eisenstein series terms is O T 3`ǫP θ`ǫ with θ 7 1 . “ 64 ă 6 ` ˘ Acknowledgment

The authors would like to thank Joseph Hundley, without whom this work would not be possible. The authors would like to thank Valentin Blomer for reading the manuscript. The authors would like to thank Wenzhi Luo and Yiannis Sakellaridis.

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Fan Zhou Mathematics & Statistics University of Maine Orono, ME