DISTRIBUTION OF HECKE EIGENVALUES FOR GL(n)

JASMIN MATZ

Abstract. The purpose of this survey is to briefly summarize and explain the resultsof [Mat] and joint work with N. Templier [MT] about the asymptotic distribution of eigen-values of Hecke operators on cusp forms for GL(n). We also to sketch some motivation andpotential extensions of our results.

1. Introduction

Let F be a number field with ring of adeles AF , and let n ≥ 2 be an integer. Let G = GL(n),and let G(AF )1 := g ∈ G(AF ) | | det g|AF = 1 where | · |AF denotes the adelic absolutevalue on A×F . One is interested in the spectral decomposition of the space L2(G(F )\G(AF )1)under the right regular representation of G(AF ). Under G(AF ) the space L2(G(F )\G(AF )1)decomposes into invariant subspaces as

L2(G(F )\G(AF )1) = L2cusp(G(F )\G(AF )1)⊕ L2

res(G(F )\G(AF )1)⊕ L2cts(G(F )\G(AF )1),

where L2cusp (resp. L2

res, resp. L2cts) denotes the cuspidal (resp. residual, resp. continuous)

part of L2 under the right regular representation of G(AF )1. The cuspidal part is the mostfundamental one in the sense that the residual and continuous parts can be described in termsof Eisenstein series and their residues attached to cuspidal representations on Levi subgroupsof G [Lan76, MW95]. It is therefore of importance to understand the spectral propertiesof the space of cusp forms. One of the most basic questions is to asymptotically count thenumber of Laplace eigenfunctions of bounded eigenvalue for the locally symmetric spacesG(F )\G(AF )1/K where K = K∞ · Kf ⊆ K is a finite index subgroup of a fixed maximalcompact subgroup K ⊆ G(AF ). The Weyl law answers this question in many cases.

Suppose F = Q for the rest of the introduction. Then O(n) is a maximal compact subgroupof G(F∞) = G(R), and G(AQ)1/O(n) ' SLn(R)/ SO(n) =: X. Let ∆ be the Laplacianon L2(X), and let Γ ⊆ SLn(R) be an arithmetic congruence subgroup. Weyl’s law in itsmost basic form counts the number of eigenvalues of ∆ in L2

cusp(Γ\X). More precisely, let0 ≤ µ1 ≤ µ2 ≤ . . . be the cuspidal eigenvalues of ∆ (with multiplicities). Then

(1) #i | µ2i ≤ Y ∼ c vol(Γ\X)Y d

as Y → ∞ for c > 0 a constant depending only on n, and d = dimRX. This was proven bySelberg for n = 2 [Sel56], by Miller for n = 3 [Mil01], and by Muller for general n [Mul07].The Weyl law also holds for more general groups, cf. [DKV79, LV07]. This in particular provesthe existence of infinitely many cusp forms in L2(Γ\X) but stills gives only crude informationon the spectral properties of Γ\X. Apart from ∆ there are many more naturally occuringoperators on L2

cusp(Γ\X), and one can study the distribution of their (joint) eigenvalues aswell.

Let D(X) be the algebra of SLn(R)-invariant differential operators on X. It is isomorphicto the Weyl group invariants Z(sln(C))W ' aWC of the center Z(sln(C)) of the universal

1

2 JASMIN MATZ

enveloping algebra of the Lie algebra sln(C) of SLn(C). Here a is the Lie algebra of themaximal diagonal torus in SLn(R) which can be identified with all vectors (X1, . . . , Xn) ∈ Rnsuch that

∑iXi = 0, and aC is its complexification. If π ⊆ L2

cusp(X) is an irreduciblecomponent, elements of D(X) act by a scalar on π by Schur’s Lemma so that π defines acharacter λπ : aWC −→ C (the infinitesimal character), that is, λπ is a W -invariant element inthe dual space a∗C of aC. In generalization of (1) one can ask how the λπ distribute if one takeslarger and larger subsets of a∗C. This question was answered by Lapid and Muller [LM09], seealso below.

Apart from the algebra of differential operators, there is a second family of operators actingon L2

cusp(Γ\X), namely, the algebra of Hecke operators. The Hecke algebra is commutativeand preserves the eigenspaces of D(X). Suppose Tnn∈N is a family of Hecke operators,and let ψ1, ψ2, . . . be a joint eigenbasis for L2

cusp(Γ\X) for D(X) and Tnn∈N. For every ilet λi ∈ a∗C/W be the infinitesimal character of the irreducible representation generated byψi, and let ai(n) ∈ [−‖Tn‖, ‖Tn‖] be the eigenvalue of ψi under Tn. Here ‖Tn‖ denotes theoperator norm of Tn. Then

Λi := (λi, ai(1), ai(2), . . .)

defines a point in the space

A := a∗C/W ×∏n∈N

[−‖Tn‖, ‖Tn‖],

and one can ask how these Λi distribute in A (with respect to the chosen ordering of thebasis). This question was studied in [Sar87] for n = 2, and in [ST15] for groups G for whichG(R) has discrete series, cf. also [SST16].

2. Results

2.1. Notation. Recall that n ≥ 2 and G = GL(n) over a fixed number field F . AF denotesthe ring of adeles of F , and AF,f the finite part of AF,f . Let OF be the ring of integers of F .If v is a non-archimedean place of F , we write OFv for the ring of integers in Fv, qv ⊆ OFvfor the maximal ideal in OFv , $v ∈ qv for a fixed uniformizing element of Fv, and qv for thecardinality of the residue field at v. Let T0 ⊆ G be the maximal torus consisting of diagonalmatrices, and let P0 = T0U0 be the usual minimal parabolic subgroup of upper triangularmatrices with U0 the unipotent radical of P0. We write Z ⊆ G for the center of G. We alsoidentify Z(F ) with a subgroup of the finite part Z(AF,f ) ⊆ G(AF,f ).

We fix the usual maximal compact subgroup K ⊆ G(AF ), K =∏v Kv, with

Kv =

O(n) if v is a real place,

U(n) if v is a complex place,

G(OFv) if v is non-archimedean.

For a non-archimedean place v and an integer m ≥ 0 let

Kv(qmv ) = ker (Kv −→ G(OFv/qmv ))

be the principal congruence subgroup of level qmv . If a ⊆ OF is an ideal with prime factoriza-tion a =

∏v<∞ qmvv , we put

Kf (a) =∏v<∞

Kv(qmvv ),

DISTRIBUTION OF HECKE EIGENVALUES 3

and K(a) = K∞ · Kf (a) with K∞ =∏v|∞Kv ⊆ G(F∞) =

∏v|∞G(Fv). If F = Q and

N ∈ Z≥1, we also write Kf (N) = Kf (NZ) and K(N) = K(NZ).Let Πcusp(G(AF )1) denote the set of irreducible cuspidal automorphic representations of

G(AF )1, and for π = π∞ · πf ∈ Πcusp(G(AF )1) let λπ∞ ∈ a∗C/W denote the infinitesimalcharacter of π∞. If convenient, we identify π with its representation space so that we maywrite dimπK for the dimension of the subspace of K-fixed vectors in the representation spaceof π.

For simplicity of the statements below we choose the Haar measure on G(Fv) such that itgives Kv volume 1 for every v. We then take the product measure on G(AF ), and fix themeasure on G(AF )1 via the exact sequence

1 −→ G(AF )1 → G(AF ) −→ R>0 −→ 1

where we take the usual multiplicative Lebesgue measure on R>0. The maximal compactsubgroup K then has volume 1 with respect to the measure on G(AF )1. If Ξ ⊆ G(AF )(resp. Ξ ⊆ G(AF )1, resp. Ξ ⊆ G(AF,f )) is a measurable subset, we write vol(Ξ) for the volumeon Ξ with respect to the measure on G(AF ) (resp. G(AF )1, resp. G(AF,f )). For differentchoices of measures one might need to adjust some of the constants below accordingly.

2.2. Weyl law with remainder term for SLn(R). Let F = Q, and let K = K∞ ·Kf withKf ⊆ Kf a finite index subgroup such Kf ⊆ Kf (N) for some N ≥ 3. This last requirementensures that Kf does not have any non-trivial element of finite order.

Lapid and Muller [LM09] proved a refined version of the Weyl law for G(F )\G(AF )1/K:If Ω ⊆ ia∗ is a W -invariant bounded domain with piecewise C2-boundary, then

(2)∑

π∈Πcusp(G(AQ)1):λπ∞∈tΩ

dimπK =vol(G(Q)\G(AQ)1/Kf )

|W |

∫tΩ

c(λ)−2 dλ+O(td−1(log t)max3,n),

as t → ∞, where c(λ) denotes the Harish-Chandra c-function for SLn(R) so that c(λ)−2dλis the spherical Plancherel measure for SLn(R). In more classical terms this gives the as-

ymptotic distribution (with weight factor dimπK(N)) of the infinitesimal characters of cuspforms on Γ(N)\X, N ≥ 3. This is because the quotient G(Q)\G(AQ)1/K(N) is isomorphicto (Z/NZ)×-copies of Γ(N)\X for Γ(N) = γ ∈ SLn(Z) | γ ≡ 1 mod N the principalcongruence subgroup of level N . Taking Ω to be the unit ball in ia∗, one recovers the usualWeyl law (1) together with an upper bound for the error term.

Let Bt(0) denote the ball of radius t in a∗C. According to [LM09] one also has

(3)∑

π∈Πdisc(G(AQ)1):λπ∞∈Bt(0)\ia∗

dimπK = O(td−2)

i.e. the number of non-tempered π ∈ Πcusp(G(AQ)1) (which are supposed to be non-existentaccording to the generalized archimedean Ramanujan Conjecture) is at most of lower orderthan the number of tempered representations.

2.3. Traces of Hecke operators. We now turn to the main results of [Mat, MT]. Let Fbe an imaginary quadratic number field (that is, a quadratic field extension of Q with onecomplex place) or F = Q. The first case is covered in [Mat] while the second case is thesubject of [MT].

4 JASMIN MATZ

2.3.1. Hecke algebra. For every non-archimedean place v of F consider the spherical Heckealgebra Hv = C∞c (G(Fv) // Kv) of locally constant, compactly supported bi-Kv-invariantfunctions. This is a commutative C-algebra under convolution for which the characteristicfunction of Kv is the unit element. For λ = (λ1, . . . , λn) ∈ Zn let τv,λ ∈ Hv denote the

characteristic function of the double coset Kv$λvKv, where

$λv :=

$λ1v

. . .$λnv

.

The set of functions τλ | λ ∈ Zn, λ1 ≥ . . . ≥ λn generates Hv as a C-algebra. We writeZn,+ for the set of tuples (λ1, . . . , λn) ∈ Zn with λ1 ≥ . . . ≥ λn. If λ ∈ Zn, we write

‖λ‖ = (∑λ2i )

1/2 for the usual Euclidean norm of λ. If κ ≥ 0, we let H≤κv be the sub-vector-space of Hv generated (as a vector space over C) by the functions τv,λ with ‖λ‖ ≤ κ. If S

is a finite set of non-archimedean places, we put HS =∏v∈SHv, and H≤κS =

∏v∈SH≤κvv

if κ = (κv)v∈S is a sequence of non-negative numbers. If τS ∈ HS we also identify τSwith a function τ ∈ C∞c (G(AF,f ) // Kf ) by putting τ = τS · 1KS∪S∞ where S∞ is the set

of archimedean places of F , and 1KS∪S∞ : G(AS∪S∞F ) −→ C the characteristic function of

KS∪S∞ =∏v 6∈S∪S∞ Kv.

If κ = (κv)v∈S is a sequence of non-negative numbers, we set

Πκ =∏v∈S

qκvv .

This number provides an upper bound for the “degrees” (that is, L1-norms) of the Hecke

operators in H≤κS : There exists a > 0 such that for every τS ∈ HκS with |τS | ≤ 1 we have

‖τ‖L1(G(AF,f )) = ‖τS‖L1(G(FS)) ≤ Πaκ.

2.3.2. Distribution of traces of Hecke operators. Let F = π ∈ Πcusp(G(AF )1) | πK 6= 0 bethe sepctral set of all everywhere unramified cuspidal representations with trivial K∞-type,cf. [SST16]. Let Ω ⊆ ia∗ be as before. We use the infinitesimal character and the domain Ωto put an order on the set F : For t > 0 let

F(t) = FΩ(t) = π ∈ F | λπ∞ ∈ tΩ.

According to the generalized archimedean Ramanujan conjecture, every element of F shouldeventually appear in F(t) for t sufficiently large if Ω is ”thick enough”, that is, if Ω is suchthat

⋃t>0 tΩ = ia∗. In any case, the estimate (3) from [LM09] shows (for F = Q; but one

can show that a similar statement is true for F imaginary quadratic) that one does not miss”too many” elements.

Theorem 2.1. (i) As t→∞ we have

|F(t)| ∼ |O×F |vol(G(F )\G(AF )1/Kf )

|W |

∫tΩ

c(λ)−2 dλ

in the sense that the difference of the left and right hand side tends to 0 as t → ∞.Here |O×F | is the number of multiplicative units in OF .

(ii) There exist constants a, b, δ > 0 (depending only on n, Ω, and F ) such that thefollowing holds: For every finite set of non-archimedean places S0, every sequence of

DISTRIBUTION OF HECKE EIGENVALUES 5

non-negative numbers κ = (κv)v∈S0 and every τS0 ∈ H≤κS0

with |τ | ≤ 1 we have

(4) limt→∞|F(t)|−1

∑π∈F(t)

trπf (τ) = limt→∞|F(t)|−1

∑π∈F(t)

trπS0(τS0) =∑

z∈Z(F )/Z(OF )

τ(z),

and

(5)

∣∣∣∣∣∣|F(t)|−1∑

π∈F(t)

trπf (τ)−∑

z∈Z(F )/Z(OF )

τ(z)

∣∣∣∣∣∣ ≤ aΠbκt−δ

for every t ≥ 1.

Remark 2.2. (i) The number |O×F | is finite by our assumption that F has only onearchimedean place.

(ii) Taking S0 = ∅ so that τ is the characteristic function of Kf , the second part ofTheorem 2.1 also gives an upper bound for the remainder term of the asymptotic ofthe first part. Hence for F = Q we obtain the analogue of [LM09] but for the fullmodular group Γ = SLn(Z) (which was excluded in [LM09] for technical reasons) -however with a slightly worse error term.

(iii) Taking τS0 =∏v∈S0

τv,λv in the above theorem, we see that the main term, that is theright hand side of (4), vanishes for many sequences of λv. More precisely, the mainterm vanishes unless

(6) λv,1 = . . . = λv,n

for every v < ∞. In this situation, τ corresponds to an ideal a ⊆ OF defined by

a =∏v∈S0

qλv,1v and τ(z) 6= 0 if and only if z (identified with an element in F×)

generates a so that a needs to be principal. Hence if for every v (6) is satisfied and ifthe sequence of λv,1, v ∈ S0, corresponds to a principal ideal, we get

limt→∞|F(t)|−1

∑π∈F(t)

trπf (τ) = 1,

and the left hand side vanishes in all other cases. In general, any τS0 ∈ HκS0

is a linearcombination of characteristic functions of double cosets so that this consideration canbe applied to an arbitrary τS0 .

If F = Q we can reformulate the above result in more measure theoretic terms, namely interms of measures on the unitary dual of PGLn(QS). (For F 6= Q one can make a similarreformulation but one has to be more careful with central characters.) Let H = PGLn. Let

FH = π0 ∈ Πcusp(H(AQ)) | πKH 6= 0 for KH = K ∩ H(AQ) the usual maximal compactsubgroup of H(AQ), and put FH(t) = π0 ∈ FH | λπ∞ ∈ tΩ. The sets F and FH aswell as F(t) and FH(t) can be canonically identified with each other since every π ∈ Fhas trivial central character wo that ot can be identified with an element of FH . Hence ifτ = τS0 ⊗ 1KS0 ∈ C∞c (G(AQ,f )) is bi-Kf -invariant,

trπf (τ) =

∫G(AQ,f )

τ(x)ϕ(x) dx

where ϕ is a normalized spherical matrix coefficient for πf . This equals∫Z(AQ,f )\G(AQ,f )

∫Z(AQ,f )

τ(zg) dz ϕ(g) dg =

∫Z(AQ,f )\G(AQ,f )

∑γ∈Z(Q)/Z(Z)

τ(γg)ϕ(g) dg

6 JASMIN MATZ

Hence the above equals∫Z(AQ,f )\G(AQ,f )

∑γ∈Z(Q)/Z(Z)

τ(γg)

ϕ(g) dg = trπf (τ)

where

τ(x) =∑

γ∈Z(Q)/Z(Z)

τ(γx) =∑

γ∈Z(Z[S−10 ])/Z(Z)

τS0(γx) = τS0(x)

with Z[S−10 ] = Z[p−1 | p ∈ S0]. In particular,∑

π∈F(t)

trπS0(τS0) =∑

π∈FH(t)

trπS0(τS0).

Since |F(t)| = |FH(t)| we get by Theorem 2.1

(7) limt→∞

∣∣FH(t)∣∣−1 ∑

π∈FH(t)

trπS0(τS0) = τS0(1).

Each π0 ∈ FH defines a point in

Aur :=∏v<∞

H(Qv)ur,

as well as its projection to the S0-component

AurS0

:=∏v∈S0

H(Qv)ur

where H(Qv)ur

denotes the unramified unitary dual of H(Qv). Hence we can ask how theset FH(t), considered as a subset of Aur or Aur

S0, distributes in Aur or Aur

S0. For n = 2 this

question was studied in [Sar87] for F = Q, and in [IR10] for F imaginary quadratic; forgroups with discrete series at ∞ this question was studied in [Ser97, CDF97, Shi12, ST15].

For πv ∈ H(Qv)ur

let δπv denote the Dirac measure supported at πv, and let δπS0=∏

v∈S0δπv . Put

µS0count,t =

∣∣FH(t)∣∣−1 ∑

π0∈FH(t)

δπS0.

For each v <∞ we also have the spherical Plancherel measure µPl,v on H(Qv)ur

. Let µPl,S0 =∏v∈S0

µPl,v. Then (7) says that

µcount,t( τS0) −→ µS0Pl (τS0)

as t→∞ for every τS0 ∈ HS0 , and it also gives an upper bound for the error term.

Here τS0 =∏v∈S0

τv with τv defined byτv(πv) = trπv(τv)

for every tempered πv ∈ H(Qv)ur

. By Sauvageot’s density principle [Sau97] (cf. also [Shi12,

ST15, FLM15]) this is enough to prove that µS0count,t −→ µS0

Pl since the bi-KHS0

-invariant func-tions on H(FS0) are contained in the image of HS0 under the map τS0 7→ τS0 .

DISTRIBUTION OF HECKE EIGENVALUES 7

2.3.3. Standard L-functions. The above theorem gives information on the coefficients of L-functions attached to unramified cuspidal representations: If π ∈ F , there is a standardL-function L(s, π) associated to π for <s sufficiently large. The L-function can be written asa Dirichlet series

L(s, π) =∑a⊆OF

Aa(π)N(a)−s

for suitable coefficients Aa(π) ∈ C, where the sum runs over all integral ideals in OF , andN(a) = |OF /a| denotes the norm of the ideal a. Moreover, for each a there exists an elementin the Hecke algebra τS0 ∈ HS0 (with S0 the set of places dividing a) such that Aa(π) =trπS0(τS0) for all π ∈ F . More precisely, this τ is a linear combination of those

∏v<∞ τλv

with N(a) =∏v<∞ q

∑i λv,i

v and λv,1 ≥ . . . ≥ λv,n ≥ 0. In the case of F = Q and a a principalideal NZ, then τ = TN is the usual Hecke operator attached to N [Gol06, §9].

Then the above theorem implies that there exist a, b, δ > 0 such that for every ideal a ⊆ OFwe have

limt→∞|F(t)|−1

∑π∈F(t)

Aa(π) =

1 if a = bn for some principal ideal b ⊆ OF ,0 else,

and further, ∣∣∣∣∣∣|F(t)|−1∑

π∈F(t)

Aa(π)− δn(a)

∣∣∣∣∣∣ ≤ aN(a)bt−δ, t ≥ 1,

where δn(a) = 1 if a is the nth power of some principal ideal in OF , and δn(a) = 0 otherwise.Using Hecke relations, one can similarly compute the asymptotics for higher moments∑π∈F(t)Aa(π)k for any k ∈ Z≥0.

2.4. The relevance of the error term. Since much work needs to be invested to prove theestimate (5), we want to indicate briefly a motivation for it: As explained above, the traces ofHecke operators are closely related to standard L-functions of automorphic representations.Our spectral set of representations F(t) defines a family of L-functions L(s, π), π ∈ F(t).There has been much recent interest in the distribution of low-lying zeros of families of L-functions, cf. [KS99, ILS00, ST15, SST16]. More precisely, one is interested in the k-leveldensities

(8) |F(t)|−1∑

π∈F(t)

∑γπj1

,...,γπjk

Φ

(γπj1 log t

2π, . . . ,

γπjk log t

2π

),

where Φ is a Schwartz–Bruhat function on Rk whose Fourier transform has compact support,and the ρπj1 = 1

2 + iγπj1 , . . . , ρπjk

= 12 + iγπjk run over all pairwise different k-tuples of zeros of

L(s, π). Since we do not assume GRH, the γπj may happen to be complex, and we identify Φ

with its holomorphic extension to Ck. (Similar expressions can be studied for other familiesof L-functions of course.)

It is conjectured that the low–lying zeros of families of L-functions are distributed accordingto certain symmetry types associated with the families (cf. [SST16, Conjecture 2]). This meansthat for any Schwartz-Bruhat function Φ the limit of (8) as t→∞ is supposed to equal

(9)

∫Rk

Φ(x)W (x) dx

8 JASMIN MATZ

where W (x) is is a certain density attached to the conjectured symmetry type.One can attack this problem by using the explicit formula for L-functions (cf. [ST15]). To

control unwanted terms in the explicit formula one then uses the estimate (5) among otherthings. In particular, one can show that for the family of L-functions attached to F(t) theexpression (8) approaches (9) as t →∞ for any Schwartz-Bruhat functions Φ whose Fouriertransform has sufficiently small support, see [MT]. The quality of the estimate (5) controlsthe allowed size of the support of the Fourier transform of Φ.

There is another application of our results, see [MT, Corollary 1.6, Corollary 1.7], namely,we can give a bound towards the p-adic Ramanujan conjecture on average (see [LM09] for anaverage bound towards the archimedean Ramanujan conjecture). If π ∈ F , then for every fi-

nite prime p we can identify πp with its Satake parameter in απ(p) = diag(α(1)π (p), . . . , α

(n)π (p)) ∈

T0(C)/W . The p-adic Ramanujan conjecture asserts that in fact απ(p) ∈ T0(C)1/W for allπ ∈ F and all finite prime p where T0(C)1 denotes the group of all complex diagonal matriceswith entries of absolute value 1. From our results we can now deduce the following: Forθ, t > 0 define

R(p, t, θ) = |π ∈ F(t) | max1≤j≤n

logp |α(j)π (p)| > θ|.

Hence the p-adic Ramaujan conjecture asserts that R(p, t, θ) = 0 for every θ > 0. Note thatit is known that R(p, t, θ) = 0 whenever θ > 1

2 −1

n2+1by [LRS99]. Then we can deduce on

the one hand that there are constants c, ω > 0 such that for all t ≥ 1, all θ > 0 and all finiteprimes p we have

R(p, t, θ) ≤ Ctd−cθ+ω

log p

for some C > 0, a constant which depends on p and θ. On the other hand we can show thatif we are given a finite set S0 of finite primes, then for every θ > 0 there exists a constantρ > 0 such that

R(p, t, θ) ≤ C ′td−ρ

for all t ≥ 1. Here C ′ > 0 is again a constant depending only on S0 and θ.

3. Idea of proof

The main tool for proving Theorem 2.1 is the Arthur-Selberg trace formula for the groupG = GL(n) over F (again, F = Q or F is imaginary quadratic in this section). It is acommon approach to use various kinds of trace formulae to prove the Weyl law in its differentforms, cf. [Sel56, DKV79, Mil01, Mul07, LV07, LM09, Mul16]. In fact, one motivation forSelberg to develop the trace formula was to prove the Weyl law for locally symmetric spacesΓ\ SL2(R)/SO(2) for Γ ⊆ SL2(R) an arithmetic congruence subgroup.

Recall that the Arthur-Selberg trace formula is an identity of distributions

Jgeom(f) = Jspec(f)

of the so-called geometric and spectral side on the space of smooth, compactly-supportedtest functions f ∈ C∞c (G(AF )1), cf. [Art05]. The main strategy is then as follows: For anappropriate choice of test function (or rather a family of test function - see below for details),it is not too hard to show that

∑π: λπ∞∈tΩ trπf (τ) is the main part of the spectral side (or

rather of some integral over tΩ of the spectral side) as t→∞. Similarly, it can be shown that|F(t)|−1

∑z∈Z(F )/Z(OF ) τ(z) is the main part of the (integral over tΩ of the) geometric side.

The main difficulty is to obtain an upper bound for the error term, and in particular, to prove

DISTRIBUTION OF HECKE EIGENVALUES 9

its effectiveness in τ . This is achieved by analyzing the remaining parts of the geometric andspectral side of the trace formula.

Finding good upper bounds for the remaining parts of the geometric side of the traceformula is the most difficult part. Bounding the remaining parts on the spectral side isvery similar to the proof in [LM09], and we will not go into further details. Many of theproblems on the geometric side which we need to consider do not appear in the treatment ofthe geometric side in [LM09]. This is because in [LM09] the non-archimedean test functionis fixed in contrast to the fact that we want to vary our S0 and τS0 . In fact, in [LM09] it canbe achieved that only the unipotent part of the geometric side of the trace formula remainsto study (see also below for a short reminder of the coarse expansion of Jgeom(f)).

To explain the proof in some more detail we first need to explain our choice of test functions.

3.1. Test functions. The family of test functions used in our proof is constructed with thespectral side in mind: It is of the form Fµ,τ = (fµ∞ · τ)|G(AF )1 for a suitable family of bi-K∞-

invariant functions fµ∞ ∈ C∞c (G(F∞)1 // K∞) depending on the spectral parameter µ ∈ a∗C.The choice of the non-archimedean part of the test function suggests itself from what wewant to get from the cuspidal part of the trace formula, and it is the same as in [LM09].More precisely, it is chosen such that trπ∞(fµ∞) only contributes if λπ∞ is very close to µ. Inparticular, the integral

(10)

∫tΩ

∑π∈F

trπ(Fµ,τ ) dµ

basically captures only those π ∈ F with λπ∞ ∈ tΩ, that is, it equals∑π∈F : λπ∞∈tΩ

trπf (τ)

up to an error term which can be estimated, cf. also [LM09].The family fµ∞ is constructed following the ideas of [DKV79]. By the Paley–Wiener Theo-

rem the diagram

C∞c (G(F∞)1 //K∞)H //

A ((

P(a∗C)W

C∞c (aC)W

F

OO

is commutative and all maps are isomorphisms. Here:

• P(a∗C)W is the space of Weyl group invariant Paley-Wiener functions on a∗C,• H denotes the spherical Fourier transform (= Harish Chandra transform),• A is the Abel transform, and• F the Fourier transform.

Hence the inverses A−1 and H−1 are well-defined. If h ∈ C∞c (aC)W and µ ∈ a∗C we put

hµ(X) := h(X)e−〈µ,X〉 where 〈·, ·〉 denotes the pairing on a∗C × aC. One then fixes an appro-

priate choice of h ∈ C∞c (aC)W as in [DKV79] (cf. [LM09]) and puts fµ∞ := A−1(hµ). Moreprecisely,

(11) fµ∞(g) = |W |−1

∫ia∗F(hµ)(λ)φλ(g)c(λ)−2 dλ,

10 JASMIN MATZ

where

φλ(g) =

∫K∞

e〈λ+ρ,H0(kg)〉 dk

is the elementary spherical function of parameter λ, and c(λ) denotes the Harish-Chandrac-function for G(F∞).

3.2. Expansions of the geometric side. The starting point for the analysis of Jgeom(Fµ,τ )is its coarse expansion, see [Art78], [Art05, §10]: Two elements g1, g2 ∈ G(F ) are calledgeometrically equivalent if their semisimple parts (in the Jordan decomposition) are conjugatein G(F ). Since G = GL(n), this amounts to saying that g1 and g2 have the same characteristicpolynomial. G(F ) then decomposes into a disjoint union of geometric equivalence classesunder this relation, and we write O for the set of all these equivalence classes.

Example 3.1. The variety of unipotent elements ounip in G(F ) constitutes one of the equiv-alence classes in O. Similarly, for any central element γ ∈ Z(F ), the geometric equivalenceclass generated by γ equals γounip.

Arthur shows that there exist distributions Jo : C∞c (G(AF )1) −→ C, o ∈ O, such that

Jgeom(f) =∑o∈O

Jo(f),

see [Art05, §10]. For a fixed compactly supported test function, all but finitely many Jo(f)vanish so that the coarse expansion is in fact a finite sum. More precisely, the distribution Johas support in ⋃

γ∈oAdG(AF ) · γ,

where AdG(AF) · γ is the G(AF )-conjugacy class of γ.Each of the distributions Jo has a finer expansion (cf. [Art05, §19]): Let o ∈ O and let S

be a sufficiently large set of places of F depending on f and o as explained in [Art86, §7]. Inparticular, S must contain the archimedean place of F , and it has to be so large that f canbe written as fS ⊗ 1KS with fS ∈ C∞c (G(FS)1) and 1KS ∈ C∞c (G(ASF )) the characteristicfunction of KS =

∏v 6∈S Kv. Then

(12) Jo(f) =∑M

|WM ||WG|

∑γ

aM (S, γ)JGM (γ, fS),

where

• M runs over all F -Levi subgroups of G containing the maximal torus T0 of diagonalmatrices,• WM denotes the Weyl group of the pair (T0,M),• γ runs over a (arbitrary) set of representatives for the M(F )-conjugacy classes inM(F ) ∩ o,• aM (S, γ) ∈ C are certain ”global” coefficient that are independent of f ,• JGM (fS , γ) are certain S-adic weighted orbital integrals, and• aM (S, γ) ∈ C and JGM (fS , γ) depend only on the M(F )-conjugacy class of γ.

Since there are only finitely many M(F )-conjugacy classes in M(F ) ∩ o, this fine expansionof Jo(f) is a finite sum. One should note that the sum over γ in general needs to be takenover a set of representatives for a certain equivalence relation on M(F ) ∩ o that depends on

DISTRIBUTION OF HECKE EIGENVALUES 11

S. It is a special feature of G = GL(n) that this equivalence relation reduces to conjugacyand thus is independent of S.

Using our family of test functions Fµ,τ in the geometric side of the trace formula andintegrating over µ ∈ tΩ (hence mirroring the integral (10) on the geometric side of the traceformula), we need to consider for each o ∈ O the sum-integral∑

M

|WM ||WG|

∑γ

aM (S, γ)

∫tΩJGM (fµ∞ · τS\∞, γ) dµ.

The pairs (M,γ) ∈ G × Z(F ) are exactly those which contribute to the main term: IfM = G and γ ∈ Z(F ), one has

aG(S, γ) = vol(G(F )\G(AF )1)

and ∫tΩJGM (fµ∞ · τS\∞, γ) dµ = τ(z)

∫tΩfµ∞(1) dµ.

Using Plancherel inversion, one can show that this last integral equals |W |−1∫tΩ c(λ)−2 dλ up

to a contribution to the error term (see [LM09]).Hence it remains to show that the rest of the geometric side only contributes to the error

term in (5). The remaining main steps in the proof of Theorem 2.1 in [Mat, MT] are thereforeas follows

(1) Find the (finitely many) classes o ∈ O for which Jo(Fµ,τ ) 6= 0, and keep track of how

they depend on τ .(2) Find a sufficiently large set of places S such that the fine expansion (12) holds for any

o from step (1). Keep track of the dependence of S on τ .(3) For any pair (M,γ) 6∈ G × Z(F ) with γ ∈ o ∩ M(F ) find an upper bound for

aM (S, γ) for any o from step (1) and S from step (2). Keep track of the dependenceon τ .

(4) For any pair (M,γ) 6∈ G × Z(F ) with γ ∈ o ∩M(F ) find an upper bound for theintegral |

∫tΩ J

GM (Fµ,τS , γ) dµ| for any o from step (1) and S from step (2). Keep track

of the dependence on τ .

We will not comment any further on steps (1) and (2) but explain the relevance and maindifficulties in the last two steps.

3.3. Global coefficients. The global coefficients aM (S, γ) are in general only understood insome special cases, although there has been some recent progress [CL, Cha]. If γ is semisimple,aM (S, γ) is independent of S and equals

aM (S, γ) = vol(Mγ(F )\Mγ(AF )1),

where Mγ(F ) is the centralizer of γ in M(F ) [Art86, Theorem 8.2]. If γ is not semisimple,exact expressions for aM (S, γ) are only known in a few low-rank examples [JL70, Fli82, HW].For GL(n) there exists at least an upper bound which is sufficiently good to prove the errorestimate in (5) [Mat15]: There exist a, b > 0 depending only on n and the degree of F overQ such that

aM (S, γ) ≤ aDbF

∑(sv)v∈S∈Z

|S|≥0:∑

v sv≤n−1

∏v∈S\S∞

∣∣∣∣∣ζ(sv)Fv

(1)

ζFv(1)

∣∣∣∣∣ ,

12 JASMIN MATZ

where ζFv(s) = (1 − q−sv )−1 denotes the local Dedekind zeta function, and ζ(sv)Fv

(s) its svth

derivative. For certain types of γ the upper bound for aM (γ, S) has recently been improvedin [Cha].

3.4. Erratum to [Mat15]. There is a mistake in the volume formula for G(F )\G(AF )1 asstated in [Mat15] which has some effect on the formulation of a conjecture in that paper.

In fact, in the normalization of measures in [Mat15] the adelic quotient G(F )\G(AF )1 hasvolume

vol(G(F )\G(AF )1) = Dn(n−1)

4F res

s=1ζF (s)

n∏k=2

ζF (k),

where ζF is the Dedekind zeta function of F , and if n = 1, the empty product is interpreted

as 1. This formula was incorrectly stated in [Mat15] where the factor Dn(n−1)

4F was missing on

the right hand side. This does not have any effect on the statement or proof of the resultsof [Mat15]. However, the statement of the first part of [Mat15, Conjecture 1.3] needs to bemodified by the obvious power of the discriminant of DF .

More precisely, the inequality (4) in [Mat15, Conjecture 1.3] should read

∣∣aM (V, S)∣∣ ≤ CDNM+κ

F

∑sv∈Z≥0,v∈Sfin:∑

sv≤η

∏v∈Sfin

∣∣∣∣ζ(sv)F,v (1)

ζF,v(1)

∣∣∣∣,see [Mat15] for the missing notation. Here the number NM is defined as follows: There is apartition (n1, . . . , nr) of n such that M is over F isomorphic to GL(n1)× . . .×GL(nr). Wethen define NM =

∑ri=1 ni(ni − 1)/4.

3.5. Weighted orbital integrals. To attack step (4), one first needs to better understandthe weighted orbital integrals. The first step is to reduce the S-adic integral JGM (γ, fS)to a linear combination of products of v-adic integrals for v ∈ S. This can be done byusing Arthur’s splitting formula for weighted orbital integrals [Art88, §9]. It reduces step (4)basically to two different problems, namely, to bound for every Levi L ⊇M

• the archimedean integral:

(13)

∣∣∣∣∫tΩJLM (γ, fµ∞) dµ

∣∣∣∣ ,• the non-archimedean integrals

∣∣JLM (γ, τv)∣∣ for v ∈ S\∞.

For the non-archimedean integrals, it was shown in [Mat] by using explicit computationson the Bruhat-Tits building as in [ST15, §7] combined with bounds for unweighted orbitalintegrals [ST15, §7, Appendix B] that there exist a, b, c > 0 depending only on n and theglobal field F such for any non-archimedean v, any κv ≥ 0, and any τv ∈ H≤κvv , |τv| ≤ 1, wehave ∣∣JLM (γ, τv)

∣∣ ≤ qa+bκvv ∆−v (γ)c

where

∆−v (γ) :=∏α

max1, |1− α(γ)|−1F (γ)

with α running over all positive roots of (T0, G), F (γ)/F the splitting field of γ, and γ ∈T0(F (γ)) a diagonal matrix having the same eigenvalues as γ (in F (γ)). Note that ∆−v (γ)

DISTRIBUTION OF HECKE EIGENVALUES 13

is well-defined since the entries of γ are unique up to permutation (i.e. γ is unique up toconjugation by Weyl group elements).

To estimate (13), we require a good pointwise upper bound for the elementary sphericalfunctions φλ. This task is significantly easier if F is imaginary quadratic than if F = Q.In the former case F∞ = C, the elementary spherical functions φλ for GLn(C) are well

understood and can be expressed as rational functions in e〈λ,H0(·)〉 and e〈ρ,H0(·)〉. In thelatter case, F∞ = R, the elementary spherical functions for GLn(R) can only be expressed asintegrals, but not as rational functions of elementary functions as in the complex case. It isnot easy to obtain a non-trivial estimate for these functions which is effective in the spectralparameter as well as the group parameter. Recently a sufficiently good upper bound for thesespherical functions was proven in [BP, MT]. There were several preceding upper bounds forspherical functions, cf. [DKV83, Mar], but they always required at least one of the variables(the spherical parameter or the group element) to stay in a bounded set and away from thesingular set.

4. Further directions

4.1. Improving the error term. As explained in Section 2.4, the effective dependence ofthe bound (5) on τ makes Theorem 2.1 applicable in proving certain conjectures about low-lying zeros of families of L-functions. It would be desirable to improve the bound (5) or atleast to control the constant b in terms of n as this would lead to a better understanding ofhow large the support of the Fourier transform of the test function Φ in (8) may be.

The main obstacles when trying to give an upper bound for b are bounding the non-archimedean weighted orbital integrals JLM (γ, τv), and bounding the global coefficients aM (γ, S).In principle, the upper bounds for both quantities can be at least made effective in n, butwith our types of proofs only very crude bounds would arise. Recent work [Cha] gives goodbounds for the global coefficients in some special cases.

4.2. General number fields. The purpose of this section is to formulate the analogue ofour main theorem over a general number field (see (14)), and to explain what points then needto be changed in the proof of the theorem. In particular, the construction of the archimedeantest function needs to be modified.

Suppose F is a number field of degree d = [F : Q] with r1 real and r2 complex places sothat d = r1 + 2r2. For each v|∞ let a∗v,0 = X(T0/Q)Q ⊗ R ' Rn where X(T0/Q)Q denotes

the group of rational characters T0 −→ GL(1) for T0 considered as a group over Q, andav,0 = HomR(a∗v,0,R). Similarly, let a∗0 = X(ResF/Q T0)Q ⊗ R and a0 = HomR(a∗0,R) with

X(ResF/Q T0)Q the group of Q-characters of T0 as a group over F . Then

a0 '⊕v|∞

av,0, and a∗0 '⊕v|∞

a∗v,0.

We define av,G, a∗v,G, aG, and a∗G similarly with G in place of T0. We let av, a∗v, a, and a∗ be

the spaces such that av,0 = av ⊕ av,G, a∗v,0 = a∗v ⊕ a∗v,G, and so on. Let

a∞ = X = (Xi)1≤i≤n(r1+r2) ∈ a0 |∑i

Xi = 0,

and

a∞,∗ = λ = (λi)1≤i≤n(r1+r2) ∈ a∗0 |∑i

λi = 0.

14 JASMIN MATZ

Further, leta∞G = aG ∩ a∞, and a∞,∗G = a∗G ∩ a∞,∗.

If π ∈ Πcusp(G(AF )1), the infinitesimal character λπ∞ is now an element in a∞,∗C . It has aunique decomposition

λπ∞ = λξπ∞ +∑v|∞

λπ′v ∈ a∞,∗G,C ⊕⊕v|∞

a∗v,C

where λξπ∞ corresponds to the central character ξπ∞ of π∞, π′∞ = ξ−1π∞π∞, and π′∞ =

∏v|∞ π

′v.

Let Ω ⊆ a∞,∗C be a nice bounded set. For simplicity we assume that Ω is of the form

Ω = ΩZ ⊕⊕v|∞

Ωv

for suitable nice bounded subsets ΩZ ⊆ a∞,∗G,C, and Ωv ⊆ a∗v,C.

For each v|∞ let fµvv , µv ∈ a∗v,C be constructed as before. Let fZ : Z(F∞)/R>0(Z(F∞) ∩K∞) −→ C be a compactly supported function with fZ(1) = 1, and let fZ denote its Fourier

transform on a∞,∗G,C. For µZ ∈ a∞,∗G,C let fµZZ be such that fµZZ (λZ) = fZ(λZ − µZ). We thendefine

fµ∞(g) = fµZZ (z)∏v|∞

fµvv (g′v)

for g ∈ G(F∞)1 with z its central component in Z(F∞) ∩G(F∞)1, g′ = z−1g =∏v|∞ g

′v, and

µ = µZ +∑

v|∞ µv ∈ a∞,∗G,C ⊕⊕

v|∞ a∗v,C = a∞,∗C .

This choice of test function allows us to essentially reduce the analysis of the trace formulato the previously considered cases for F = Q or F imaginary quadratic. In particular, theintegral over the cuspidal part of the spectral side of the trace formula for the test functionfµ∞ · τ ∫

tΩJcusp(fµ∞ · τ) dµ

should equal, up to an error term, ∑π∈F(t)

trπf (τ),

where F = π ∈ Πcusp(G(AF )1) | πK 6= 0 and F(t) = π ∈ F | λπ∞ ∈ tΩ as before. Onthe other hand, if z ∈ Z(F ), then it should follow similarly as in the other cases that up to anegligible error term we have∑

z∈Z(F )

∫tΩfµ∞ · τ(z) dµ = Λ(t)

∑z∈Z(F1)

τ(z)

with

Λ(t) = vol(G(F )\G(AF )1/Kf )∏v|∞

|W |−1

∫tΩv

cv(λ)−2 dλ

where cv denotes the Harish-Chandra c-function for G(Fv)1, and F1 the set of all elements

in F× which lie in the kernel of the composite map

F× −→ aG −→ a∞G .

Here the first map is given by x 7→ (log |xv|v)v|∞, and the second map is the orthogonalprojection onto a∞G . That z ∈ Z(F )\Z(F1) only contribute to the error term can be seen by

DISTRIBUTION OF HECKE EIGENVALUES 15

Fourier inversion and integration by parts. The remaining parts of the trace formula againshould only contribute to the error term.

Hence the final statement is expected to be

(14) limt→∞

Λ(t)−1∑

π∈F(t)

trπS0(τS0) =∑

z∈Z(F1)

τ(z).

4.3. General level. In the previous section we only considered the family F of everywhereunramified cuspidal representations. If Kf ⊆ Kf is a finite index subgroup, one can moregenerally consider the family

FK = π ∈ Πcusp(G(AF )1) | πK 6= 0

of cuspidal representations having a K-fixed vector for K := K∞ ·Kf . We can accordinglyput FK(t) = π ∈ FK | λπ∞ ∈ tΩ. For F = Q and Kf contained in Kf (N) for some N ≥ 3,the Weyl law was proven in [LM09] as explained above. However, the dependence of theestimate of the error term on Kf was left unspecified in [LM09]. It might be interesting tomake this dependence explicit as this might also allow to study families of representationswith varying level.

Remark 4.1. The method of using the trace formula to prove the Weyl law has the disad-vantage that one counts the representations in FK(t) with a certain weight factor, namely thedimension dimπK of the K-fixed space of π. If K = K is the maximal compact subgroup,then by multiplicity-one one has dimπK = 1 for every π ∈ F(t) = FK(t) so that in this caseone indeed counts the number of representations in FK(t). It would be interesting to seewhether one can count the number of π ∈ FK(t) of conductor K, or at least the number ofnewforms over π ∈ FK(t).

In [Mat] the upper bound in the error term was made effective in Kf if the ground fieldF is imaginary quadratic (the same can be done for F = Q). More precisely, we prove thefollowing in [Mat]: Let Kf ⊆ Kf be a finite index subgroup and put K = K∞ · Kf . TheWeyl law then becomes (cf. (2) from [LM09] for F = Q)

(15) ΛK(t) :=∑

π∈FK(t)

dimπK ∼ |Z(F ) ∩Kf |vol(G(F )\G(AF )1/K)

|W |

∫tΩ

c(λ)−2 dλ

as t → ∞. (Recall that c denotes the Harish-Chandra c-function on G(F∞), that is, here itis the c-function for GLn(C).) Moreover, there exist constants a, b, c, δ > 0 depending onlyon n, F , and Ω such that the following holds: Let Ξ ⊆ G(AF,f ) be an open compact subsetwhich is bi-Kf -invariant (that is, k1Ξk2 = Ξ for all k1, k2 ∈ Kf ), and let τΞ ∈ C∞c (G(AF,f ))be the characteristic function of Ξ normalized by vol(Kf )−1. Then

limt→∞

ΛK(t)−1∑

π∈FK(t)

trπf (τΞ) =∑

z∈Z(F )/Z(F )∩Kf

τΞ(z) = |(Z(F ) ∩ Ξ)/(Z(F ) ∩Kf )| ,

and

(16)

∣∣∣∣∣∣ΛK(t)−1∑

π∈FK(t)

trπf (τΞ)− |(Z(F ) ∩ Ξ)/(Z(F ) ∩Kf )|

∣∣∣∣∣∣ ≤ a[K : K]b vol(Ξ)ct−δ

for every t ≥ 1.

16 JASMIN MATZ

Remark 4.2. (i) Taking Ξ = Kf , (16) also provides an upper bound for the error termin (15).

(ii) If Kf = Kf , the upper bound for the remainder term in (16) is the same a inTheorem 2.1: In the situation of Theorem 2.1 we may assume that τS0 =

∏v∈S0

τvwith τv ∈ H≤κvv the characteristic function of Ξv := Kv$

λvv Kv for suitable λv with

‖λv‖ ≤ κv. But then the volume of Ξ =∏v<∞ Ξv (which equals the degree of the

Hecke operator τ) is ≤ Πaκ for some a > 0 depending only on n and F .

4.4. General K∞-type. So far we only considered representations with trivial K∞-type,that is, π such that πK∞∞ 6= 0. Suppose σ is an irreducible unitary representation of K∞with representation space Vσ. One can consider π ∈ Πcusp(G(AF )1) which have K∞-typeσ, that is, for which σ occurs in the decomposition of the restriction of π∞ to K∞ intoirreducibles. For F = Q and Kf ⊆ Kf a finite index subgroup, the main term of the Weyllaw for representations with Kf -fixed vector and K∞-type σ (with σ(−1) = id if −1 ∈ Kf )was proven in [Mul07]. More precisely, taking Ω = B1(0) the unit ball in ia∗ [Mul07] provesthat as t→∞∑

π∈FK(t)

dimπKff dim (Hπ∞ ⊗ Vσ)K∞ ∼

δKf dimσ

vol(Kf )

vol(G(Q)\G(AQ)1)

(4π)d/2Γ(d/2 + 1)td

where Hπ∞ denotes the representation space of π∞, and δKf equals 1 or 2 depending onwhether −1 6∈ Kf or −1 ∈ Kf .

The method of proof of [Mul07] is not applicable if one wants to obtain a bound on the errorterm. It might, however, be possible to modify the proof of [LM09, Mat, MT] to incorporatemore general K∞-types. Already in [MT] we use a particular non-trivial K∞-type to obtainodd Maass forms. In general, however, one major obstacle in carrying this approach over toarbitrary σ is that the inversion formula (11) for the spherical Harish-Chandra transform isin general not valid. For certain K∞-types it still holds (cf. [HS94, Chapter I, §5]), but ingeneral one needs to take into account the residues arising in the proof of the Paley-Wienertheorem when changing the contour of certain integrals [Del82, Art83, Shi94].

Suppose for simplicity that σ is one-dimensional, and consider for λ ∈ ia∗C the elementaryσ-spherical function

Φσ,λ(g) =

∫K∞

e〈λ+ρ,H0(kg)〉σ(k−1κ(kg)) dk, g ∈ G(F∞),

where κ(kg) denotes the K∞-component of kg in its Iwasawa decomposition kg = tuk1 ∈T0(R)U0(R)K∞. Then Φσ,λ(g) ∈ EndVσ, and Φσ,λ satisfies the invariance properties

Φσ,λ(k1gk2) = σ(k1k2)Φσ,λ(g)

for all k1, k2 ∈ K∞, g ∈ G(F∞). The Harish-Chandra transform gives a map

f 7→ H(f)(λ) :=

∫G(F∞)1

f(g)Φσ−1,λ(g) dg, λ ∈ a∗C

for f ∈ C∞c (G(F∞)1, σ), the space of all f ∈ C∞c (G(F∞)1) satisfying f(k1gk2) = σ(k1k2)f(g)for all k1, k2 ∈ K∞ and g ∈ G(F∞). The resulting function is a holomorphic function on a∗C.However, the inversion formula (11) for H is only valid for certain σ.

DISTRIBUTION OF HECKE EIGENVALUES 17

References

[Art78] James Arthur, A trace formula for reductive groups. I. Terms associated to classes in G(Q), DukeMath. J. 45 (1978), no. 4, 911–952. MR 518111 (80d:10043)

[Art83] , A Paley-Wiener theorem for real reductive groups, Acta Mathematica 150 (1983), no. 1,1–89.

[Art86] , On a family of distributions obtained from orbits, Canad. J. Math. 38 (1986), no. 1, 179–214.MR 835041 (87k:11058)

[Art88] , The invariant trace formula. I. Local theory, J. Amer. Math. Soc. 1 (1988), no. 2, 323–383.MR 928262 (89e:22029)

[Art05] , An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimuravarieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 1–263. MR 2192011(2007d:11058)

[BP] Valentin Blomer and Anke Pohl, The sup-norm problem for the Siegel modular space of rank two,arXiv:1402.4635v2.

[CDF97] J. B. Conrey, W. Duke, and D. W. Farmer, The distribution of the eigenvalues of Hecke operators,Acta Arith. 78 (1997), no. 4, 405–409. MR 1438595 (98k:11047)

[Cha] Pierre-Henri Chaudouard, Sur la contribution unipotente dans la formule des traces d’Arthur pourles groupes generaux lineaires, arXiv:1411.3005.

[CL] Pierre-Henri Chaudouard and Gerard Laumon, Un theoreme du support pour la fibration de Hitchin,arXiv:1407.3562.

[Del82] Patrice Delorme, Theoreme de type Paley-Wiener pour les groupes de Lie semi-simples reels avecune seule classe de conjugaison de sous groupes de Cartan, J. Funct. Anal. 47 (1982), no. 1, 26–63.

[DKV79] J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Spectra of compact locally symmetricmanifolds of negative curvature, Invent. Math. 52 (1979), no. 1, 27–93. MR 532745 (82a:58050a)

[DKV83] , Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in realsemisimple Lie groups, Compositio Math. 49 (1983), no. 3, 309–398. MR 707179 (85e:58150)

[Fli82] Yuval Z. Flicker, The trace formula and base change for GL(3), Lecture Notes in Mathematics, vol.927, Springer-Verlag, Berlin, 1982. MR 663002 (84d:10035)

[FLM15] Tobias Finis, Erez Lapid, and Werner Muller, Limit multiplicities for principal congruence subgroupsof GL(n) and SL(n), J. Inst. Math. Jussieu 14 (2015), no. 03, 589–638.

[Gol06] Dorian Goldfeld, Automorphic forms and L-functions for the group GL(n,R), Cambridge Studies inAdvanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006.

[HS94] Gerrit Heckman and Henrik Schlichtkrull, Harmonic analysis and special functions on symmetricspaces, Perspectives in Mathematics, vol. 16, Academic Press, Inc., San Diego, CA, 1994.

[HW] Werner Hoffmann and Satoshi Wakatsuki, On the geometric side of the arthur trace formula for thesymplectic group of rank 2, arXiv:1310.0541.

[ILS00] Henryk Iwaniec, Wenzhi Luo, and Peter Sarnak, Low lying zeros of families of L-functions, Inst.

Hautes Etudes Sci. Publ. Math. (2000), no. 91, 55–131 (2001). MR 1828743 (2002h:11081)

[IR10] Ozlem Imamoglu and Nicole Raulf, On the behaviour of eigenvalues of Hecke operators, Math. Res.Lett. 17 (2010), no. 1, 51–67. MR 2592727 (2011e:11094)

[JL70] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol.114, Springer-Verlag, Berlin, 1970. MR 0401654 (53 #5481)

[KS99] Nicholas Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, vol. 45,American Mathematical Soc., 1999.

[Lan76] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes inMathematics, Vol. 544, Springer-Verlag, Berlin, 1976. MR 0579181 (58 #28319)

[LM09] Erez Lapid and Werner Muller, Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n),Duke Math. J. 149 (2009), no. 1, 117–155.

[LRS99] Wenzhi Luo, Zeev Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for GL(n),Proceedings of Symposia in Pure Mathematics, vol. 66, Providence, RI; American MathematicalSociety; 1998, 1999, pp. 301–310.

[LV07] Elon Lindenstrauss and Akshay Venkatesh, Existence and Weyl’s law for spherical cusp forms, Geom.Funct. Anal. 17 (2007), no. 1, 220–251.

18 JASMIN MATZ

[Mar] Simon Marshall, Lp norms of higher rank eigenfunctions and bounds for spherical functions, J. Eur.Math. Soc., to appear.

[Mat] Jasmin Matz, Weyl’s law for Hecke operators on GL(n) over imaginary quadratic number fields,arXiv:1310.6525.

[Mat15] , Bounds for global coefficients in the fine geometric expansion of Arthurs trace formula forGL(n), Israel J. of Math. 205 (2015), no. 1, 337–396.

[Mil01] Stephen D. Miller, On the existence and temperedness of cusp forms for SL3(Z), J. Reine Angew.Math. 533 (2001), 127–169.

[MT] Jasmin Matz and Nicolas Templier, Sato-Tate equidistribution for the family of Hecke-Maass formson SL(n,Z), arXiv:1505.07285.

[Mul07] Werner Muller, Weyl’s law for the cuspidal spectrum of SLn, Ann. of Math. (2) 165 (2007), no. 1,275–333.

[Mul16] , Asymptotics of automorphic spectra and the trace formula, these proceedings.[MW95] C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in

Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995, Une paraphrase de l’Ecriture[A paraphrase of Scripture]. MR 1361168 (97d:11083)

[Sar87] Peter Sarnak, Statistical properties of eigenvalues of the Hecke operators, Analytic number theoryand Diophantine problems (Stillwater, OK, 1984), vol. 70, Birkhauser Boston, Boston, MA, 1987,pp. 321–331.

[Sau97] Francois Sauvageot, Principe de densite pour les groupes reductifs, Compositio Mathematica 108(1997), no. 02, 151–184.

[Sel56] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaceswith applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.

[Ser97] Jean-Pierre Serre, Repartition asymptotique des valeurs propres de l’operateur de Hecke Tp, J. Amer.Math. Soc. 10 (1997), no. 1, 75–102. MR 1396897 (97h:11048)

[Shi94] Nobukazu Shimeno, The Plancherel formula for spherical functions with a one-dimensional K-typeon a simply connected simple Lie group of Hermitian type, J. Funct. Anal. 121 (1994), no. 2, 330–388.

[Shi12] Sug Woo Shin, Automorphic Plancherel density theorem, Israel J. of Math. 192 (2012), no. 1, 83–120.[SST16] Peter Sarnak, Sug-Woo Shin, and Nicolas Templier, Families of L-functions and their Symmetry,

these proceedings.[ST15] Sug Woo Shin and Nicolas Templier, Sato–Tate theorem for families and low-lying zeros of automor-

phic L-functions, Invent. Math. (2015), 1–177.

Universitat Leipzig, Mathematisches Institut, Postfach 100920, 04009 Leipzig, GermanyE-mail address: matz@math.uni-leipzig.de