Weak Weyl’s law for congruence subgroups

Werner Müller

Mathematisches Institut Universität Bonn

Joint work with Jean-Pierre Labesse

Montreal, May 5, 2004

1. The problem

• G connected semisimple algebraic group over ℚ

• Γ ⊂ G(ℚ) arithmetic subgroup

• K_{∞} ⊂ G(ℝ) maximal compact subgroup ϕ ∈ L^{2}(Γ\G(ℝ)) is called cusp form ⇔

1) Dϕ = λ(D)ϕ, D ∈ Z(𝔤_{ℂ});

2) ϕ is K_{∞}-finite; 3)

∫_{N_{P} (ℝ)∩Γ\N_{P} (ℝ)} ϕ(ng) dn = 0

for all proper rational parabolic subgroups P ⊂ G, N_{P} the unipotent radical of P. Problem: Existence and construction of cusp forms • A convenient way to count cusp forms is to count the Casimir eigenvalues of cusp forms containing a fixed K_{∞}-type.

• L_{cusp}^{2} (Γ\G(ℝ)) ⊂ L^{2}(Γ\G(ℝ)) space of cusp forms • σ : K_{∞} → GL(V_{σ}) irreducible unitary representation.

H_{cusp}^{Γ} (σ) := (L_{cusp}^{2} (Γ\G(ℝ)) ⊗ V_{σ}) ^{K_{∞}}

Space of Γ-cusp forms of ”weight” σ.

• Ω ∈ Z(𝔤_{ℂ}) Casimir element.

• Δ_{σ} selfadjoint operator in H_{cusp}^{Γ} (σ) induced by −ρ_{∞}(Ω) ⊗ Id, ρ_{∞} regular representation of G(ℝ). Geometric interpretation: Assume that Γ is torsion free. Let

X = G(ℝ)/K_{∞}

be the Riemannian symmetric space and let Ẽ_{σ} → X be the homogeneous vector bundle attached to σ. Set

E_{σ} = Γ\Ẽ_{σ} → Γ\X.

Then

(L^{2}(Γ\G(ℝ)) ⊗ V_{σ}) ^{K_{∞}} ≅ L^{2}(Γ\X, E_{σ})

and

Δ_{σ} = (∇^{σ})^{∗} ∇^{σ} − λ_{σ}Id,

where ∇^{σ} is the canonical invariant connection of Ẽ_{σ} and λ_{σ} the Casimir eigenvalue of σ.

• Δ_{σ} has pure point spectrum in H_{cusp}^{Γ} (σ):

λ_{0} ≤ λ_{1} ≤ λ_{2} ≤ ⋯ → ∞.

• cuspidal spectrum of ”weight” σ. Counting function:

N_{cusp}^{Γ} (λ, σ) := # {i: |λ_{i}| ≤ λ} .

Let

X = G(ℝ)/K_{∞}  and d = dim X.

Weyl’s constant:

C_{Γ} := \frac{Vol(Γ\X)}{(4π)^{d/2}Γ(d/2 + 1)} .

Conjecture(Sarnak, 1984):

N_{cusp}^{Γ} (λ, σ) ∼ dim (σ)C_{Γ} λ^{d/2}

as λ → ∞.

2. Results

1. Special cases The conjecture has been proved in the following cases:

• A. Selberg, 1954: Γ ⊂ SL (2, ℝ) congruence subgroup, σ = 1.

• I. Efrat, 1987: Γ ⊂ SL (2, ℝ)^{n} Hilbert modular group, σ = 1.

• A. Reznikov, 1993: Γ ⊂ SO _{0}(n, 1) congruence subgroup, σ = 1.

• St. Miller, 2001: Γ = SL (3, ℤ), σ = 1.

• M., 2003: Γ ⊂ SL (n, ℤ) principal congruence subgroup, σ arbitrary.

2. General results

Theorem(Donnelly). G semisimple algebraic group over ℚ and Γ ⊂ G(ℝ) lattice. Then

lim sup_{λ→∞} \frac{N_{cusp}^{Γ} (λ, σ)}{λ^{d/2}} ≤ dim (σ)C_{Γ}.

Theorem(Piatetski-Shapiro). Let σ = 1. For every Γ there exists a normal subgroup of finite index Γ′ of Γ such that

lim_{λ→∞} N_{cusp}^{Γ′} (λ, 1) = ∞.

• A.B. Venkov, G = SL (2).

• Let S be a finite set of primes containing at least two finite primes. There exists C_{Γ}(S) ≤ 1 with 0 < C_{Γ}(S) for Γ a deep enough congruence subgroup.

Theorem(Labesse-M.). Let G be almost simple, connected and simply connected such that G(ℝ) is non compact. For every congruence subgroup Γ ⊂ G(ℝ) and every σ such that σ|_{Z_{Γ}} =Id we have

dim (σ)C_{Γ}C_{Γ}(S) ≤ lim inf_{λ→∞} \frac{N_{cusp}^{Γ} (λ, σ)}{λ^{d/2}} .

3. Methods

The method is a combination of the Arthur trace formula and the heat equation method.

a) G = SL (2).

• Selberg’s method

• Γ ⊂ SL (2, ℝ) discrete, co-finite area, σ = 1.

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) .

Δ: C_{c}^{∞}(Γ\H) → L^{2}(Γ\H)

essentially self-adjoint.

Spec_{pp} (Δ) : 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯

_{i}^{∑} e^{−tλ_{i}} = Tr (e^{−tΔ} ↾ L_{disc}^{2} (Γ\H)

• Use Selberg trace formula to compute the trace • C(s) = (C_{ij}(s)) scattering matrix

ϕ(s) := det C(s).

Selberg trace formula applied to the heat kernel gives

_{j}^{∑} e^{−tλ_{j}} − \frac{1}{4π} ∫_{ℝ} \frac{ϕ′}{ϕ}(1/2 + ir)e^{−(1/4+r^{2})t} dr

= \frac{Area (Γ\H)}{4π} t^{−1} + O (\frac{log t}{√t})

as t → 0+.

• Karamata’s Theorem implies Weyl’s law

N_{disc}^{Γ} (λ) − \frac{1}{4π} ∫_{−√λ}^{√λ} \frac{ϕ′}{ϕ}(1/2 + ir) dr ∼ \frac{Area (Γ\H)}{4π} λ

as λ → ∞.

N_{disc}^{Γ} (λ) = N_{cusp}^{Γ} (λ) + N_{res}^{Γ} (λ), N_{res}^{Γ} (λ) ≤ C.

• Γ = SL (2, ℤ):

ϕ(s) = √π\frac{Γ(s − 1/2)ζ(2s − 1)}{Γ(s)ζ(2s)} .

|\frac{ζ′(1 + ir)}{ζ(1 + ir)} | ≤ C log (|r|)^{6}, |r| ≥ 2.

This implies

N_{cusp}^{Γ} (λ) ∼ \frac{Area (Γ\H)}{4π} λ, λ → ∞.

• similar for Γ = Γ(N).

b) G = SL (n)

• Selberg trace formula is replaced by (noninvariant) Arthur trace formula

_{χ∈𝔛}^{∑} J_{χ}(f) = _{𝔬∈O}^{∑} J_{𝔬}(f), f ∈ C_{c}^{∞}(G(𝔸)^{1}). The following facts are needed to prove Weyl’s law:

• Weak version of Ramanujan conjecture (Luo, Rudnick, Sarnak)

• Analytic properties of Rankin-Selberg L-functions (Jacquet, Piatetski-Shapiro, Shalika, Shahidi, Mœglin, Waldspurger,...), bounds on the logarithmic derivatives.

A weaker result suffices: Let π_{i}, i = 1, 2, be a cuspidal automorphic representation of GL_{n_{i}}(𝔸). Set

Λ(s, π_{1}, π_{2}) = \frac{L(s, π_{1} × π̃_{2})}{L(1 + s, π_{1} × π̃_{2})ϵ(s, π_{1} × π̃_{2})} .

Then

∫_{−T}^{T} |\frac{Λ′}{Λ} (ir, π_{1}, π_{2})| dr ≤ C T log (T + ν(π_{1} × π̃_{2}))

for T > 0, where ν(π_{1} × π̃_{2}) is the analytic conductor.

• Description of the residual spectrum (Mœglin, Waldspurger).

• At present it seems to be out of reach to extend these results to other groups. c) Weaker result Define

T_{−1} : L^{2}(Γ\H) → L^{2}(Γ\H)

by

T_{−1}(z) = −\overline{z}.

Then T_{−1}^{2} = Id and

L^{2}(Γ\H) = L_{+}^{2} (Γ\H) ⊕ L_{−}^{2}(Γ\H)

• decomposition in even and odd functions.

• T_{−1}E(z, s) = E(z, s).

• Δ has pure point spectrum in L_{−}^{2}(Γ\H). Correponds to Dirichlet problem on one half F_{+} of the fundamental domain. One can use the same methods as in the case of compact surfaces.

N_{−}^{Γ}(λ) ∼ \frac{Area (F_{+} )}{4π} λ, λ → ∞.

b) General case.

• G connected and simply connected algebraic group over ℚ.

• Use simple trace formula which avoids continuous spectrum.

• Adèlic framework: G(𝔸) = ∏′_{p≤∞} G(ℚ_{p})

K_{fin} = _{p<∞}^{∏} K_{p}, K_{p} ⊂ G(ℚ_{p})

K_{fin} ⊂ G(𝔸_{fin}) decomposable open compact subgroup,

Γ = K_{fin} ∩ G(ℚ).

Strong approximation:

G(ℚ)\G(𝔸)/K_{fin} ≅ Γ\G(ℝ).

• S finite set of primes, |S| ≥ 2.

L_{cusp}^{2} (G(ℚ)\G(𝔸), S) ⊂ L_{cusp}^{2} (G(ℚ)\G(𝔸)) spanned by cusp form orthogonal to 1 and on which

G_{S} = _{p∈S}^{∏} G(ℚ_{p})

acts by the Steinberg representation. Put

H_{cusp}^{Γ,S} := L_{cusp}^{2} (G(ℚ)\G(𝔸), S)^{K_{fin}}.

Let K = K_{∞}K_{fin}. Set H_{cusp}^{Γ} (σ, S) := (L_{cusp}^{2} (G(ℚ)\G(𝔸), S) ⊗ V_{σ}) ^{K} .

• Λ_{cusp} (σ, S) spectrum of −ρ_{∞}(Ω) ⊗ Id in H_{cusp}^{Γ} (σ, S).

• h_{t} ∈ C^{∞}(G(ℝ)) kernel of e^{−tΔ_{σ}} .

• ρ_{∞} regular representation of G(ℝ) in H_{cusp}^{Γ} .

Tr (ρ_{∞}(h_{t}) ↾ H_{cusp}^{Γ,S} ) = _{λ∈Λ_{cusp} (σ,S)}^{∑} m(λ)e^{−tλ}.

• Apply simple version of Arthur’s trace formula to compute the trace and to determina the asymptotic bahaviour as t → 0+.

3. Simple trace formula

a) Adèlic version Let

e_{fin} = \frac{1}{Vol(K_{fin})} χ_{K_{fin}}.

For f_{∞} ∈ C_{c}^{∞}(G(ℝ)):

Tr (ρ_{∞}(f_{∞}) ↾ H_{cusp}^{Γ,S} )

= Tr (ρ_{∞}(f_{∞} ⊗ e_{fin}) ↾ L_{cusp}^{2} (G(ℚ)\G(𝔸), S)) .

• ρ regular representation of G(𝔸).

b) Steinberg representation

• G_{p} = G(ℚ_{p}), P_{p} ⊂ G_{p} minimal parabolic. π_{St} ⊂ Ind_{P_{p}}^{G_{p}}(1)

unique irreducible subrepresentation.

• f_{p} ∈ C_{c}^{∞}(G_{p}) pseudo-coefficient of Steinberg representation, if

Tr (π_{p}(f_{p})) = 0, unless π_{p} = \begin{cases}π_{St}; \\ 1. \end{cases}

Tr (π_{St}(f_{p})) = 1, Tr (1_{p}(f_{p})) = (−1)^{q} for some integer q. Existence of pseudo-coefficients: Kazdan, Kottwitz, Euler-Poincaré functions

• Kottwitz: O_{γ}(f_{p}) = 0, for γ ∈ G(ℚ) non elliptic.

• π_{S} Steinberg representation of G_{S}. Set

C(K_{S}) = dim (H_{π_{S}}^{K_{S}}) , e_{S} = \frac{1}{Vol(K_{S})} χ_{K_{S}}.

Tr π_{S}(e_{S}) = C(K_{S})Tr π_{S}(f_{S})

• Replace e_{S} by f_{S}.

c) Assume: S contains two different finite primes.

Set

f = f_{∞} ⊗ f_{S} ⊗ e_{fin,S}.

• Then f is cuspidal at two places in the sense of Arthur.

Then Arthur’s trace formula

_{χ∈𝔛}^{∑} I_{χ}(f) = _{𝔬∈O}^{∑} I_{𝔬}(f)

is reduced to

\begin{aligned}Tr (ρ(f) ↾ L_{cusp}^{2} & (G(ℚ)\G(𝔸), S)) + 1(f) \\ & = _{γ∈𝔊_{e}}^{∑} a(γ)O_{γ}(f). \end{aligned}

• 𝔊_{e} set of representatives of conjugacy classes of semisimple elliptic elements in G(ℚ).

O_{γ}(f) = ∫_{G(𝔸)_{γ}\G(𝔸)} f(g^{−1}γg) dg.

Final formula

\begin{aligned}Tr (ρ_{∞} & (f_{∞}) ↾ H_{cusp}^{Γ,S} ) \\ & = C(K_{S}) (_{γ∈𝔊_{e}}^{∑} a(γ)O_{γ}(f) − 1(f)) . \end{aligned}

• Advantage: avoids all difficulties due to the continous spectrum.

• Disadvantage: C(K_{S}) ≠ 0, only if Γ is a deep enough congruence subgroup.

4) Application to the heat kernel

a) The heat kernel

• σ : K_{∞} → GL(V_{σ}) irreducible unitary representation.

• E_{σ} → X = G(ℝ)/K_{∞} associated homogeneous vector bundle.

Δ_{σ}: C^{∞}(X, E_{σ}) → C^{∞}(X, E_{σ})

elliptic differential operator induced by −Ω ⊗ Id.

Δ_{σ}: C_{c}^{∞}(X, E_{σ}) → L^{2}(X, E_{σ})

essentially self-adjoint.

• e^{−tΔ_{σ}} smoothing operator.

(e^{−tΔ_{σ}} φ) (g) = ∫_{G(ℝ)} H_{t}(g^{−1}g_{1})φ(g_{1}) dg_{1},

• H_{t} ∈ (C^{1}(G(ℝ)) ⊗ End (V_{σ})) ^{K_{∞}×K_{∞}}. Set

h_{t}(g) := tr H_{t}(g), g ∈ G(ℝ), t ≥ 0.

• π irreducible unitary representation of G(ℝ)

Tr π(h_{t}) = e^{−tπ(Ω)} dim Hom _{K_{∞}}(H(π), V_{σ}^{∗}). b) Modified heat kernel

• h_{t} ∉ C_{c}^{∞}(G(ℝ)) needs modification. Let

φ ∈ C_{c}^{∞}(ℝ), φ(u) = \begin{cases}1, & |u| < 1/2; \\ 0, & |u| > 1. \end{cases}

Let d(x, y) be the geodesic distance of x, y ∈ X. Set

φ_{t}(g) := φ (d^{2}(gx_{0}, x_{0})/√t) , g ∈ G(ℝ), t > 0. The modified heat kernel is defined by

h̃_{t}(g) := φ_{t}(g)h_{t}(g), g ∈ G(ℝ), t > 0.

Proposition.

|Tr (ρ_{∞}(h_{t})) − Tr (ρ_{∞}(h̃_{t})) | ≤ Ce^{−c/√t}

for 0 < t ≤ 1.

Set

f_{t} = h̃_{t} ⊗ e_{fin,S} ⊗ f_{S},

where f_{S} is a pseudo-coefficient of the Steinberg representation.

The simple trace formula combined with the proposition yields

\begin{aligned}Tr (ρ_{∞}(h_{t}) ↾ H_{cusp}^{Γ,S} ) & = C(K_{S}) _{γ∈𝔊_{e}}^{∑} a(γ)O_{γ}(f_{t}) \\ & + O(1) \end{aligned}

as t → 0+.

c) Geometric side

We have

O_{γ}(f_{t}) = O_{γ}(h̃_{t})O_{γ}(e_{fin,S})O_{γ}(f_{S}).

where

O_{γ}(h̃_{t}) = ∫_{G(ℝ)_{γ}\G(ℝ)} h̃_{t}(g_{∞}^{−1}γg_{∞}) dg_{∞}.

• supp h̃_{t} → K_{∞} as t → 0+. For h ∈ G(ℝ) let

C_{h} = {ghg^{−1} | g ∈ G(ℝ)}.

Let γ ∉ Z_{G}(ℝ). Then

C_{γ} ∩ K_{∞} ⊂ C_{γ}

is a proper submanifold. By dominated convergence:

t^{d/2}O_{γ}(h̃_{t}) ⟶ 0, t → 0.

Assume: z ∈ Z_{G}(ℝ) . Then

O_{γ}(h̃_{t}) = h_{t}(z)

and

t^{d/2}h_{t}(z) ⟶ \frac{trσ(z)}{(4π)^{d/2}vol(K_{∞})} , t → 0.

Assume: σ ↾ Z_{Γ} = Id. Set d_{S} = f_{S}(z) and

C_{S}(Γ) = C(K_{S})d_{S}Vol(K_{S}).

Then

lim_{t→0} t^{d/2}Tr (ρ_{∞}(h_{t}) ↾ H^{Γ,S} )

= C_{S}(Γ)\frac{dim (σ)Vol(Γ\X)}{(4π)^{d/2}} .

Let λ_{π} = π(Ω) be the Casimir eigenvalue of π. Then Tr (ρ_{∞}(h_{t}) ↾ H_{cusp}^{Γ,S} ) = _{π∈Π(G(ℝ))}^{∑} e^{tλ_{π}}m_{Γ}(π, S) dim Hom _{K_{∞}}(H(π), V_{σ}^{∗}).

Set

N_{cusp}^{Γ} (T, σ, S)
= _{|λ_{π}|≤T}^{∑} m_{Γ}(π, S) dim Hom _{K_{∞}}(H(π), V_{σ}^{∗}).

Then Karamata’s theorem implies

lim_{T →∞} \frac{N_{cusp}^{Γ} (T, σ, S)}{T^{d/2}} = dim (σ)C_{S}(Γ)C_{Γ}.

• C_{S}(Γ) ≠ 0 if and only if C(K_{S}) ≠ 0.

• C(K_{S}) ≠ 0, if the Steinberg representation contains a non zero K_{S}-invariant vector. This is the case if K_{p} ⊂ I_{P}, a minimal parahoric subgroup for p ∈ S.