ON THE SPACE OF CUSP FORMS FOR GL(n)

WERNER MÜLLER

(joint work with Birgit Speh)

Let 𝔸 be the ring of adèles of ℚ and let G = GL(n). Let Z = Z_{G}(ℝ)^{0}. Let R be the regular representation of G(𝔸) in L^{2}(ZG(ℚ)\G(𝔸)), defined by

(R(g)f)(g′) = f(g′g).

This is a unitary representation and one of the fundamental problems in the theory of automorphic forms is to study the decomposition of R into irreducible unitary representations of G(𝔸). Note that every irreducible representation of G(𝔸) is a rstricted tensor product

π = ⊗_{v≤∞}π_{v} = π_{∞} ⊗ ⊗_{p}π_{p}

of irresucible representations of the local groups GL(n, ℚ_{v}). By Langlands’ theory of Eisenstein series there exists a decomposition

L^{2}(ZG(ℚ)\G(𝔸)) = L_{d}^{2}(ZG(ℚ)\G(𝔸)) ⊕ L_{c}^{2}(ZG(ℚ)\G(𝔸)),

into invariant subspaces, where L_{d}^{2}(ZG(ℚ)\G(𝔸)) is the smallest closed subspace containing all irresucible subrepresentations of R. Its orthogonal complement L_{c}^{2}(ZG(ℚ)\G(𝔸)) can be described in terms of Eisenstein series. Let

L2_{cusp}(ZG(ℚ)\G(𝔸)) ⊂ L_{d}^{2}(ZG(ℚ)\G(𝔸))

be the space of cusp forms. Then

L_{d}^{2}(ZG(ℚ)\G(𝔸)) = L2_{cusp}(ZG(ℚ)\G(𝔸)) ⊕ L2_{res}(ZG(ℚ)\G(𝔸)).

The residual subspace has been completely determined by Moeglin and Waldspurger. Let

L2_{cusp}(ZG(ℚ)\G(𝔸)) = ⨁_{π∈Ĝ(𝔸)}m(π)H_{π}.

Then the main issue is to gain information about the multiplicities. The main toll to study cusp forms is the Arthur-Selberg trace formula. Let G(𝔸)^{1} be the intersection of the kernels of the maps x ↦ |χ(x)|, x ∈ G(𝔸), where χ ranges over the group X(G)_{ℚ} of characters of G defined over ℚ. Then the (noninvariant) trace formula of Arthur is an identidy

Date: June, 2001.

∑_{𝔬∈𝔒}J_{𝔬}(f) = ∑_{χ∈𝔛}J_{χ}(f), f ∈ C_{0}^{∞}(G(𝔸)^{1}),

between distributions on G(𝔸)^{1}. The left hand side is the geometric side and the right hand side the spectral side of the trace formula. The distributions J_{𝔬} are parametrized by semisimple conjugacy in G(ℚ) and are closely related to weighted orbital integrals on G(𝔸)^{1}. In this talk we study the spectral side of the trace formula which we try to use to study the asymptotic distribution of the multiplicities. The main result is the following theorem.

Theorem 0.1. The spectral side of the Arthur trace formula for GL(n) is absolutely convergent.

This theorem will facilitate the application of the trace formula and will also open up new directions for its application. Of particular importance in the proof is the investigation of the analytic properties of the intertwining operators. Let P ⊂ G be a maximal parabolic subgroup with P = MN and

M = GL(n_{1}) × GL(n_{2}).

Let π = π_{1} ⊗ π_{2} be a cuspidal automorphic representation of M(𝔸). For s ∈ ℂ let χ_{s} : 𝔸^{×}/ℚ^{×} → ℂ^{×} be the character χ_{s}(a) = |a|^{s}. Extend χ_{s} to GL(m, 𝔸) by χ_{s}(g) = χ_{s}(det(g)). Let

π_{s} = π_{1} · χ_{s} ⊗ π_{2}χ_{−s}.

Let I_{P}^{G}(π_{s}) be the corresponding induced repersentation of G(𝔸) acting in H_{P} (π_{s}). Let H_{P}^{0} (π_{s}) be the subspace of K-finite vectors and let

M(π, s) : H_{P}^{0} (π_{s}) → H0 ; \overline{P} (π_{s})

be the standard intertwining operator. It is a meromorphic operator valued function of s. Let L(s, π_{1} × π̌_{2}) and ϵ(s, π_{1} × π̌_{2}) be the Rankin-Selberg L-function and the ϵ-factor. Then

M(π, s) = \frac{L(s, π_{1} × π̌_{2})}{L(1 + s, π_{1} × π̌_{2})ϵ(s, π_{1} × π̌_{2})} ⨂_{v∈S} N(π_{v}, s), (0.1)

where S is a finite set of places containing the Archimedean one, and N(π_{v}, s) are normalized local intertwining operators. Let m(π, s) denote the L-factor in (0.1). Let ∥ π ∥ be the norm of the infinitesimal character of π_{∞}. Then we have Theorem 0.2. For every pair of open compact subgroups K_{i,f} ⊂ GL(m_{i}, 𝔸_{f}), i = 1, 2, there exists C > 0 such that

∫_{ℝ} |\frac{m′}{m} (π, iλ)| (1 + λ^{2})^{−2} ≤ C ∥ π ∥

for all π = π_{1} × π_{2} such that π_{i} ∈ Π_{cusp}(GL(m_{i}, 𝔸) has a nonzero K-invariant vector.

In fact we conjecture that the following holds Conjecture. There exist C > 0, k ∈ ℕ, such that

|\frac{L′(1 + iλ, π_{1} × π̌_{2})}{L(1 + iλ, π_{1} × π̌_{2})}| ≤ C(log λ)^{k}

for |λ| ≥ 1 and all π_{i} ∈ Π_{cusp} GL(m_{i}, 𝔸).

Finally we establish a certain weak form of the Ramanujan conjecture.

Universität Bonn, Mathematisches Institut, Beringstrasse 1, D – 53115 Bonn, Germany

E-mail address: mueller@math.uni-bonn.de