TAILIEUCHUNG - Đề tài "Weyl’s law for the cuspidal spectrum of SLn "

Let Γ be a principal congruence subgroup of SLn (Z) and let σ be an Γ irreducible unitary representation of SO(n). Let Ncus (λ, σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to σ. In this paper we Γ prove that the counting function Ncus (λ, σ) satisfies Weyl’s law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SLn (Z). Contents 1. Preliminaries 2. Heat kernel estimates 3. Estimations of the discrete spectrum 4 | Annals of Mathematics Weyl s law for the cuspidal spectrum of SLn By Werner M uller Annals of Mathematics 165 2007 275 333 Weyl s law for the cuspidal spectrum of SLn By Werner Muller Abstract Let r be a principal congruence subgroup of SLn Z and let Ơ be an irreducible unitary representation of SO n . Let Njủs A ơ be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for r which transform under SO n according to Ơ. In this paper we prove that the counting function N s A ơ satisfies Weyl s law. Especially this implies that there exist infinitely many cusp forms for the full modular group SLn Z . Contents 1. Preliminaries 2. Heat kernel estimates 3. Estimations of the discrete spectrum 4. Rankin-Selberg L-functions 5. Normalizing factors 6. The spectral side 7. Proof of the main theorem References Let G be a connected reductive algebraic group over Q and let r c G Q be an arithmetic subgroup. An important problem in the theory of automorphic forms is the question of the existence and the construction of cusp forms for r. By Langlands theory of Eisenstein series La cusp forms are the building blocks of the spectral resolution of the regular representation of G R in L2 r G R . Cusp forms are also fundamental in number theory. Despite their importance very little is known about the existence of cusp forms in general. In this paper we will address the question of existence of cusp forms for the group G SLn. The main purpose of this paper is to prove that cusp forms exist in abundance for congruence subgroups of SLn Z n 2. 276 WERNER MULLER To formulate our main result we need to introduce some notation. For simplicity assume that G is semisimple. Let K be a maximal compact subgroup of G R and let X G R ỊKtx be the associated Riemannian symmetric space. Let Z gc be the center of the unviersal enveloping algebra of the com-plexification of the Lie algebra g of G R . Recall that a cusp form for r in the sense of La is a .

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