~ ~ -c O H O M O L O ~ ~ and Automorphic Representations

This article is an introduction to the L~-cohomology of arithmetic symmetric spaces. We shall describe properties of automorphic representations which lead to a remarkable interplay among Hodge structures, Lefschetz structures, and Hecke operators. In order to make the discussion slightly more concrete, we shall focus on the special case of Siegel moduli space. Resume. Cet article est une introduction a la cohomologie L2 des espaces symetriques arithmetiques. On dkcrit des propriktks des representations automorphes ou interagissent de facon remarquahle les structures de Hodge et de Lefschetz et les opkrateurs de Hecke. Pour fixer les idkes, on se restreint au cas particulier des espaces de modules de Siegel. I was asked by the editors to submit a general article on the trace formula. There are severalsuch papers already [4], [22], [61, [231, [IS], and I was not sure I could add anything. I decided to write a survey article on a different topic, one to which the trace formula has been applied and will certainly be applied further. The article is on the relationship between L2-cohomology and automorphic representations. It is an introduction of sorts to the papers [3] and [S], especially [8,§91 I shall try to Illustrate some general phenomena by looking at a basic case-Siegel moduli space S(N) of level N. The phenomena, partly known (Theorem 1) and partly conjectural (Conjecture 2), concern the interplay among Hodge structures, Lefschetz structures and Hecke operators. Taken together, they may be regarded as a reciprocity law for S(N) at the Archimedean place. For a general discussion of reciprocity laws at the unramified finite places, see the papers [26] and [211. Fix a positive integer n and let 3 f be the Siegel upper half space of genus n. Then 3 f is the space of ( n x n)-complex matrices of the form where Xis a real symmetric matrix and Y is a real positive-definite matrix. Observe that 3f is an open subset of the complexvector space of complex Supported in part by an NSERC Research Grant.