Cohomology of Representations and Langlands Functoriality

Abstract

Let G be a real semi-simple Lie group. Let A be an arithmetic subgroup of the group G. Suppose that F is a finite-dimensional representation of G. One of the objects of interest is the cohomology group H (A, F). In particular, determining when these groups are non-zero and computing cohomology classes of these groups. It is well known that these groups have interpretations using relative Lie algebra cohomology of the group G with respect to a compact subgroup K. This interpretation gives us a relation between the cohomology groups H (A, F) and a finite subset of the set of representations of G. Here we obtain some non-vanishing results for the cohomology classes for the group GL(N). We use the principle of Langlands functoriality to compute these classes. We start with V, a ‘nice’ representation of a classical group G, and use Local Langlands correspondence to transfer V to a representation, i(V), of an appropriate GL(N) and ask whether i(V) contribute to the cohomology groups H (A, F). We characterize when a tempered representation of a classical group G transfers to a cohomological representation of GL(n). This is summarized in Theorem 4.2.5. We also start with a cohomological representation of Sp(4,R) and ask when the transferred representation of GL(5,R) is cohomological. We obtain a complete result in the case of representations with trivial coefficients. This is summarized in Theorem 5.5.2.