Cuspidal Cohomology for GL(N) over Number Fields

Abstract

Let GL(N) be the algebraic group over a number field F. We are interested in a subspace (known as cuspidal cohomology) of the sheaf cohomology of locally symmetric ad`elic space in the coefficient system of a finite-dimensional representation M_λ of Res_{F/Q} GL(N) with the highest weight λ. Our study focuses on establishing a non-vanishing property of cuspidal cohomology. We prove the non-vanishing of a Lefschetz number to prove the non-vanishing of cuspidal cohomology for SL(N) when F is Galois over maximal totally real subfield and the highest weight is strongly pure. It also proves the non-vanishing of cuspidal cohomology for GL(N). Given an irreducible representation of SL_2(F_q) for an odd prime q, we find the dimension of the space of cusp forms with respect to the full modular group taking values into certain representation spaces. The dimension equals the multiplicity of the representation in the space of classical cusp forms with respect to the principal congruence subgroup of level q.