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.
