Characters of Lie Groups

Abstract

We consider a connected Lie group with co-compact radical G. Let g be the Lie algebra of G. Let 1 be in the dual of g. Under the assumption that g(l) is commutative and reductive in g,we construct an application ϕ → F_l,ϕ from D(G) to the space of C^∞functions on an open dense subset of G(l). If G is compact, F_l,ϕ is — up to a scalar — the invariant integral of ϕ relative to the Cartan subgroup G(l) of G. Using this, we obtain a formula for the trace of the operator T(l, G)(ϕ), where T(l, G) is the unitary representation of G associated to l.