Higher Duality 3

Introduction. Let G be a reductive Lie group with maximal compact subgroup K and let Γ be a cocompact discrete subgroup of G. We consider the induced representation IndΓ (C), which is the canonical representation on the space L2(Γ\G). The duality theorem of Gelfand [6] states that there is a Frobenius reciprocity as follows: for any irreducible unitary representation π of G it holds HomG,cont(Ind G Γ (C), π) ∼= HomΓ(C, π ), where π is a certain extension of π. The right hand side can also be written as H0(Γ, π). On the left hand side one gets rid of topology by switching to the (g,K)-modules, i.e. one replaces the left hand side by Homg,K(C (Γ\G)K , πK), where the index K means K-finite vectors. The main result of this note is the generalization to higher degrees: