Invariant Tensor Product

In this paper, we define invariant tensor product and study invariant tensor products associated with discrete series representations. Let G(V1) × G(V2) be a pair of classical groups diagonally embedded in G(V1 ⊕ V2). Suppose that dimV1 < dimV2. Let π be a discrete series representation of G(V1 ⊕ V2). We prove that the functor π⊗G(V1) maps unitary representations of G(V1) to unitary representations of G(V2). Here we enlarge the definition of unitary representations by including the zero dimensional “representation ”. 1 Invariant Tensor Products Various forms of invariant tensor products appear in the literature. In many cases, they are employed to study the space HomG(π1, π2) where one of the representations π1 and π2 is irreducible. In this paper, we formulate the concept of invariant tensor product uniformly. We also study the invariant tensor functor associated with discrete series representations for classical groups. Definition 1 Let G be a locally compact topological group and dg be a left invariant Haar measure. Let (π,Hπ) and (π1,Hπ1) be two unitary representations of G. Let V and V1 be two dense subspaces of Hπ and Hπ1 . Formally, define the averaging operator L : V ⊗ V1 → (V ⊗ V1)R as follows, ∀ u, v ∈ V, u1, v1 ∈ V1, L(v ⊗ v1)(u⊗ u1) = ∫ G ( (π ⊗ π1)(g)(v ⊗ v1), (u⊗ u1) ) dg (1)