A Theorem on Induced Representations

In [l], we proved a criterion for the disjointness of two induced representations UL and UM of a Lie group G, where L and M are finite-dimensional unitary representations of compact subgroups H and R, respectively, of G. The purpose of this paper is to improve this theorem by getting a stronger conclusion, while dropping the conditions that G be a Lie group and H be compact, and that L and M he finite-dimensional. Moreover, the restriction on R is weakened to read: G has arbitrarily small neighborhoods of the identity invariant under the adjoint action of R on G. Finally, the proof given below is fairly elementary, while the proof in [l] is quite involved. Notations and conventions: Let libe a topological vector space. C(G, 11) will denote the space of continuous functions from G to tU, equipped with the topology of uniform convergence on compact subsets of G while C0(G, 11 ) will denote the space of those fEC(G, It) with compact support. If 11 is omitted it is understood that 11 = C. If 111 is another topological vector space, £(11, 1li) will denote the space of continuous linear maps from 11 into Hi equipped with the topology of bounded convergence. All integrations are with respect to right Haar measure. For any locally compact group G,ho will denote its modular function. If/, gEC0(G),f o g will denote the convolution of/ and g, and /* is defined by f*(x) =ôa(x)~1f(x~l)~. If L and M are representations of G, R(L, M) will denote the space of intertwining operators for L and M (see [3]), while I(L, M) will denote the dimension of R(L, M). For the definition of induced representation used below, see [l]. Finally, for any function / on the group G and any xEG, fx and fx are defined by fx(y) =f(x~ly) and fx(y) =/(yx).