On C∞-vectors and intertwining bilinear forms for representations of Lie groups

AbstractThe main result of this paper is a structure theorem for continuous intertwining bilinear forms on the Fréchet spaces of C∞-vectors of two Banach space representations of a Lie group. Using elliptic regularity theory it is shown that such forms can be identified with a certain class of closed densely defined intertwining operators. As an application of this result it is shown that all the usual criteria for equivalence and irreducibility of unitary representations remain valid for the corresponding differentiable representations on the spaces of C∞-vectors. The results are used to study families of representations having a common space of C∞-vectors, and the theory is illustrated by some examples. The spaces of C∞-vectors of the regular representations in Lp(G) are described, and the result is used to characterize a certain class of closed translation invariant operators from Lp(G) to Lq(G). The same technique is used to characterize the space of C∞-vectors of an arbitrary induced representation of a Lie group