Generalized Matrix Coefficients for Infinite Dimensional Unitary Representations

Let (π,H) be a unitary representation of a Lie group G. Classically, matrix coefficients are continuous functions on G attached to a pair of vectors in H and H∗. In this note, we generalize the definition of matrix coefficients to a pair of distributions in (H−∞, (H∗)−∞). Generalized matrix coefficients are in D(G), the space of distributions on G. By analyzing the structure of generalized matrix coefficients, we prove that, fixing an element in (H∗)−∞, themap H−∞ →D(G) is continuous. This effectively answers the question about computing generalized matrix coefficients. For the Heisenberg group, our generalized matrix coefficients can be considered as a generalization of the Fourier-Wigner transform.