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∗)-∞, the map 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.
Publication Source (Journal or Book title)
Journal of the Ramanujan Mathematical Society
He, H. (2014). Generalized matrix coefficients for infinite dimensional unitary representations. Journal of the Ramanujan Mathematical Society, 29 (3), 253-272. Retrieved from https://digitalcommons.lsu.edu/mathematics_pubs/473