Generalized matrix coefficients for infinite dimensional unitary representations

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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.

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Journal of the Ramanujan Mathematical Society

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