On representations of real Jacobi groups

Abstract

We consider a category of continuous Hilbert space representations and a category of smooth Frechet representations, of a real Jacobi group $G$. By Mackey's theory, they are respectively equivalent to certain categories of representations of a real reductive group $\tilde{L}$. Within these categories, we show that the two functors of taking smooth vectors for $G$, and for $\tilde{L}$, are consistent with each other. By using Casselman-Wallach's theory of smooth representations of real reductive groups, we define matrix coefficients for distributional vectors of certain representations of $G$. We also formulate Gelfand-Kazhdan criteria for Jacobi groups which could be used to prove the multiplicity one theorem for Fourier-Jacobi models.

References in corpus (2)

• Smooth Frechet Globalizations of Harish-Chandra Modules
• Multiplicity one theorems for Fourier-Jacobi models