On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fréchet 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 that take 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 real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.