Abstract
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.
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Sun, B. On representations of real Jacobi groups. Sci. China Math. 55, 541–555 (2012). https://doi.org/10.1007/s11425-011-4333-3
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DOI: https://doi.org/10.1007/s11425-011-4333-3