The Fourier-Jacobi map and small representations

Weissman, Martin

doi:10.1090/S1088-4165-03-00197-3

The Fourier-Jacobi map and small representations
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by Martin H. Weissman

Represent. Theory 7 (2003), 275-299

DOI: https://doi.org/10.1090/S1088-4165-03-00197-3

Published electronically: July 28, 2003

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Abstract:

We study the “Fourier-Jacobi” functor on smooth representations of split, simple, simply-laced $p$-adic groups. This functor has been extensively studied on the symplectic group, where it provides the representation-theoretic analogue of the Fourier-Jacobi expansion of Siegel modular forms. Our applications are different from those studied classically with the symplectic group. In particular, we are able to describe the composition series of certain degenerate principal series. This includes the location of minimal and small (in the sense of the support of the local character expansion) representations as spherical subquotients.

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