Generalized and degenerate Whittaker quotients and Fourier coefficients

Abstract

The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In order to encompass other representations, one attaches a degenerate (or a generalized) Whittaker model $W_{\mathcal{O}}$, or a Fourier coefficient in the global case, to any nilpotent orbit $\mathcal{O}$. In this note we survey some classical and some recent work in this direction - for Archimedean, p-adic and global fields. The main results concern the existence of models. For a representation $\pi$, call the set of maximal orbits $\mathcal{O}$ with $W_{\mathcal{O}}$ that includes $\pi$ the Whittaker support of $\pi$. The two main questions discussed in this note are: (1) What kind of orbits can appear in the Whittaker support of a representation? (2) How does the Whittaker support of a given representation $\pi$ relate to other invariants of $\pi$, such as its wave-front set?