Generalized and degenerate Whittaker quotients and Fourier coefficients

TL;DR

This paper surveys the theory of generalized Whittaker models attached to nilpotent orbits, exploring their existence, properties, and relation to invariants like wave-front sets across various fields.

Contribution

It provides a comprehensive overview of classical and recent results on the existence and structure of degenerate Whittaker models for different types of fields.

Findings

Existence criteria for generalized Whittaker models.

Relationship between Whittaker support and wave-front sets.

Classification of orbits appearing in Whittaker supports.

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?