Degenerate principal series of metaplectic groups and Howe correspondence

TL;DR

This paper extends the analysis of degenerate principal series representations from real symplectic groups to metaplectic groups, highlighting their role in Howe correspondence and related automorphic forms.

Contribution

It provides new results on the structure of degenerate principal series of metaplectic groups, complementing previous symplectic group results and clarifying their role in Howe correspondence.

Findings

Structures anticipated by natural subrepresentations from Howe correspondence

Key role in understanding Howe correspondence in the archimedean case

Implications for Siegel-Weil formula and its generalizations

Abstract

The main purpose of this article is to supplement the authors' results on degenerate principal series representations of real symplectic groups with the analogous results for metaplectic groups. The basic theme, as in the previous case, is that their structures are anticipated by certain natural subrepresentations constructed from Howe correspondence. This supplement is necessary as these representations play a key role in understanding the basic structure of Howe correspondence (and its complications in the archimedean case), and their global counterparts play an equally essential part in the proof of Siegel-Weil formula and its generalizations (work of Kudla-Rallis). The full results in the metaplectic case also shed light on the seeming peculiarities, when the results in the symplectic case are viewed in their isolation.