Criteria for the Existence of Cuspidal Theta Representations

TL;DR

This paper investigates the conditions under which cuspidal theta representations exist on r-fold metaplectic covers of general linear groups, extending known results beyond low-rank cases.

Contribution

It establishes necessary conditions for the existence of cuspidal theta representations on higher-rank metaplectic covers of GL(n).

Findings

Necessary conditions for existence are derived.

Results extend known cases from low-rank to arbitrary rank.

Provides a framework for future existence proofs.

Abstract

Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of $GL_j$, $j\leq 3$. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the $r$-fold metaplectic cover of the general linear group of arbitrary rank.