R-group and Whittaker space of some genuine representations

TL;DR

This paper investigates the R-group and Whittaker space dimensions of genuine principal series representations of covering groups, providing formulas and conjectures that enhance understanding of their structure and are verified in specific cases.

Contribution

It establishes a formula linking the R-group to Whittaker space dimensions and proposes a conjectural simplified formula for saturated covers, verified in certain cases.

Findings

Derived a formula relating R-group to Whittaker space dimension.

Proposed a conjectural simplified formula for saturated covers.

Verified the conjecture in cases including symplectic group covers.

Abstract

For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for saturated covers of a semisimple simply-connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.