R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

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 certain 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.

Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications

“…The remaining part of the paper concerns (P3). We apply Theorem 1.2 to determine certain Whittaker dimensions and verify several formulas speculated in [Gao20,Gaoa].…”

“…Relations between these two and also the metaplectic W -representation have been hinted at in several works including [McN16] [CG10] [CO13] via the presence of the local scattering matrices, which describe the intertwining operators between the Whittaker models of principal series I(χ) and I( w χ). (P3) Verify some speculative formulas regarding the Whittaker dimension of some Iwahori-spherical representations, especially those irreducible constituents of a regular or unitary unramified principal series, see [Gao20,Gaoa,Gaob]. A solution to (P1) would provide a valuable tool to investigate (P2) and (P3).…”