Gelfand pairs for affine Weyl groups

TL;DR

This paper provides a unified approach to determine when certain pairs involving affine Weyl groups are Gelfand pairs, extending previous results and addressing cases where traditional methods do not apply.

Contribution

It introduces a new criterion linking Gelfand properties of subgroups in finite Weyl groups to those in affine Weyl groups, covering all irreducible types.

Findings

Characterization of Gelfand pairs for all irreducible affine Weyl groups.

Identification of conditions under which subgroup pairs are Gelfand pairs.

Extension of previous results beyond types C_n and B_n.

Abstract

This paper is motivated by several combinatorial actions of the affine Weyl group of type $C_n$. Addressing a question of David Vogan, it was shown in an earlier paper that these permutation representations are essentialy multiplicity-free~\cite{arXiv:2009.13880}. However, the Gelfand trick, which was indispensable in~\cite{arXiv:2009.13880} to prove this property for types $C_n$ and $B_n$, is not applicable for other classical types. Here we present a unified approach to fully answer the analogous question for all irreducible affine Weyl groups. Given a finite Weyl group $W$ with maximal parabolic subgroup $P\leq W$, there corresponds to it a reflection subgroup $H$ of the affine Weyl group $\widetilde W$. It turns out that while the Gelfand property of $P\leq W$ does not imply that of $H\leq \widetilde{W}$, but $Q=N_W(P)\leq W$ has the Gelfand property if and only if $K=QH\leq \widetilde{W}$ has. Finally, for each irreducible type we describe when $(W,Q)$ is a Gelfand pair.