On involutions of minuscule Kirillov algebras induced by real structures

TL;DR

This paper explores involutions on minuscule Kirillov algebras induced by real structures, analyzing their action on spectra and fixed points via equivariant cohomology, and applies this to recover Stembridge's $q=-1$ phenomenon.

Contribution

It provides a geometric characterization of involutions on minuscule Kirillov algebras and links them to real partial flag varieties, revealing new insights into their structure.

Findings

Describes involutions on spectra of minuscule Kirillov algebras induced by real structures.

Models fixed points using equivariant cohomology of real partial flag varieties.

Recovers Stembridge's $q=-1$ phenomenon through geometric methods.

Abstract

We study Kirillov algebras attached to minuscule highest weight representations of semisimple Lie algebras. They can be viewed as equivariant cohomology algebras of partial flag varieties. Real structures on the varieties then induce involutions of these algebras. We describe how these involutions act on the spectra of minuscule Kirillov algebras, and model the fixed points via the equivariant cohomology of real partial flag varieties. We then use this model to characterise freeness of the fixed point coordinate ring over the appropriate base. As an application, we recover a $q=-1$ phenomenon of Stembridge in the minuscule case by geometric means.