On certain varieties attached to a Weyl group element

TL;DR

This paper studies algebraic varieties associated with elliptic Weyl group elements, showing they form affine algebraic varieties and describing their structure, especially for classical groups, including automorphisms.

Contribution

It introduces a new geometric construction of varieties linked to elliptic Weyl group elements and characterizes their structure for classical groups.

Findings

The set of G-orbits forms an affine algebraic variety.

For classical groups, the variety is an affine space modulo a finite group.

Constructs nontrivial automorphisms of the variety.

Abstract

Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of G, B is a Borel subgroup of G and B,gBg^{-1} are in relative position w. Then G acts naturally on X. Assume that w has minimal length in its conjugacy class. We show that the set of G-orbits in X has a well defined structure of an affine algebraic variety V. When G is a classical group we show that this variety is an affine space modulo the action of a finite diagonalizable group. In this case we also construct some nontrivial automorphisms of X.