On certain varieties attached to a Weyl group element

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.

30 pages, new material added

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