  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.
