From conjugacy classes in the Weyl group to unipotent classes, II

TL;DR

This paper investigates the relationship between conjugacy classes in Weyl groups and unipotent classes in reductive algebraic groups, establishing canonical inverse maps and connections to special representations.

Contribution

It proves the existence of a canonical one-sided inverse for the map from Weyl group conjugacy classes to unipotent classes and relates this to special representations.

Findings

Existence of a canonical one-sided inverse for p.

Relation  = r p for a unique map r.

Construction of a surjective map from conjugacy classes to special representations.

Abstract

Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map \Phi_p from the set \underline{W} of conjugacy classes in the Weyl group W to the set of unipotent classes in G. Here we prove three results about \Phi_p. First we show that \Phi_p has a canonical one sided inverse. Next we show that \Phi_0 =r\Phi_p for a unique map r. Finally we construct a natural surjective map from \underline{W} to the set of special representations of W which is the composition of \Phi_0 with another natural map; we show that this map depends only on the Coxeter group structure of W. We also define the special conjugacy classes in W (in 1-1 correspondence with the special representations of W) and describe them explicitly for each simple type.