Demazure product of the affine Weyl groups

TL;DR

This paper explores the Demazure product in extended affine Weyl groups, revealing connections to the quantum Bruhat graph and providing explicit formulas for key structures in Lie theory.

Contribution

It establishes a novel link between Demazure products of affine Weyl groups and the quantum Bruhat graph, with explicit formulas for Newton points and the Lusztig-Vogan map.

Findings

Connection between Demazure product and quantum Bruhat graph

Explicit formulas for Newton points in affine Weyl groups

Formulas for the Lusztig-Vogan map

Abstract

The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell/shrunken Weyl chambers of $\tilde W$, and obtain an explicit formula on the Lusztig-Vogan map from the coweight lattice to the set of dominant coweights.