Infinite-dimensional reductive monoids associated to highest weight representations of Kac-Moody groups

TL;DR

This paper constructs an infinite-dimensional monoid from a highest weight representation of a Kac-Moody group, revealing structural similarities to reductive algebraic monoids and analyzing its algebraic and combinatorial properties.

Contribution

It introduces a new class of monoids associated with Kac-Moody groups, extending reductive monoid theory to infinite-dimensional settings.

Findings

The monoid is unit regular.

It admits a Bruhat decomposition.

Its idempotent lattice matches the face lattice of a convex hull.

Abstract

Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highest weight.