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

TL;DR

This paper constructs a new class of infinite-dimensional monoids from highest weight representations of Kac-Moody groups, revealing structural similarities to reductive algebraic monoids and providing insights into their Bruhat decompositions.

Contribution

It introduces a novel construction of monoids associated to Kac-Moody groups and analyzes their algebraic and combinatorial properties, extending reductive monoid theory to infinite dimensions.

Findings

Monoid has properties similar to J-irreducible reductive monoids

The monoid is unit regular and admits a Bruhat decomposition

The idempotent lattice corresponds to 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.