Fully commutative elements and spherical nilpotent orbits

TL;DR

This paper explores the relationship between fully commutative elements in Weyl groups and spherical nilpotent orbits in simple Lie algebras, extending previous classifications beyond simply laced types.

Contribution

It provides a new characterization of fully commutative elements in Weyl groups in terms of spherical nilpotent orbit closures, generalizing prior work to non-G_2 types.

Findings

Fully commutative elements correspond to subalgebras in the closure of spherical nilpotent orbits.

Characterization extends to ad-nilpotent ideals of the Borel subalgebra.

Results connect Weyl group combinatorics with geometric orbit classifications.

Abstract

Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their connections with the spherical nilpotent orbits in g. If g is not of type G_2, it is shown that an element w in W is fully commutative if and only if the subalgebra of b determined by the inversions of w lies in the closure of a spherical nilpotent orbit. A similar characterization is also given for the ad-nilpotent ideals of b, which are parametrized by suitable elements in the affine Weyl group of g thanks to the work of Cellini and Papi.