Fully commutative elements and spherical nilpotent orbits

Let g be a complex simple Lie algebra, with fixed Borel subalgebra b subset of g 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 G2, it is shown that an element w is an element of 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.(c) 2022 Elsevier Inc. All rights reserved.