Fully commutative elements and spherical nilpotent orbits

2021/11/22 by Gandini, Jacopo
#17B08 #20F55 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2111.11273

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 G₂, 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.

Related

• Some amazing properties of spherical nilpotent orbits
• On adjoint orbits in nilpotent ideals of a Borel subalgebra
• Spherical Nilpotent Orbits in Positive Characteristic
• On some representations of nilpotent Lie algebras and superalgebras
• Ad-nilpotent Ideals and Equivalence Relations