Homology of analogues of Heisenberg Lie algebras

TL;DR

This paper computes the homology of three families of 2-step nilpotent Lie (super)algebras related to classical groups, completing previous work and introducing new combinatorial and representation-theoretic methods.

Contribution

It provides a comprehensive calculation of homology for these algebras using elementary combinatorics and stable representation theory, extending prior results by Getzler.

Findings

Homology formulas for symplectic, orthogonal, and general linear Lie algebras.

Elementary combinatorial approach based on Weyl groups and partitions.

A shorter, more conceptual method using stable representation theory.

Abstract

We calculate the homology of three families of 2-step nilpotent Lie (super)algebras associated with the symplectic, orthogonal, and general linear groups. The symplectic case was considered by Getzler and the main motivation for this work was to complete the calculations started by him. In all three cases, these algebras can be realized as the nilpotent radical of a parabolic subalgebra of a simple Lie algebra, and our first approach relies on a theorem of Kostant, but is otherwise elementary and involves combinatorics of Weyl groups and partitions which may be of independent interest. Our second approach is an application of (un)stable representation theory of the classical groups in the sense of recent joint work of the author with Snowden, which is shorter and more conceptual.