Zhelobenko Invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford Algebra Conjecture

TL;DR

This paper proves an analogue of the Kostant Clifford Algebra Conjecture using Zhelobenko invariants and Bernstein-Gelfand-Gelfand operators, extending previous results to a generalized Harish-Chandra map.

Contribution

It introduces a new approach to the Kostant conjecture by replacing the Harish-Chandra map with a generalized version involving Zhelobenko invariants and BGG operators.

Findings

Proves an analogue of the Kostant conjecture with a generalized Harish-Chandra map.

Connects Zhelobenko invariants to the BGG operators in the context of the conjecture.

Extends previous results from type A to a broader setting.

Abstract

Let g be a complex simple Lie algebra and h a Cartan subalgebra. The Clifford algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of the simple 2m+1-dimensional module of the principal s-triple obtained from the Langlands dual. Bazlov settled this conjecture positively in type A. The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of the Harish-Chandra map for the enveloping algebra U(g) composed with evaluation at the half sum of the positive roots. Here an analogue of the Kostant conjecture is settled by replacing the Harish-Chandra map by a "generalized Harish-Chandra" map which had been studied notably by Zhelobenko. The proof involves a symmetric algebra version of the Kostant conjecture (settled in works of Alekseev-Bazlov-Rohr), the Zhelobenko invariants in the adjoint case and surprisingly the Bernstein-Gelfand-Gelfand operators introduced in their study of the cohomology of the flag variety.