Analogue Zhelobenko Invariants and the Kostant Clifford Algebra Conjecture

TL;DR

This paper proves that Zhelobenko invariants can describe the image of the Harish-Chandra map for complex simple Lie algebras, extending previous results related to the Kostant conjecture and its analogues.

Contribution

It demonstrates that analogue Zhelobenko invariants characterize the Harish-Chandra map's image, generalizing earlier work on Kostant's conjecture.

Findings

Zhelobenko invariants describe the Harish-Chandra map's image.

Extension of Kostant conjecture to a broader algebraic context.

Proof of the analogue invariants' role in the Harish-Chandra map.

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 of g. 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. In an earlier work we settled an analogue of the Kostant conjecture obtained by replacing the Harish-Chandra map by a "generalized Harish-Chandra" map whose image is described via Zhelobenko invariants. Here we show that there are analogue Zhelobenko invariants which describe the image of the Harish-Chandra map. Following this a similar proof goes through.