Kirillov polynomials for the exceptional Lie algebra $\mathfrak g_2$

TL;DR

This paper extends the theory of Kirillov polynomials, originally developed for classical Lie algebras, to the exceptional Lie algebra g_2, linking polynomial coefficients to Springer correspondence.

Contribution

It develops the theory of Kirillov polynomials for g_2, connecting polynomial coefficients to Springer correspondence, a novel extension for exceptional Lie algebras.

Findings

Polynomials with properties related to type A representation theory.

Leading coefficient expressed via Springer correspondence.

Advancement in understanding orbit method for g_2.

Abstract

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of $q$ elements and fixed Jordan type. One obtains polynomials with respect to $q$ with many interesting properties and close relation to type A representation theory. In the present work we develop the corresponding theory for the exceptional Lie algebra $\mathfrak g_2$. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.