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

Martin T. Luu

TL;DR

This work extends Kirillov's polynomial counting from type A to the exceptional Lie algebra $\mathfrak{g}_2$ over finite fields. Using a faithful $7$-dimensional representation obtained by folding from $\mathfrak{so}_8$, the authors construct explicit Kirillov polynomials $P_{\\lambda}(q)$ counting elements by fixed Jordan type in $\mathfrak{g}_2$, and show these decompose as $P_{\\lambda}(q)=q^{a}(q-1)^{b}R_{\\lambda}(q)$ with irreducible $R_{\\lambda}(q)$; they identify five nonzero polynomials corresponding to the five nilpotent orbits. A detailed rank-sequence analysis confirms the explicit forms, such as $P_{7}(q)=q^{4}(q-1)^{2}P_{3,3,1}(q)$ and $P_{1^{7}}(q)=1$, and the leading coefficients are shown to equal dimensions of Weyl-group representations arising from the Springer correspondence. This establishes a direct link between Kirillov polynomials for $\mathfrak{g}_2$, nilpotent orbits, and Springer theory, providing a concrete exceptional-type example and a template for further extensions.

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.

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

TL;DR

This work extends Kirillov's polynomial counting from type A to the exceptional Lie algebra over finite fields. Using a faithful -dimensional representation obtained by folding from , the authors construct explicit Kirillov polynomials counting elements by fixed Jordan type in , and show these decompose as with irreducible ; they identify five nonzero polynomials corresponding to the five nilpotent orbits. A detailed rank-sequence analysis confirms the explicit forms, such as and , and the leading coefficients are shown to equal dimensions of Weyl-group representations arising from the Springer correspondence. This establishes a direct link between Kirillov polynomials for , nilpotent orbits, and Springer theory, providing a concrete exceptional-type example and a template for further extensions.

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 elements and fixed Jordan type. One obtains polynomials with respect to 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 . In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

Paper Structure

This paper contains 3 sections, 2 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The case of $\mathfrak g_2$
  3. Completion of the proof

Key Result

Theorem 1

Consider a finite field $\mathbb{F}_{q}$ of characteristic $p>3$. The Kirillov polynomials $P_{\lambda}(q)$ for the exceptional Lie algebra $\mathfrak g_{2}$ with respect to its $7$-dimensional faithful representation exist and are given by Write for $a,b$ in $\mathbb{Z}^{\ge 0}$ and $R_{\lambda}(0)R_{\lambda}(1)$ non-zero. The constant term of $R_{\lambda}(q)$ equals $1$, all other coefficients

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 2.1