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Character theory at a torsion element

Santosh Nadimpalli, Santosha Pattanayak, Dipendra Prasad

Abstract

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have significance. An important input is to analyse torsion elements of order d in an adjoint group with minimal dimensional centraliser, and to prove that in most cases when d divides the Coxeter number of G, this gives rise to a unique conjugacy class.

Character theory at a torsion element

Abstract

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have significance. An important input is to analyse torsion elements of order d in an adjoint group with minimal dimensional centraliser, and to prove that in most cases when d divides the Coxeter number of G, this gives rise to a unique conjugacy class.

Paper Structure

This paper contains 22 sections, 26 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Restriction to the Principal ${\rm SL}_2(\mathbb C)$: the product formula
  4. Character values on principal elements
  5. The constant $d_m$ for the group ${\rm G}_\lambda(m)$
  6. Examples
  7. Explicit calculation for ${\rm G}_2$
  8. Character at a general torsion element
  9. Adjoint representation restricted to principal $\rm{SL}_2(\mathbb C)$
  10. Restriction to principal ${\rm SL}_2(\mathbb{C})$ determines the representation?
  11. Review of Kac-coordinates
  12. Classical groups
  13. ${\rm GL}_n(\mathbb{C})$
  14. ${\rm Sp}_{2n}(\mathbb{C})$
  15. ${\rm SO}_{2n+1}(\mathbb{C})$
  16. ...and 7 more sections

Key Result

Lemma 3.2

Let ${\rm G}$ be a connected reductive group, with a maximal torus ${\rm T}$ contained in a Borel subgroup ${\rm B}$ of ${\rm G}$, and the associated root datum. Let $\rho$ be half the sum of positive roots of ${\rm T}$ on ${\rm B}$, thus $\rho \in X^*({\rm T}) \otimes \mathbb{Q}$. For any root $\al Similarly, if $\rho^\vee$ is half the sum of positive coroots, then,

Theorems & Definitions (45)

  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.4
  • Corollary 3.5
  • Lemma 3.6
  • Corollary 3.7
  • Proposition 3.8
  • proof
  • ...and 35 more