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Borel-de Siebenthal Positive Root Systems

Pampa Paul

TL;DR

This work classifies Borel-de Siebenthal positive root systems for equi-rank real forms and links the classification to the BD discrete series representations of $G$ with a fixed infinitesimal character. It develops a graded decomposition of $ rak g$ via the BD root data, shows every BD root system containing $P_ rak k$ is either the base system $P$ or obtained by a controlled root replacement, and relates the count of such systems to the covering index of $ ext{Int}( rak g)$. The results yield the precise number of unitary inequivalent BD discrete series in the non-Hermitian case, and connect to holomorphic/anti-holomorphic BD discrete series in the Hermitian case through BD root orders. Overall, the paper deepens the understanding of equi-rank representations and provides explicit combinatorial data for discrete series classified by BD root systems.

Abstract

Let $G$ be a connected simple Lie group with finite centre, $K$ be a maximal compact subgroup of $G,$ and rank$(G)=$ rank$(K).$ Let $\frak{g}_0=$Lie$(G), \frak{k}_0=$Lie$(K) \subset \frak{g}_0, \frak{t}_0$ be a maximal abelian subalgebra of $\frak{k}_0, \frak{g}=\frak{g}_0^\mathbb{C}, \frak{k}=\frak{k}_0^\mathbb{C},$ and $\frak{h}=\frak{t}_0^\mathbb{C}.$ The existence of a Borel-de Siebenthal positive root system of $Δ(\frak{g}, \frak{h})$ is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of $Δ(\frak{g}, \frak{h}),$ assuming the existence. As an application, we have determined the number of unitary equivalence classes of all Borel-de Siebenthal discrete series representations of $G$ (if $G/K$ is not Hermitian symmetric) with a fixed infinitesimal character.

Borel-de Siebenthal Positive Root Systems

TL;DR

This work classifies Borel-de Siebenthal positive root systems for equi-rank real forms and links the classification to the BD discrete series representations of with a fixed infinitesimal character. It develops a graded decomposition of via the BD root data, shows every BD root system containing is either the base system or obtained by a controlled root replacement, and relates the count of such systems to the covering index of . The results yield the precise number of unitary inequivalent BD discrete series in the non-Hermitian case, and connect to holomorphic/anti-holomorphic BD discrete series in the Hermitian case through BD root orders. Overall, the paper deepens the understanding of equi-rank representations and provides explicit combinatorial data for discrete series classified by BD root systems.

Abstract

Let be a connected simple Lie group with finite centre, be a maximal compact subgroup of and rank rank Let LieLie be a maximal abelian subalgebra of and The existence of a Borel-de Siebenthal positive root system of is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of assuming the existence. As an application, we have determined the number of unitary equivalence classes of all Borel-de Siebenthal discrete series representations of (if is not Hermitian symmetric) with a fixed infinitesimal character.
Paper Structure (4 sections, 6 theorems, 4 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Borel-de Siebenthal positive root system
  3. Case by case consideration:
  4. Discrete series representations

Key Result

Theorem 1.1

Assume that $\frak{k}$ is semisimple. Let $\phi' \in \Phi_0$ be such that $n_{\phi'}(\delta)=1,$ and $\phi \in \Phi_0$ be such that $w_{\frak{l}_0}^0 (\phi') = -\phi.$ Let $\frak{l}'_0$ be the reductive subalgebra of $\frak{k}$ containing $\frak{h}$ and the Dynkin diagram of $[\frak{l}'_0, \frak{l}'

Figures (1)

  • Figure 1: Diagram of the weights of the $\frak{k}$-module $\frak{p}$ for $\frak{g}_0 = \frak{so}(2p,2l-2p+1)$

Theorems & Definitions (10)

  • Theorem 1.1
  • Corollary 1.1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5