Borel-de Siebenthal Positive Root Systems

2023/09/20 by Pampa Paul, Paul, Pampa
Mathematics · #17B10 #17B20 #17B22 #17B25 #22E46 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2309.11099

openalex publication_date 2023/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \frakg₀=Lie(G), \frakk₀=Lie(K) ⊂ \frakg₀, \frakt₀ be a maximal abelian subalgebra of \frakk₀, \frakg=\frakg₀^ℂ, \frakk=\frakk₀^ℂ, and \frakh=\frakt₀^ℂ. The existence of a Borel-de Siebenthal positive root system of Δ(\frakg, \frakh) is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of Δ(\frakg, \frakh), 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.

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