$ {L}_{0}$-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series

Pampa Paul; K.N. Raghavan; Parameswaran Sankaran

doi:10.1016/j.crma.2012.11.009

Lie Algebras

L_{0}

-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series

Presented by: the Editorial Board

Pampa Paul ¹; K.N. Raghavan ¹; Parameswaran Sankaran ¹

¹ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1007-1009.

Abstract
Résumé

Let $G_{0}$ be a simply connected non-compact real simple Lie group and let $K_{0}$ be a maximal compact subgroup of $G_{0}$ . Suppose that $K_{0}$ is semisimple and that $rank (K_{0}) = rank (G_{0})$ . Let $Δ^{+}$ be a Borel–de Siebenthal positive root system and let $π_{λ}$ be the Borel–de Siebenthal discrete series of $G_{0}$ with Harish-Chandra parameter λ. One has a certain subgroup $L_{0} \subset K_{0}$ so that $K_{0} / L_{0}$ is an irreducible Hermitian symmetric space. Also, there is a holomorphic discrete series $π_{λ^{'}}$ of $K_{0}^{⁎}$ , the non-compact dual of $K_{0}$ , with Harish-Chandra parameter $λ^{'} : = λ - (1 / 2) \sum α$ , where the sum is over non-compact roots in $Δ^{+}$ . We prove that there are infinitely many $L_{0}$ -types common to $π_{λ}$ and $π_{λ^{'}}$ under certain hypotheses.

Soit $G_{0}$ un groupe de Lie simple réel simplement connexe non-compact et soit $K_{0}$ un sous-groupe compact maximal de $G_{0}$ . Supposons que $K_{0}$ soit semisimple, et que $rang (K_{0}) = rang (G_{0})$ . Supposons que $Δ^{+}$ soit un système positif de racines de Borel–de Siebenthal de $G_{0}$ . Soit $π_{λ}$ la représentation de la série discrète de Borel–de Siebenthal de $G_{0}$ avec paramètre de Harish-Chandra λ. Il existe un sous-groupe connexe $L_{0} \subset K_{0}$ tel que $K_{0} / L_{0}$ soit un espace Hermitien symétrique irréductible. Soit $K_{0}^{⁎}$ le dual non-compact de $K_{0}$ par rapport à $L_{0}$ . On a une série discrète holomorphe $π_{λ^{'}}$ de $K_{0}^{⁎}$ avec paramètre de Harish-Chandra $λ^{'} : = λ - (1 / 2) \sum α$ où α parcourt les racines non-compactes de $Δ^{+}$ . On montre quʼil existe une infinité de $L_{0}$ -types communs à $π_{λ}$ et $π_{λ^{'}}$ sous certaines hypothèses.

Received: 2012-10-01
Accepted: 2012-11-13
Published online: 2012-11-20

DOI: 10.1016/j.crma.2012.11.009

Author's affiliations:

Pampa Paul ¹; K.N. Raghavan ¹; Parameswaran Sankaran ¹

¹ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

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     title = {$ {L}_{0}$-types common to a {Borel{\textendash}de} {Siebenthal} discrete series and its associated holomorphic discrete series},
     journal = {Comptes Rendus. Math\'ematique},
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Pampa Paul; K.N. Raghavan; Parameswaran Sankaran. $ {L}_{0}$-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series. Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1007-1009. doi : 10.1016/j.crma.2012.11.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.11.009/

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[5] P. Paul; K.N. Raghavan; P. Sankaran $L_{0}$ -types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series | arXiv

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