On bounded generalized Harish-Chandra modules

Ivan Penkov[1]; Vera Serganova[2]

  • [1] Jacobs University Bremen School of Engineering and Science Campus Ring 1 28759 Bremen (Germany)
  • [2] University of California Berkeley Department of Mathematics Berkeley CA 94720 (USA)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 2, page 477-496
  • ISSN: 0373-0956

Abstract

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Let 𝔤 be a complex reductive Lie algebra and 𝔨 ⊂ 𝔤 be any reductive in 𝔤 subalgebra. We call a ( 𝔤 , 𝔨 ) -module M bounded if the 𝔨 -multiplicities of M are uniformly bounded. In this paper we initiate a general study of simple bounded ( 𝔤 , 𝔨 ) -modules. We prove a strong necessary condition for a subalgebra 𝔨 to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded ( 𝔤 , 𝔨 ) -module, and then establish a sufficient condition for a subalgebra 𝔨 to be bounded (Theorem 5.1). As a result we are able to classify the maximal bounded reductive subalgebras of 𝔤 = sl ( n ) .

How to cite

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Penkov, Ivan, and Serganova, Vera. "On bounded generalized Harish-Chandra modules." Annales de l’institut Fourier 62.2 (2012): 477-496. <http://eudml.org/doc/251052>.

@article{Penkov2012,
abstract = {Let $\mathfrak\{g\}$ be a complex reductive Lie algebra and $\mathfrak\{k\}\subset \mathfrak\{g\}$ be any reductive in $\mathfrak\{g\}$ subalgebra. We call a $(\mathfrak\{g\},\mathfrak\{k\})$-module $M$ bounded if the $\mathfrak\{k\}$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(\mathfrak\{g\},\mathfrak\{k\})$-modules. We prove a strong necessary condition for a subalgebra $\mathfrak\{k\}$ to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded $(\mathfrak\{g\},\mathfrak\{k\})$-module, and then establish a sufficient condition for a subalgebra $\mathfrak\{k\}$ to be bounded (Theorem 5.1). As a result we are able to classify the maximal bounded reductive subalgebras of $\mathfrak\{g\}=\mathrm\{sl\}(n)$.},
affiliation = {Jacobs University Bremen School of Engineering and Science Campus Ring 1 28759 Bremen (Germany); University of California Berkeley Department of Mathematics Berkeley CA 94720 (USA)},
author = {Penkov, Ivan, Serganova, Vera},
journal = {Annales de l’institut Fourier},
keywords = {Generalized Harish-Chandra module; bounded $(\mathfrak\{g\},\mathfrak\{k\})$-module; generalized Harish-Chandra module; bounded -module},
language = {eng},
number = {2},
pages = {477-496},
publisher = {Association des Annales de l’institut Fourier},
title = {On bounded generalized Harish-Chandra modules},
url = {http://eudml.org/doc/251052},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Penkov, Ivan
AU - Serganova, Vera
TI - On bounded generalized Harish-Chandra modules
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 477
EP - 496
AB - Let $\mathfrak{g}$ be a complex reductive Lie algebra and $\mathfrak{k}\subset \mathfrak{g}$ be any reductive in $\mathfrak{g}$ subalgebra. We call a $(\mathfrak{g},\mathfrak{k})$-module $M$ bounded if the $\mathfrak{k}$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(\mathfrak{g},\mathfrak{k})$-modules. We prove a strong necessary condition for a subalgebra $\mathfrak{k}$ to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded $(\mathfrak{g},\mathfrak{k})$-module, and then establish a sufficient condition for a subalgebra $\mathfrak{k}$ to be bounded (Theorem 5.1). As a result we are able to classify the maximal bounded reductive subalgebras of $\mathfrak{g}=\mathrm{sl}(n)$.
LA - eng
KW - Generalized Harish-Chandra module; bounded $(\mathfrak{g},\mathfrak{k})$-module; generalized Harish-Chandra module; bounded -module
UR - http://eudml.org/doc/251052
ER -

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