Decomposition of reductive regular Prehomogeneous Vector Spaces

Hubert Rubenthaler[1]

  • [1] Université de Strasbourg et CNRS Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex (France)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 2183-2218
  • ISSN: 0373-0956

Abstract

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Let ( G , V ) be a regular prehomogeneous vector space (abbreviated to P V ), where G is a reductive algebraic group over ℂ . If V = ⊕ i = 1 n V i is a decomposition of V into irreducible representations, then, in general, the PV’s ( G , V i ) are no longer regular. In this paper we introduce the notion of quasi-irreducible P V (abbreviated to Q -irreducible), and show first that for completely Q -reducible P V ’s, the Q -isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of Q -irreducible P V ’s. Finally we classify the Q -irreducible PV’s of parabolic type.

How to cite

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Rubenthaler, Hubert. "Decomposition of reductive regular Prehomogeneous Vector Spaces." Annales de l’institut Fourier 61.5 (2011): 2183-2218. <http://eudml.org/doc/219784>.

@article{Rubenthaler2011,
abstract = {Let $(G,V)$ be a regular prehomogeneous vector space (abbreviated to $PV$), where $G$ is a reductive algebraic group over $\mathbb\{C\}$. If $V= \oplus _\{i=1\}^\{n\}V_\{i\}$ is a decomposition of $V$ into irreducible representations, then, in general, the PV’s $(G,V_\{i\})$ are no longer regular. In this paper we introduce the notion of quasi-irreducible $PV$ (abbreviated to $Q$-irreducible), and show first that for completely $Q$-reducible $PV$’s, the $Q$-isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of $Q$-irreducible $PV$’s. Finally we classify the $Q$-irreducible PV’s of parabolic type.},
affiliation = {Université de Strasbourg et CNRS Institut de Recherche Mathématique Avancée 7 rue René Descartes 67084 Strasbourg Cedex (France)},
author = {Rubenthaler, Hubert},
journal = {Annales de l’institut Fourier},
keywords = {reductive groups; prehomogeneous vector spaces; relative invariants; prehomogeneous vector spaces of parabolic type},
language = {eng},
number = {5},
pages = {2183-2218},
publisher = {Association des Annales de l’institut Fourier},
title = {Decomposition of reductive regular Prehomogeneous Vector Spaces},
url = {http://eudml.org/doc/219784},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Rubenthaler, Hubert
TI - Decomposition of reductive regular Prehomogeneous Vector Spaces
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 2183
EP - 2218
AB - Let $(G,V)$ be a regular prehomogeneous vector space (abbreviated to $PV$), where $G$ is a reductive algebraic group over $\mathbb{C}$. If $V= \oplus _{i=1}^{n}V_{i}$ is a decomposition of $V$ into irreducible representations, then, in general, the PV’s $(G,V_{i})$ are no longer regular. In this paper we introduce the notion of quasi-irreducible $PV$ (abbreviated to $Q$-irreducible), and show first that for completely $Q$-reducible $PV$’s, the $Q$-isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of $Q$-irreducible $PV$’s. Finally we classify the $Q$-irreducible PV’s of parabolic type.
LA - eng
KW - reductive groups; prehomogeneous vector spaces; relative invariants; prehomogeneous vector spaces of parabolic type
UR - http://eudml.org/doc/219784
ER -

References

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