Let be a regular prehomogeneous vector space (abbreviated to ), where is a reductive algebraic group over . If is a decomposition of into irreducible representations, then, in general, the PV’s are no longer regular. In this paper we introduce the notion of quasi-irreducible (abbreviated to -irreducible), and show first that for completely -reducible ’s, the -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 -irreducible ’s. Finally we classify the -irreducible PV’s of parabolic type.
Soit un espace préhomogène (en abrégé ) régulier, où est un groupe algébrique réductif, défini sur . Si est une décomposition de en représentations irréductibles, alors, en général, les espaces préhomogènes ne sont pas réguliers. Dans cet article nous introduisons la notion de quasi-irréductible (en abrégé -irréducible), et nous montrons d’abord que pour les complètement -réductibles, les composantes -isotypiques sont définies de manière intrinsèque, comme en théorie ordinaire des représentations. Nous montrons également que, dans un sens approprié, tout régulier est une somme directe de quasi-irréductibles. Finalement nous classifions les de type parabolique qui sont -irréductibles.
Keywords: reductive groups, prehomogeneous vector spaces, relative invariants, prehomogeneous vector spaces of parabolic type
Mot clés : Groupes réductifs, espaces préhomogènes, invariants relatifs, espaces préhomogènes de type parabolique
@article{AIF_2011__61_5_2183_0, author = {Rubenthaler, Hubert}, title = {Decomposition of reductive regular {Prehomogeneous} {Vector} {Spaces}}, journal = {Annales de l'Institut Fourier}, pages = {2183--2218}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {5}, year = {2011}, doi = {10.5802/aif.2670}, zbl = {1250.11100}, mrnumber = {2961852}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2670/} }
TY - JOUR AU - Rubenthaler, Hubert TI - Decomposition of reductive regular Prehomogeneous Vector Spaces JO - Annales de l'Institut Fourier PY - 2011 SP - 2183 EP - 2218 VL - 61 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2670/ DO - 10.5802/aif.2670 LA - en ID - AIF_2011__61_5_2183_0 ER -
%0 Journal Article %A Rubenthaler, Hubert %T Decomposition of reductive regular Prehomogeneous Vector Spaces %J Annales de l'Institut Fourier %D 2011 %P 2183-2218 %V 61 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2670/ %R 10.5802/aif.2670 %G en %F AIF_2011__61_5_2183_0
Rubenthaler, Hubert. Decomposition of reductive regular Prehomogeneous Vector Spaces. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2183-2218. doi : 10.5802/aif.2670. http://www.numdam.org/articles/10.5802/aif.2670/
[1] Local zeta functions attached to the minimal spherical series for a class of symmetric spaces, Mem. Amer. Math. Soc., Volume 174 (2005) no. 821, pp. viii+233 | MR | Zbl
[2] Semi-simple subalgebras of semi-simple Lie algebras, Amer. Math. Soc. Transl., Volume 6 (1957), pp. 111-224 | Zbl
[3] A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications, J. Algebra, Volume 83 (1983) no. 1, pp. 72-100 | DOI | MR | Zbl
[4] Introduction to prehomogeneous vector spaces, Translations of Mathematical Monographs, 215, American Mathematical Society, Providence, RI, 2003 (Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author) | MR | Zbl
[5] Universal transitivity of simple and -simple prehomogeneous vector spaces, Ann. Inst. Fourier (Grenoble), Volume 38 (1988) no. 2, pp. 11-41 | DOI | Numdam | MR | Zbl
[6] A classification of -simple prehomogeneous vector spaces of type , J. Algebra, Volume 114 (1988) no. 2, pp. 369-400 | DOI | MR | Zbl
[7] Some P.V.-equivalences and a classification of -simple prehomogeneous vector spaces of type , Trans. Amer. Math. Soc., Volume 308 (1988) no. 2, pp. 433-494 | DOI | MR | Zbl
[8] Classification des espaces préhomogènes de type parabolique réguliers et de leurs invariants relatifs, Travaux en Cours [Works in Progress], 40, Hermann, Paris, 1991 | MR | Zbl
[9] Espaces vectoriels préhomogènes, sous-groupes paraboliques et -triplets, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980) no. 3, p. A127-A129 | MR | Zbl
[10] Espaces préhomogènes de type parabolique, Lectures on harmonic analysis on Lie groups and related topics (Strasbourg, 1979) (Lectures in Math.), Volume 14, Kinokuniya Book Store, Tokyo, 1982, pp. 189-221 | MR | Zbl
[11] Espaces préhomogènes de type parabolique, Université de Strasbourg, 1982 (Thèse d’État) | MR | Zbl
[12] Algèbres de Lie et espaces préhomogènes, Travaux en Cours [Works in Progress], 44, Hermann Éditeurs des Sciences et des Arts, Paris, 1992 (With a foreword by Jean-Michel Lemaire) | MR | Zbl
[13] Convergence of the zeta functions of prehomogeneous vector spaces, Nagoya Math. J., Volume 170 (2003), pp. 1-31 http://projecteuclid.org/getRecord?id=euclid.nmj/1114631874 | DOI | MR | Zbl
[14] Zeta functions in several variables associated with prehomogeneous vector spaces. III. Eisenstein series for indefinite quadratic forms, Ann. of Math. (2), Volume 116 (1982) no. 1, pp. 77-99 | DOI | MR | Zbl
[15] Zeta functions with polynomial coefficients associated with prehomogeneous vector spaces, Comment. Math. Univ. St. Paul., Volume 45 (1996) no. 2, pp. 177-211 | MR | Zbl
[16] Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J., Volume 120 (1990), pp. 1-34 http://projecteuclid.org/getRecord?id=euclid.nmj/1118782193 (Notes by Takuro Shintani, Translated from the Japanese by Masakazu Muro) | MR | Zbl
[17] A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., Volume 65 (1977), pp. 1-155 | MR | Zbl
[18] On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2), Volume 100 (1974), pp. 131-170 | DOI | MR | Zbl
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