Decomposition of reductive regular Prehomogeneous Vector Spaces  [ Décomposition des espaces préhomogènes réductifs réguliers ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2183-2218.

Soit (G,V) un espace préhomogène (en abrégé PV) régulier, où G est un groupe algébrique réductif, défini sur . Si V= i=1 n V i est une décomposition de V en représentations irréductibles, alors, en général, les espaces préhomogènes (G,V i ) ne sont pas réguliers. Dans cet article nous introduisons la notion de PV quasi-irréductible (en abrégé Q-irréducible), et nous montrons d’abord que pour les PV complètement Q-réductibles, les composantes Q-isotypiques sont définies de manière intrinsèque, comme en théorie ordinaire des représentations. Nous montrons également que, dans un sens approprié, tout PV régulier est une somme directe de PV quasi-irréductibles. Finalement nous classifions les PV de type parabolique qui sont Q-irréductibles.

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a reductive algebraic group over . 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 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.

DOI : https://doi.org/10.5802/aif.2670
Classification : 11S90,  20G05,  17B20
Mots 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'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     doi = {10.5802/aif.2670},
     mrnumber = {2961852},
     zbl = {1250.11100},
     language = {en},
     url = {www.numdam.org/item/AIF_2011__61_5_2183_0/}
}
Rubenthaler, Hubert. Decomposition of reductive regular Prehomogeneous Vector Spaces. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2183-2218. doi : 10.5802/aif.2670. http://www.numdam.org/item/AIF_2011__61_5_2183_0/

[1] Bopp, Nicole; Rubenthaler, Hubert 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 2115051 | Zbl 1076.11059

[2] Dynkin, E. B. Semi-simple subalgebras of semi-simple Lie algebras, Amer. Math. Soc. Transl., Volume 6 (1957), pp. 111-224 | Zbl 0077.03404

[3] Kimura, Tatsuo A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplications, J. Algebra, Volume 83 (1983) no. 1, pp. 72-100 | Article | MR 710588 | Zbl 0533.14024

[4] Kimura, Tatsuo Introduction to prehomogeneous vector spaces, Translations of Mathematical Monographs, Volume 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 1944442 | Zbl 1035.11060

[5] Kimura, Tatsuo; Kasai, S.; Hosokawa, H. Universal transitivity of simple and 2-simple prehomogeneous vector spaces, Ann. Inst. Fourier (Grenoble), Volume 38 (1988) no. 2, pp. 11-41 | Article | Numdam | MR 949009 | Zbl 0606.14037

[6] Kimura, Tatsuo; Kasai, Shin-ichi; Inuzuka, Masaaki; Yasukura, Osami A classification of 2-simple prehomogeneous vector spaces of type I, J. Algebra, Volume 114 (1988) no. 2, pp. 369-400 | Article | MR 936979 | Zbl 0658.20026

[7] Kimura, Tatsuo; Kasai, Shin-ichi; Taguchi, Masanobu; Inuzuka, Masaaki Some P.V.-equivalences and a classification of 2-simple prehomogeneous vector spaces of type II , Trans. Amer. Math. Soc., Volume 308 (1988) no. 2, pp. 433-494 | Article | MR 951617 | Zbl 0666.14021

[8] Mortajine, A. Classification des espaces préhomogènes de type parabolique réguliers et de leurs invariants relatifs, Travaux en Cours [Works in Progress], Volume 40, Hermann, Paris, 1991 | MR 1096494 | Zbl 0717.22007

[9] Rubenthaler, Hubert Espaces vectoriels préhomogènes, sous-groupes paraboliques et 𝔰𝔩 2 -triplets, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980) no. 3, p. A127-A129 | MR 563957 | Zbl 0428.22011

[10] Rubenthaler, Hubert 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 683471 | Zbl 0561.17005

[11] Rubenthaler, Hubert Espaces préhomogènes de type parabolique, Université de Strasbourg, 1982 (Thèse d’État) | MR 683471 | Zbl 0546.22019

[12] Rubenthaler, Hubert Algèbres de Lie et espaces préhomogènes, Travaux en Cours [Works in Progress], Volume 44, Hermann Éditeurs des Sciences et des Arts, Paris, 1992 (With a foreword by Jean-Michel Lemaire) | MR 2412335 | Zbl 0840.17007

[13] Saito, Hiroshi 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 | Article | MR 1994885 | Zbl 1045.11083

[14] Sato, Fumihiro 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 | Article | MR 662121 | Zbl 0497.10012

[15] Sato, Fumihiro Zeta functions with polynomial coefficients associated with prehomogeneous vector spaces, Comment. Math. Univ. St. Paul., Volume 45 (1996) no. 2, pp. 177-211 | MR 1416190 | Zbl 0874.11063

[16] Sato, Mikio 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 1086566 | Zbl 0715.22014

[17] Sato, Mikio; Kimura, Tatsuo A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., Volume 65 (1977), pp. 1-155 | MR 430336 | Zbl 0321.14030

[18] Sato, Mikio; Shintani, Takuro On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2), Volume 100 (1974), pp. 131-170 | Article | MR 344230 | Zbl 0309.10014