Is the Luna stratification intrinsic?

Kuttler, Jochen; Reichstein, Zinovy

Is the Luna stratification intrinsic?
[La stratification de Luna, est-elle intrinsèque ?]

Annales de l'Institut Fourier, Tome 58 (2008), p. 689-721 / Harvested from Numdam

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Résumé
Abstract

Soit $G \to GL (V)$ une représentation d’un groupe algébrique réductif $G$ , définie sur un corps algébraiquement clos de caractéristique zéro. D’après D. Luna, le quotient catégorique $X = V // G$ comporte une stratification naturelle. L’article présente les deux questions suivantes :

(i) La stratification de $X$ est-elle intrinsèque ? Plus précisément, l’image d’une strate par un automorphisme de $X$ quelconque est-elle avec strate ?

(ii) Les strates individuelles de $X$ , sont-elles intrinsèques ? C’est-à-dire, est-il vrai que toute strate est invariante par tous les automorphismes de $X$ ?

D’une manière générale, la stratification de Luna n’est pas intrinsèque. Néanmoins, pour des familles de représentations intéressantes les questions (i) et (ii) ont des réponses positives.

Let $G \to GL (V)$ be a representation of a reductive linear algebraic group $G$ on a finite-dimensional vector space $V$ , defined over an algebraically closed field of characteristic zero. The categorical quotient $X = V // G$ carries a natural stratification, due to D. Luna. This paper addresses the following questions:

(i) Is the Luna stratification of $X$ intrinsic? That is, does every automorphism of $V // G$ map each stratum to another stratum?

(ii) Are the individual Luna strata in $X$ intrinsic? That is, does every automorphism of $V // G$ map each stratum to itself?

In general, the Luna stratification is not intrinsic. Nevertheless, we give positive answers to questions (i) and (ii) for interesting families of representations.

Publié le : 2008-01-01

MR 2410387 | Zbl 1145.14047

DOI : https://doi.org/10.5802/aif.2365
Classification: 14R20, 14L30, 14B05
Mots clés: quotient catégorique, stratification de Luna, invariants de matrices, type de representation

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     title = {Is the Luna stratification intrinsic?},
     journal = {Annales de l'Institut Fourier},
     volume = {58},
     year = {2008},
     pages = {689-721},
     doi = {10.5802/aif.2365},
     zbl = {1145.14047},
     mrnumber = {2410387},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2008__58_2_689_0}
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Kuttler, Jochen; Reichstein, Zinovy. Is the Luna stratification intrinsic?. Annales de l'Institut Fourier, Tome 58 (2008) pp. 689-721. doi : 10.5802/aif.2365. http://gdmltest.u-ga.fr/item/AIF_2008__58_2_689_0/

Bibliographie

[1] Artin, M. On Azumaya algebras and finite dimensional representations of rings, J. Algebra, Tome 11 (1969), pp. 532-563 | Article | MR 242890 | Zbl 0222.16007

[2] Bass, H.; Haboush, W. Linearizing certain reductive group actions, Trans. Amer. Math. Soc., Tome 292 (1985) no. 2, pp. 463-482 | Article | MR 808732 | Zbl 0602.14047

[3] Borel, A. Linear Algebraic Groups, Springer-Verlag, New York, Second edition. Graduate Texts in Mathematics, Tome 126 (1991) | MR 1102012 | Zbl 0726.20030

[4] Colliot-Thélène, J.-L.; Sansuc, J.-J. Fibrés quadratiques et composantes connexes réelles, Math. Ann., Tome 244 (1979) no. 2, pp. 105-134 | Article | MR 550842 | Zbl 0418.14016

[5] Drensky, V.; Formanek, E. Polynomial identity rings, Birkhäuser Verlag, Basel, Advanced Courses in Mathematics – CRM Barcelona (2004) | MR 2064082 | Zbl 1077.16025

[6] Formanek, E. The polynomial identities and invariants of $n \times n$ matrices., CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, Tome 78 (1991) | MR 1088481 | Zbl 0714.16001

[7] Grace, J. H.; Young, A. The Algebra of Invariants, Cambridge University Press (1903)

[8] Kraft, H. Geometrische Methoden in der Invariantentheorie, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics, D1 (1984) | MR 768181 | Zbl 0569.14003

[9] Kuttler, J.; Reichstein, Z. Is the Luna stratification intrinsic? ()

[10] Le Bruyn, L.; Procesi, C. Étale local structure of matrix invariants and concomitants, in Algebraic groups Utrecht 1986, Lecture Notes in Math., Tome 1271 (1987), pp. 143-175 | Article | MR 911138 | Zbl 0634.14034

[11] Le Bruyn, L.; Reichstein, Z. Smoothness in algebraic geography, Proc. London Math. Soc. (3), Lecture Notes in Math., Tome 79 (1999) no. 1, pp. 158-190 | Article | MR 1687535 | Zbl 1032.16012

[12] Lorenz, M. On the Cohen-Macaulay property of multiplicative invariants, Trans. Amer. Math. Soc., Tome 358 (2006) no. 4, pp. 1605-1617 | Article | MR 2186988 | Zbl 02242546

[13] Luna, D. Slices étales, Sur les groupes algébriques, Soc. Math. France, Mémoire 33, Paris (1973), pp. 81-105 | Numdam | MR 318167

[14] Luna, D.; Richardson, R. W. A generalization of the Chevalley restriction theorem, Duke Math. J., Tome 46 (1979) no. 3, pp. 487-496 | Article | MR 544240 | Zbl 0444.14010

[15] Mumford, D. The red book of varieties and schemes, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1358 (1988) | MR 971985 | Zbl 0658.14001

[16] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, Berlin (1994) | MR 1304906 | Zbl 0797.14004

[17] Popov, V. L. Criteria for the stability of the action of a semisimple group on a factorial manifold, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., Tome 34 (1970), pp. 523-531 (English transl.: Math. USSR-Izv. 4 (1971), pp. 527–535) | MR 262416 | Zbl 0261.14011

[18] Popov, V. L. Generically multiple transitive algebraic group actions, Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces (2004) (TIFR, Mumbai, India, to appear. Preprint available at www.arxiv.org/math.AG/0409024) | Zbl 1135.14038

[19] Popov, V. L.; Vinberg, E. B. Invariant Theory, Algebraic Geometry IV, Encyclopedia of Mathematical Sciences, Springer, Tome 55 (1994), pp. 123-284 | Zbl 0789.14008

[20] Prill, D. Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J., Tome 34 (1967), pp. 375-386 | Article | MR 210944 | Zbl 0179.12301

[21] Procesi, C. The invariant theory of $n \times n$ matrices, Advances in Math., Tome 19 (1976) no. 3, pp. 306-381 | Article | MR 419491 | Zbl 0331.15021

[22] Reichstein, Z. On automorphisms of matrix invariants, Trans. Amer. Math. Soc., Tome 340 (1993) no. 1, pp. 353-371 | Article | MR 1124173 | Zbl 0820.16021

[23] Reichstein, Z. On automorphisms of matrix invariants induced from the trace ring, Linear Algebra Appl., Tome 193 (1993), pp. 51-74 | Article | MR 1240272 | Zbl 0802.16017

[24] Reichstein, Z.; Vonessen, N. Group actions on central simple algebras: a geometric approach, J. Algebra, Tome 304 (2006) no. 2, pp. 1160-1192 | Article | MR 2265511 | Zbl 05077840

[25] Richardson, R. W.; Jr. Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math., Tome 16 (1972), pp. 6-14 | Article | MR 294336 | Zbl 0242.14010

[26] Richardson, R. W.; Jr. Conjugacy classes of $n$ -tuples in Lie algebras and algebraic groups, Duke Math J., Tome 57 (1988) no. 1, pp. 1-35 | Article | MR 952224 | Zbl 0685.20035

[27] Schwarz, G. W. Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., Tome 51 (1980), pp. 37-135 | Article | Numdam | MR 573821 | Zbl 0449.57009