Is the Luna stratification intrinsic?

Kuttler, Jochen; Reichstein, Zinovy

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 689-721.

Soit $G\to GL\left(V\right)$ 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 $G\to GL\left(V\right)$ 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.

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

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

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

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

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

[6] Formanek, E. The polynomial identities and invariants of $n\times n$ matrices., CBMS Regional Conference Series in Mathematics, Volume 78 (1991) | MR | Zbl

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

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

[9] Kuttler, J.; Reichstein, Z. Is the Luna stratification intrinsic? ( math.AG/0610669)

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

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

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

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

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

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

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

[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., Volume 34 (1970), pp. 523-531 English transl.: Math. USSR-Izv. 4 (1971), pp. 527–535 | MR | Zbl

[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

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

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

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

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

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

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

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

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

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