Kuttler, Jochen; Reichstein, Zinovy
Is the Luna stratification intrinsic?  [ La stratification de Luna, est-elle intrinsèque ? ]
Annales de l'institut Fourier, Tome 58 (2008) no. 2 , p. 689-721
MR 2410387 | Zbl 1145.14047
doi : 10.5802/aif.2365
URL stable : http://www.numdam.org/item?id=AIF_2008__58_2_689_0

Classification:  14R20,  14L30,  14B05
Mots clés: quotient catégorique, stratification de Luna, invariants de matrices, type de representation
Soit GGL(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 GGL(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.

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