no. 2
Is the Luna stratification intrinsic?
[La stratification de Luna, est-elle intrinsèque ?]
Kuttler, Jochen1 ; Reichstein, Zinovy2
1 University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 (Canada) Current address: University of Alberta Department of Mathematical and Statistical Sciences Edmonton, AB T6G 2G1 (Canada)
2 University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 (Canada)
Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 689-721.

Soit G→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→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.

DOI : 10.5802/aif.2365
Classification : 14R20, 14L30, 14B05
Keywords: Categorical quotient, Luna stratification, matrix invariant, representation type
Mot clés : quotient catégorique, stratification de Luna, invariants de matrices, type de representation

Kuttler, Jochen 1 ; Reichstein, Zinovy 2

1 University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 (Canada) Current address: University of Alberta Department of Mathematical and Statistical Sciences Edmonton, AB T6G 2G1 (Canada)
2 University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 (Canada) 
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Kuttler, Jochen; Reichstein, Zinovy. Is the Luna stratification intrinsic?. Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 689-721. doi : 10.5802/aif.2365. https://aif.centre-mersenne.org/articles/10.5802/aif.2365/

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