Is the Luna stratification intrinsic?
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, p. 689-721

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.

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.

DOI : https://doi.org/10.5802/aif.2365
Classification:  14R20,  14L30,  14B05
Keywords: Categorical quotient, Luna stratification, matrix invariant, representation type
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     author = {Kuttler, Jochen and Reichstein, Zinovy},
     title = {Is the Luna stratification intrinsic?},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {689-721},
     doi = {10.5802/aif.2365},
     mrnumber = {2410387},
     zbl = {1145.14047},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_689_0}
}
Kuttler, Jochen; Reichstein, Zinovy. Is the Luna stratification intrinsic?. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 689-721. doi : 10.5802/aif.2365. http://www.numdam.org/item/AIF_2008__58_2_689_0/

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