# zbMATH — the first resource for mathematics

Is the Luna stratification intrinsic? (English) Zbl 1145.14047
Let $$V$$ be a finite dimensional $$k$$-vector space where $$k$$ is an algebraically closed field of characteristic zero, and $$G\to GL(V)$$ a representation of a reductive linear algebraic group. The categorical quotient, $$X=V//G$$ has a stratification, defined by D. Luna, where the strata are given as follows. Let $$\pi: V\to X$$ be the quotient map. For each point $$p\in X$$, the fibre $$\pi^{-1}(p)$$ has a unique closed orbit. Let $$v_p$$ be a point on such an orbit, and let $$H$$ be the isotropy group of $$v_p$$. Denote by $$(H)$$ the conjugacy class of $$H$$. Then a stratum in $$X$$ is the set $$X^{(H)}=\{p\in X\mid Stab(v_p)\in(H)\}$$. There are finitely many strata, and they are locally closed. In this article, the authors consider the following questions: (i) Does every automorphism of $$X$$ map each stratum into another stratum? (Is the stratification intrinsic?) (2) Does every automorphism of $$X$$ preserve each stratum? (Are the individual strata intrinsic?)
In general, the answer to both questions is negative. For example, there are many cases where the quotient $$X$$ is affine space, in which case the group of automorphisms is transitive. However, under some conditions, there are positive results to these questions. For example, as described in the present article, if $$G$$ is finite and contains no pseudo-groups, the answer to the first question is positive, and under some other assumptions, one can show that the strata are intrinsic. For infinite reductive groups, the authors give conditions under which the Luna stratification is intrinsic. Also for a certain family of representations, they show that even the strata are intrinsic. More precisely, it is shown that if $$G\to GL(W)$$ is a finite-dimensional representation of a reductive algebraic group $$G$$ and $$V=W^r$$, where any of the following three conditions hold : (i) $$r\geq 2\dim (W)$$; or (ii) $$G$$ preserves a nondegenerate quadratic form on $$W$$ and $$r\geq\dim(W)+1$$; or (iii) $$W={\mathbf g}$$ is the adjoint representation of $$G$$ and $$r\geq3$$, then the Luna stratification of $$X=V//G$$ is intrinsic. As for the second question, if $$V=M_n^r$$ is the space of $$r$$-tuples of $$n\times n$$ matrices, with $$r\geq 3$$, and $$G=GL_n$$ acts on $$V$$ by simultaneous conjugation, then every Luna stratum of the categorical quotient $$V//GL_n$$ is intrinsic.

##### MSC:
 14R20 Group actions on affine varieties 14L40 Other algebraic groups (geometric aspects) 14B05 Singularities in algebraic geometry
Full Text:
##### References:
  Artin, M., On Azumaya algebras and finite dimensional representations of rings, J. Algebra, 11, 532-563, (1969) · Zbl 0222.16007  Bass, H.; Haboush, W., Linearizing certain reductive group actions, Trans. Amer. Math. Soc., 292, 2, 463-482, (1985) · Zbl 0602.14047  Borel, A., Linear Algebraic Groups, 126, (1991), Springer-Verlag, New York · Zbl 0726.20030  Colliot-Thélène, J.-L.; Sansuc, J.-J., Fibrés quadratiques et composantes connexes réelles, Math. Ann., 244, 2, 105-134, (1979) · Zbl 0418.14016  Drensky, V.; Formanek, E., Polynomial identity rings, (2004), Birkhäuser Verlag, Basel · Zbl 1077.16025  Formanek, E., The polynomial identities and invariants of $$n× n$$ matrices., CBMS Regional Conference Series in Mathematics, 78, (1991), American Mathematical Society, Providence, RI · Zbl 0714.16001  Grace, J. H.; Young, A., The Algebra of Invariants, (1903), Cambridge University Press · JFM 34.0114.01  Kraft, H., Geometrische Methoden in der Invariantentheorie, (1984), Friedr. Vieweg & Sohn, Braunschweig · Zbl 0569.14003  Kuttler, J.; Reichstein, Z., Is the luna stratification intrinsic? · Zbl 1145.14047  Le Bruyn, L.; Procesi, C., Étale local structure of matrix invariants and concomitants, in Algebraic groups Utrecht 1986, Lecture Notes in Math., 1271, 143-175, (1987) · Zbl 0634.14034  Le Bruyn, L.; Reichstein, Z., Smoothness in algebraic geography, Proc. London Math. Soc. (3), Lecture Notes in Math., 79, 1, 158-190, (1999) · Zbl 1032.16012  Lorenz, M., On the Cohen-Macaulay property of multiplicative invariants, Trans. Amer. Math. Soc., 358, 4, 1605-1617, (2006) · Zbl 1129.13005  Luna, D., Sur les groupes algébriques, Slices étales, 81-105, (1973), Soc. Math. France, Mémoire 33, Paris · Zbl 0286.14014  Luna, D.; Richardson, R. W., A generalization of the Chevalley restriction theorem, Duke Math. J., 46, 3, 487-496, (1979) · Zbl 0444.14010  Mumford, D., The red book of varieties and schemes, 1358, (1988), Springer-Verlag, Berlin · Zbl 0658.14001  Mumford, D.; Fogarty, J.; Kirwan, F., Geometric invariant theory, (1994), Springer-Verlag, Berlin · Zbl 0797.14004  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., 34, 523-531, (1970) · Zbl 0261.14011  Popov, V. L., Generically multiple transitive algebraic group actions, Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces, (2004) · Zbl 1135.14038  Popov, V. L.; Vinberg, E. B., Invariant theory, Algebraic Geometry IV, Encyclopedia of Mathematical Sciences, Springer, 55, 123-284, (1994) · Zbl 0789.14008  Prill, D., Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J., 34, 375-386, (1967) · Zbl 0179.12301  Procesi, C., The invariant theory of $$n× n$$ matrices, Advances in Math., 19, 3, 306-381, (1976) · Zbl 0331.15021  Reichstein, Z., On automorphisms of matrix invariants, Trans. Amer. Math. Soc., 340, 1, 353-371, (1993) · Zbl 0820.16021  Reichstein, Z., On automorphisms of matrix invariants induced from the trace ring, Linear Algebra Appl., 193, 51-74, (1993) · Zbl 0802.16017  Reichstein, Z.; Vonessen, N., Group actions on central simple algebras: a geometric approach, J. Algebra, 304, 2, 1160-1192, (2006) · Zbl 1112.16021  Richardson, R. W.; Jr., Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math., 16, 6-14, (1972) · Zbl 0242.14010  Richardson, R. W.; Jr., Conjugacy classes of $$n$$-tuples in Lie algebras and algebraic groups, Duke Math J., 57, 1, 1-35, (1988) · Zbl 0685.20035  Schwarz, G. W., Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., 51, 37-135, (1980) · Zbl 0449.57009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.