# zbMATH — the first resource for mathematics

Reductive group actions with one-dimensional quotient. (English) Zbl 0783.14026
Let $$G$$ be a reductive complex algebraic group acting algebraically on the complex affine $$n$$-dimensional space $$X:=\mathbb{A}^ n$$. This paper arose out of the attempt to solve the linearizability problem (i.e. whether there is a $$G$$-equivariant isomorphism between $$X$$ and a $$G$$- module) in the case that $$\dim X//G=1$$. It contains a number of results on the classification of such actions. In particular, this work has led to the first examples of the nonlinearizable actions [G. W. Schwarz, C. R. Acad. Sci., Paris, Sér. I 309, No. 2, 89-94 (1989; Zbl 0688.14040)].
Let, more generally, $$X$$ be a smooth affine acyclic variety. It is proved that $$X//G \cong \mathbb{A}$$, the affine line, and either every closed orbit is a fixed point (the action is fixed-pointed) or there is a unique fixed point $$x_ 0 \in X$$. Since in the fix-pointed case it is known that one has linearizability, the second possibility is considered. It is proved that if one allows holomorphic equivalence, these actions are always linearizable. Let $$V$$ be the tangent $$G$$-module at $$x_ 0$$. It is proved that $$V^ G=\{0\}$$, $$V//G \simeq \mathbb{A}$$ and the quotient mapping $$\pi_ V:V \to \mathbb{A}$$, $$\pi_ V(0)=0$$, is given by a homogeneous polynomial. Denote its degree by $$d$$. Denote by $${\mathcal M}_{V,\mathbb{A}}$$ the set of isomorphism classes of smooth acyclic affine $$G$$-varieties $$X$$ with fixed quotient mapping $$\pi_ X:X \to \mathbb{A} \simeq X//G$$ such that (a) $$X^ G=\{x_ 0\}$$ is a single fixed point; (b) the tangent $$G$$-module at $$x_ 0$$ is isomorphic with $$V$$; (c) $$\pi_ X(x_ 0)=0 \in \mathbb{A}$$. – The isomorphism class containing $$X$$ is denoted by $$\{X\}$$. If $$\{X\} \in {\mathcal M}_{V,\mathbb{A}}$$, it is proved that
(1) the restrictions of the quotient mappings $$\pi_ X^{-1} \bigl( \mathbb{A} \backslash \{0\} \bigr) \to \mathbb{A} \backslash \{0\}$$ and $$\pi_ V^{-1} \bigl( \mathbb{A} \backslash \{0\} \bigr) \to \mathbb{A} \backslash \{0\}$$ are isomorphic $$G$$-fiber bundles with fiber $$F:=\pi_ V^{-1}(1)$$;
(2) $$\pi_ X^{-1}(U)$$ and $$\pi_ V^{-1}(U)$$ are $$G$$-isomorphic for some open set $$U\subset \mathbb{A}$$, $$0\in U$$;
(3) $$X$$ is obtained from $$\pi_ V^{-1} \bigl( \mathbb{A} \backslash \{0\} \bigr)$$ and $$\pi_ V^{-1}(U)$$ identified by a $$G$$-isomorphism over $$U\backslash \{0\}$$.
(4) there is a bijection $${\mathcal M}_{V,\mathbb{A}}{\overset\sim{}}D/ \Gamma$$ where $$\Gamma$$ denotes the $$d$$-th roots of unity and $${\mathcal D}$$ is a $$\Gamma$$-module (which makes it possible to endow $${\mathcal M}_{V,\mathbb{A}}$$ with the structure of affine variety coming from this bijection).
The $$D$$-module $$\Gamma$$ is determined, thus describing the moduli space $${\mathcal M}_{V,\mathbb{A}}$$. The criteria is given for $${\mathcal D}$$ to be trivial in terms of the Lie algebra of the group of $$G$$-equivariant automorphisms of the fiber $$F$$ and the set of all polynomial vector fields on $$V$$ annihilating $$\pi_ V$$. Along these lines it is shown that $${\mathcal M}_{V,\mathbb{A}}$$ is trivial if:
(a) the only closed $$G$$-orbits in $$V$$ are fixed points or have trivial stabilizer, (b) $$G$$ is a torus, (c) $$\dim V^{G^ 0}=1$$, (d) $$\dim V \leq 3$$, (e) $$G^ 0$$ is a simple group or (f) $$V$$ is self dual as $$G^ 0$$-module.
The examples of nontrivial moduli spaces are obtained from $$G$$-vector bundles. Namely, let $$\text{Vec}_ G(P,Q)$$ be the collection of $$G$$- vector bundles whose base point is a $$G$$-module $$P$$ with one-dimensional quotient and whose fiber at $$0\in P$$ is isomorphic to the $$G$$-module $$Q$$. Denote by $$\text{VEC}_ G(P,Q)$$ the set of $$G$$-isomorphism classes in $$\text{Vec}_ G(P,Q)$$, and by $$[E]$$ the class of $$E \in \text{Vec}_ G (P,Q)$$, The trivial class is represented by $$\Theta_ G:=P \times Q$$.
It is proved that $$\text{VEC}_ G(P,Q)$$ has a natural structure of vector group. If $$[E_ i] \in \text{VEC}_ G (P,Q)$$, $$i=1,2$$, the sum $$[E_ 3]:=[E_ 1]+[E_ 2]$$ is uniquely determined by the condition $$E_ 1\oplus E_ 2\simeq E_ 3\oplus\Theta_ G\in\text{Vec}_ G(P,Q\oplus Q)$$. It is proved that the Whitney sum induces an epimorphism of vector groups $\text{VEC}_ G(P,Q_ 1) \times \text{VEC}_ G(P,Q_ 2) \to \text{VEC}_ G(P,Q_ 1\oplus Q_ 2),$ which is bijective if $$\text{Hom} (Q_ 1,Q_ 2)^ H=\{0\}$$, where $$H$$ is a principal stabiliser of $$P$$.
If $$V:=P\oplus Q \simeq \Theta_ G$$, a natural map $$\lambda:\text{VEC}_ G(P,Q)/ \Gamma \to {\mathcal M}_{V,\mathbb{A}}$$, where $$\Gamma$$ is the group of $$d$$-th roots of unity, $$d=\deg \pi_ p$$, is defined. It is shown that if $${\mathcal M}_{P,\mathbb{A}}$$ is trivial then $$\lambda$$ is a bijection. The criterion of triviality of $$\text{VEC}_ G(P,Q)$$ in terms of the Lie algebra of $$\text{Mor} \bigl( \pi_ P^{- 1}(1),\text{GL}(Q) \bigr)^ G$$ is proved. – These results are used to obtain the examples where $$\text{VEC}_ G(P,Q)$$ is nontrivial. In its turn, this is used to obtain examples of nonlinearizable actions. It is shown that:
(a) if $$G$$ is a simple classical group, a spin group, $$G_ 2$$, $$E_ 6$$ or $$E_ 7$$, then $$G$$ has a nonlinearizable faithful action on $$\mathbb{A}^ n$$ for some $$n$$;
(b) there are nonlinearizable actions of $$O_ 2$$ on $$\mathbb{A}^ 4$$, of $$SL_ 2$$ on $$\mathbb{A}^ 7$$ and of $$SO_ 3$$ on $$\mathbb{A}^{10}$$.
Reviewer: V.L.Popov (Moskva)

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations 14L35 Classical groups (algebro-geometric aspects) 14L40 Other algebraic groups (geometric aspects) 20G99 Linear algebraic groups and related topics
Full Text:
##### References:
  H. Bass, W. Haboush, Linearizing certain reductive group actions,Trans. Amer. Math. Soc. 292 (1985), 463–482. · Zbl 0602.14047  H. Bass, W. Haboush, Some equivariant K-theory of affine algebraic group actions,Comm. Algebra 15 (1987), 181–217. · Zbl 0612.14047  G. Bredon,Introduction to Compact Transformation Groups, Pure and Applied Mathematics vol.46, Academic Press, New York, 1972. · Zbl 0246.57017  M. Demazure, P. Gabriel,Groupes Algébriques, Masson, Paris-Amsterdam, 1970.  R. Godement,Topologie Algébrique et Théorie des Faisceaux, Publ. de l’Institut de Math. de l’Université de Strasbourg XIII, Hermann, Paris, 1964.  A. Grothendieck, Torsion homologique et sections rationnelles, inSéminaire Chevalley 1958, exposé no 5.  R. Hartshorne,Algebraic Geometry, Graduate Texts in Math. vol. 52, Springer Verlag, New York-Heidelberg-Berlin, 1977. · Zbl 0367.14001  T. Kambayashi, Automorphism group of a polynomial ring and algebraic group actions on an affine space,J. Algebra 60 (1979), 439–451. · Zbl 0429.14017  F. Knop, Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen,Invent. Math. 105 (1991), 217–220. · Zbl 0739.20019  M. Koras, P. Russell, On linearizing ”good” C*-actions on C3, inGroup Actions and Invariant Theory, Can. Math. Soc. Conf. Proc. vol.10, 1989, pp. 93–102.  M. Koras, P. Russell, Codimension 2 torus actions on affinen-space, inGroup Actions and Invariant Theory, Can. Math. Soc. Conf. Proc. vol.10, 1989, pp. 103–110.  H. Kraft,Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik vol.D1, Vieweg Verlag, Braunschweig, 1984. · Zbl 0569.14003  H. Kraft, Algebraic automorphisms of affine space, inTopological Methods in Algebraic Transformation Groups, (eds. H. Kraft, T. Petrie, G. W. Schwarz), Progress in Mathematics vol.80, Birkhäuser Verlag, Basel-Boston, 1989, pp. 81–105.  H. Kraft, G-vector bundles and the linearization problem, inGroup Actions and Invariant Theory, Can. Math. Soc. Conf. Proc. vol.10, 1989, pp. 111–123.  H. Kraft, C*-actions on affine space, inOperator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Actes du colloque en l’honneur de Jacques Dixmier (eds. A. Connes, M. Duflo, A. Joseph, R. Rentschler), Progress in Mathematics vol.92, Birkhäuser Verlag, Basel-Boston, 1990, pp. 561–579. · Zbl 0747.14014  H. Kraft, T. Petrie, J. Randall, Quotient varieties,Adv. Math. 74 (1989), 145–162. · Zbl 0691.14029  H. Kraft, V. L. Popov, Semisimple group actions on the three dimensional affine space are linear,Comment. Math. Helv. 60 (1985), 446–479. · Zbl 0645.14020  H. Kraft andG. W. Schwarz, Reductive group actions on affine space with one-dimensional quotient, inGroup Actions and Invariant Theory, Can. Math. Soc. Conf. Proc. vol.10, 1989, pp. 125–132.  P. Littelmann,On spherical double cones, to appear in J. Alg. · Zbl 0823.20040  D. Luna, Slices étales,Bull. Soc. Math. France, Mémoire 33 (1973), 81–105.  D. Luna andR. W. Richardson, A generalization of the Chevalley restriction theorem,Duke Math. J. 46 (1979), 487–496. · Zbl 0444.14010  M. Masuda, T. Petrie, Equivariant algebraic vector bundles over representations of reductive groups: Theory,Proc. Nat. Acad. Sci. 88 (1991), 9061–9064. · Zbl 0753.14042  M. Masuda, L. Moser-Jauslin, T. Petrie, Equivariant algebraic vector bundles over representations of reductive groups: Applications,Proc. Nat. Acad. Sci. 88 (1991), 9065–9066. · Zbl 0753.14005  D. I. Panyushev, Semisimple automorphism groups of four-dimensional affine space,Math. USSR Izv. 23 (1984), 171–183. · Zbl 0581.14033  G. W. Schwarz, Representations of simple Lie groups with regular rings of invariants,Invent. Math. 49 (1978), 176–191. · Zbl 0391.20032  G. W. Schwarz, Representations of simple Lie groups with a free module of covariants,Invent. Math. 50 (1978), 1–12. · Zbl 0391.20033  G. W. Schwarz, Lifting smooth homotopies of orbit spaces,Publ. Math. IHES 51 (1980), 37–135. · Zbl 0449.57009  G. W. Schwarz, The topology of algebraic quotients, inTopological Methods in Algebraic Transformation Groups, (eds. H. Kraft, T. Petrie, G. W. Schwarz), Progress in Mathematics vol.10, Birkhäuser Verlag, Basel-Boston, 1989, pp. 135–152.  G. W. Schwarz, Exotic algebraic group actions,C.R. Acad. Sci. Paris 309 (1989), 89–94. · Zbl 0688.14040  J.-P. Serre,Cohomologie Galoisienne, Lecture Notes in Mathematics vol.5, Springer Verlag, Berlin-Heidelberg-New York, 1965.  J.-P. Serre,Corps Locaux, Publ. de l’Institut de Math. de l’Université de Nancago VIII, Hermann, Paris, 1968.  J.-P. serre, Espaces fibrés algébriques, inSéminaire Chevalley 1958, exposé no 5.  P. Slodowy, Der Scheibensatz für algebraische Transformationsgruppen, inAlgebraic Transformation Groups and Invariant Theory, DMV Seminar vol.13, Birkhäuser Verlag, Basel-Boston, 1989, pp. 89–113.  R. Steinberg, Regular elements of semisimple algebraic groups,Publ. Math. IHES,25 (1965), 49–80. · Zbl 0136.30002  R. Steinberg,Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc.80 (1968). · Zbl 0164.02902
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.