Oberwolfach Preprints OWP 2014 - 04 ROMAN AVDEEV AND ALEXEY PETUKHOV Spherical Actions on Flag Varieties Mathematisches Forschungsinstitut Oberwolfach gGmbH Oberwolfach Preprints (OWP) ISSN 1864-7596 Oberwolfach Preprints (OWP) Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs- Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from 1 - 200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. 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Imprint: Mathematisches Forschungsinstitut Oberwolfach gGmbH (MFO) Schwarzwaldstrasse 9-11 77709 Oberwolfach-Walke Germany Tel +49 7834 979 50 Fax +49 7834 979 55 Email [email protected] URL www.mfo.de The Oberwolfach Preprints (OWP, ISSN 1864-7596) are published by the MFO. Copyright of the content is held by the authors. SPHERICAL ACTIONS ON FLAG VARIETIES ROMAN AVDEEV AND ALEXEY PETUKHOV Abstract. For every finite-dimensional vector space V and every V -flag variety X we list all connected reductive subgroups in GL(V ) acting spherically on X. 1. Introduction Throughout this paper we fix an algebraically closed field F of characteristic 0, which is the ground field for all objects under consideration. By F£ we denote the multiplicative group of F. Let K be a connected reductive algebraic group and let X be a K-variety (that is, an algebraic variety together with a regular action of K). The action of K on X, as well as the variety X itself, is said to be spherical (or K-spherical when one needs to emphasize the acting group) if a Borel subgroup of K has an open orbit in X. Every finite-dimensional K-module that is spherical as a K-variety is said to be a spherical K- module. Spherical varieties possess various remarkable properties; a review of them can be found, for instance, in the monograph by Timashev [Tim]. Describing all spherical varieties for a fixed group K is an important and interesting problem, and by now considerable results have been achieved in solving it. A review of these results can be also found in the monograph [Tim]. Along with the problem mentioned above one can also consider an opposite problem, namely, for a given algebraic variety X find all connected reductive subgroups in the automorphism group of X that act spherically on X. In this paper we consider this problem in the case where X is a generalized flag variety. A generalized flag variety is a homogeneous space of the form G=P , where G is a connected reductive group and P is a parabolic subgroup of G. We call the variety G=P trivial if P = G and nontrivial otherwise. The following facts are well known: (1) all generalized flag varieties of a fixed group G are exactly all complete (and also all projective) homogeneous spaces of G; (2) the center of G acts trivially on G=P ; (3) the natural action of G on G=P is spherical. Let F (G) denote the set of all (up to a G-equivariant isomorphism) nontrivial gener- alized flag varieties of a fixed group G. For every generalized flag variety X we denote by Aut X its automorphism group. 2010 Mathematics Subject Classification. 14M15, 14M27. Key words and phrases. Algebraic group, flag variety, spherical variety, nilpotent orbit. The first author was supported by the RFBR grant no. 12-01-00704, the SFB 701 grant of the University of Bielefeld (in 2013), the “Oberwolfach Leibniz Fellows” programme of the Mathematisches Forschungsin- stitut Oberwolfach (in 2013), and Dmitry Zimin’s “Dynasty” Foundation (in 2014). The second author was supported by the Guest Programme of the Max-Planck Institute for Mathematics in Bonn (in 2012– 2013). 1 2 ROMAN AVDEEV AND ALEXEY PETUKHOV In [Oni, Theorem 7.1] (see also [Dem]) the automorphism groups of all generalized flag varieties were described. This description implies that, for every X 2 F (G), Aut X is an affine algebraic group and its connected component of the identity (Aut X)0 is semisimple and has trivial center. Moreover, it turns out that in most of the cases (all exceptions are listed in [Oni, Table 8], see also [Dem, § 2]), including the case G = GLn, the natural homomorphism G ! (Aut X)0 is surjective and its kernel coincides with the center of G. In any case one has X 2 F ((Aut X)0), therefore the problem of describing all spherical actions on generalized flag varieties reduces to the following one: Problem 1.1. For every connected reductive algebraic group G and every variety X 2 F (G), list all connected reductive subgroups K ½ G acting spherically on X. Since the center of G acts trivially on any variety X 2 F (G), when it is convenient, in solving Problem 1.1 the group G without loss of generality may be assumed to be semisimple. The main goal of this paper is to solve Problem 1.1 in the case G = GLn (the results are stated below in Theorems 1.6 and 1.7). Let us list all cases known to the authors where Problem 1.1 is solved. 1) G and X are arbitrary and K is a Levi subgroup of a parabolic subgroup Q ½ G. In this case the condition that the variety G=P 2 F (G) be K-spherical is equivalent to the condition that the variety G=P £ G=Q be G-spherical, where G acts diagonally (see Lemma 5.4). All G-spherical varieties of the form G=P £ G=Q are classified. In the case where P; Q are maximal parabolic subgroups this was done by Littelmann [Lit]. In the cases G = GLn and G = Sp2n the classification follows from results of the Magyar, Weyman, and Zelevinsky papers [MWZ1] and [MWZ2], respectively. (In fact, in [MWZ1] and [MWZ2] for G = GLn and G = Spn, respectively, the following more general problem was solved: describe all sets X1;:::;Xk 2 F (G) such that G has finitely many orbits under the diagonal action on X1 £:::£Xk.) Finally, for arbitrary groups G the classifica- tion was completed by Stembridge [Stem]. The results of this classification for G = GLn will be essentially used in this paper and are presented in § 5.2. 2) G and X are arbitrary and K is a symmetric subgroup of G (that is, K is the subgroup of fixed points of a nontrivial involutive automorphism of G). In this case the classification was obtained in the paper [HNOO]. 3) G is an exceptional simple group, X = G=P for some maximal parabolic subgroup P ½ G, and K is a maximal reductive subgroup in G. This case was investigated in the preprint [Nie]. Let G be an arbitrary connected reductive group, g = Lie G, and K ½ G an arbitrary connected reductive subgroup. We now discuss the key idea utilized in this paper. Let P ½ G be a parabolic subgroup and let N ½ P be its unipotent radical. Put n = Lie N. We regard the adjoint action of G on g. In view of a well-known result of Richardson (see [Rich, Proposition 6(c)]), for the induced action P : n there is an open orbit OP . We put (1.1) N (G=P ) = GOP ½ g: It is easy to see that N (G=P ) is a nilpotent (that is, containing 0 in its closure) G-orbit in g. SPHERICAL ACTIONS ON FLAG VARIETIES 3 Definition 1.2. We say that two varieties X1;X2 2 F (G) are nil-equivalent (notation: X1 » X2) if N (X1) = N (X2). It is well-known that every G-orbit in g is endowed with the canonical structure of a symplectic variety. It turns out (see Theorem 2.6) that a variety X 2 F (G) is K-spherical if and only if the action K : N (X) is coisotropic (see Definition 2.2) with respect to the symplectic structure on N (X). This immediately implies the following result. Theorem 1.3. Suppose that X1;X2 2 F (G) and X1 » X2. Then the following conditions are equivalent: (a) the action K : X1 is spherical; (b) the action K : X2 is spherical. Let [[X]] denote the nil-equivalence class of a variety X 2 F (G). The inclusion relation between closures of nilpotent orbits in g defines a partial order 4 on the set F (G)=» of all nil-equivalence classes in the following way: for X1;X2 2 F (G) the relation [[X1]] 4 [[X2]] (or [[X2]] < [[X1]]) holds if and only if the orbit N (X1) is contained in the closure of the orbit N (X2).