Older papers on Transformation Groups and Invariant Theory

  1. Hermitian characteristics of nilpotent elements, Encyclopaedia Math. Sci., Springer (2004), 132, 177-206
    Abstract. We define and study several equivariant stratifications of the isotropy and coisotropy representations of a parabolic subgroup in a complex reductive group.

  2. Self-dual projective algebraic varieties associated with symmetric spaces, with Vladimir, Encyclopaedia Math. Sci., Springer (2004), 132, 131-167
    Abstract. We discover a class of projective self-dual algebraic varieties. Namely, we consider actions of isotropy groups of complex symmetric spaces on the projectivized nilpotent varieties of isotropy modules. For them, we classify all orbit closures X such that X is isomorphic to its projective dual. We give algebraic criteria of projective self-duality for the considered orbit closures.

  3. Moore-Penrose inverse, parabolic subgroups, and Jordan pairs, Journal of Lie Theory 12 (2002), no. 2, 461-481
    Abstract. A Moore--Penrose inverse of an arbitrary complex matrix A is defined as a unique matrix A' such that AA'A=A, A'AA'=A', and AA', A'A are Hermitian matrices. We show that this definition has a natural generalization in the context of shortly graded simple Lie algebras corresponding to parabolic subgroups with aura (abelian unipotent radical) in simple complex Lie groups, or equivalently in the context of simple complex Jordan pairs. We give further generalizations and applications.

  4. Isotropic subspaces of polylinear forms, Math. Notes 69 (2001), no. 5-6, 845-852
    Abstract. We compute dimensions of maximal isotropic subspaces of generic symmetric and skew-symmetric forms of any degree.

  5. Invariant linear connections on homogeneous symplectic varieties, with Sergei, Transformation Groups 6 (2001), no. 2, 193-198
    Abstract. We find all homogeneous symplectic varieties of connected reductive algebraic groups that admit an invariant linear connection.

  6. On the Chevalley restriction theorem, Journal of Lie Theory 10 (2000), no. 2, 323-330
    Abstract. Let g be the Lie algebra of a complex semisimple Lie group G. Let s be an involution of G and let K=Gs be the subgroup of fixed points. Consider the direct sum decomposition g=k+p into the eigenspaces for s. Denote by a any maximal abelian ad-diagonalizable subalgebra of p and by W=NK(a)/ZK(a) the baby Weyl group. By the Chevalley restriction theorem for isotropy subspaces of symmetric spaces, the restriction map of algebras of invariants C[p]K->C[a]W is an isomorphism. The aim of this paper is prove that the restriction map C[pxp]K->C[axa]W is surjective, generalizing the analogous result of Joseph for the adjoint action of the reductive group.

  7. Discriminants and quasi-derivations of commutative algebras, Russian Math. Surveys 54 (1999), no. 2, 457-458

  8. Subalgebras and discriminants of anticommutative algebras, Izvestia Math. 63 (1999), no. 3, 583-595
    Abstract. The paper deals with the configuration of subalgebras in generic n-dimensional k-argument anticommutative algebras and ``regular'' anticommutative algebras.

  9. Generic algebras, Transformation Groups 1 (1996), no. 1-2, 127-151
    Abstract. We study generic commutative and anticommutative algebras of fixed dimension, their invariants, covariants and algebraic properties (e.g., the structure of subalgebras). In the case of 4-dimensional anticommutative algebras, we relate the structure of subalgebras to the configuration of 27 lines on the associated cubic surface. We prove rationality of the corresponding moduli space. In the case of 3-dimensional commutative algebras, a new proof of a recent theorem of Katsylo and Mikhailov about the 28 bitangents to the associated plane quartic is given.

  10. Unstable linear actions of connected semisimple complex algebraic groups, with Seva, Sbornik Math. 186 (1995), no. 1, 107-119
    Abstract. A linear action of a reductive group is called unstable if its generic orbits are not closed or, equivalently, if stabilizers of generic points are not reductive. The Lie algebra of a Levi subalgebra of a generic stabilizer is called the Cartan subalgebra and we compute them for all unstable linear actions of connected simple groups and for all unstable irreducible linear actions of connected semisimple groups.