In this paper we consider various problems involving the action of a reductive group G on an affine variety V . We prove some general rationality results about the G-orbits in V . In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of G for such general G-actions. We apply our general rationality results to answer a question of S...

Homogeneous bundles on P2 = SL(3)/P can be described by representations of the parabolic subgroup P . In 1966 Ramanan proved that if ρ is an irreducible representation of P then the induced bundle Eρ on P 2 is simple and even stable (see [Ram]). Since P is not a reductive group, there is a lot of indecomposable reducible representations of P and to classify homogeneous bundles on P2 and among t...

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and Pu its unipotent radical. We consider the adjoint action of P on the Lie algebra pu of Pu. Richardson’s dense orbit theorem says that there is a dense P -orbit in pu. We consider some instances when P acts with a dense orbit on terms p u of the descending central series of pu. In particular, we show (in good characteri...

We compute the algebra of differential invariants unparametrized curves in homogeneous G2 flag varieties, namely G2/P. This gives a solution to equivalence problem for such curves. consider cases integral and generic relate problems all three choices parabolic subgroup P.

Let G be an algebraic reductive group over a field of positive characteristic. Choose a parabolic subgroup P in G and denote by U its unipotent radical. Let X be a G-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on X with respect to a generic character χ : U → Ga commutes with Verdier duality. Namely, in the first e...

We will using the combinatorics of the G-stable pieces to describe the closure relation of the partition of partial flag varieties in [L3, section 3]. 1. Some combinatorics 1.1. Let k be an algebraic closure of the finite field Fq and G be a connected reductive algebraic group defined over Fq with Frobenius map F : G → G. We fix a F -stable Borel subgroup B of G and a F -stable maximal torus T ...

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical Pu, and A a closed connected unipotent subgroup of Pu which is normalized by P. We show that P acts on A with nitely many orbits provided A is abelian. This generalizes a well-known niteness result, namely the case when A is central in Pu. We also obtain an analogous result for the adjoint action of P on in...

We consider a cocompact discrete reflection group W of a CAT(0) space X . Then W becomes a Coxeter group. In this paper, we study an analogy between the Davis-Moussong complex Σ(W, S) and the CAT(0) space X , and show several analogous results about the limit set of a parabolic subgroup of the Coxeter group W .

Let MH be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X . It is a smooth variety over C equipped with a holomorphic symplectic form. Fix a projective structure P on X . Using P , we construct a quantization of a certain Zariski open dense subset of the symplectic variety MH .