On the Complexity of Reductive Group Actions over Algebraically Nonclosed Field and Strong Stability of the Actions on Flag Varieties

2019 ◽  
Vol 99 (2) ◽  
pp. 132-136 ◽  
Author(s):  
V. S. Zhgoon ◽  
F. Knop
Keyword(s):  
Group Actions ◽  
Reductive Group ◽  
Strong Stability ◽  
Flag Varieties

scholarly journals On complexity of reductive group actions over algebraically non-closed field and strong stability of the actions on flag varieties

2019 ◽  
Vol 485 (1) ◽  
pp. 22-26
Author(s):  
V. S. Zhgoon ◽  
F. Knop
Keyword(s):  
Algebraic Variety ◽  
Group Actions ◽  
Reductive Group ◽  
Strong Stability ◽  
Closed Field ◽  
Flag Varieties

We annonce the results generalizing the Vinberg's Complexity Theorem for the action of reductive group on an algebraic variety over algebraically non-closed field. Also we give new results on the strong k-stability for the actions on flag varieties.

scholarly journals Linear degenerations of flag varieties: partial flags, defining equations, and group actions

2019 ◽  
Vol 296 (1-2) ◽  
pp. 453-477
Author(s):  
Giovanni Cerulli Irelli ◽  
Xin Fang ◽  
Evgeny Feigin ◽  
Ghislain Fourier ◽  
Markus Reineke
Keyword(s):  
Group Actions ◽  
Flag Varieties

scholarly journals Braiding via geometric Lie algebra actions

2012 ◽  
Vol 148 (2) ◽  
pp. 464-506 ◽  
Author(s):  
Sabin Cautis ◽  
Joel Kamnitzer
Keyword(s):  
Lie Algebra ◽  
Braid Group ◽  
Group Actions ◽  
Derived Categories ◽  
Flag Varieties ◽  
Coherent Sheaves ◽  
Cotangent Bundles ◽  
Partial Flag Varieties

AbstractWe introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

On Orbit Closures of Symmetric Subgroups in Flag Varieties

2000 ◽  
Vol 52 (2) ◽  
pp. 265-292 ◽  
Author(s):  
Michel Brion ◽  
Aloysius G. Helminck
Keyword(s):  
Fixed Point ◽  
Borel Subgroup ◽  
Reductive Group ◽  
Double Coset ◽  
Restriction Map ◽  
Flag Varieties ◽  
Parabolic Subgroups ◽  
Involutive Automorphism ◽  
Orbit Closures ◽  
Geometric Analogue

AbstractWe study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B, ) → H0(X, ) is surjective and that Hi(X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X, ). This gives information on the restriction to K of the simple G-module H0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.

Deformations of reductive group actions

1989 ◽  
Vol 106 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Gerd Müller
Keyword(s):  
Lie Group ◽  
Tangent Space ◽  
Group Actions ◽  
Reductive Group ◽  
Analytic Space ◽  
Precise Formulation ◽  
The Given

Consider actions of a reductive complex Lie group G on an analytic space germ (X, 0). In a previous paper [16] we proved that such an action is determined uniquely (up to conjugation with an automorphism of (X, 0)) by the induced action of G on the tangent space of (X, 0). Here it will be shown that every deformation of such an action, parametrized holomorphically by a reduced analytic space germ, is trivial, i.e. can be obtained from the given action by conjugation with a family of automorphisms of (X, 0) depending holomorphically on the parameter. (For a more precise formulation in terms of actions on analytic ℂ-algebras, see Theorem 2 below. An analogue for actions on formal ℂ-algebras is given there too.)

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