Irreducible Components of Fixed Point Subvarieties of Flag Varieties

1998 ◽  
Vol 189 (1) ◽  
pp. 107-120 ◽  
Author(s):  
J. Matthew Douglass
Keyword(s):  
Fixed Point ◽  
Flag Varieties ◽  
Irreducible Components

scholarly journals A FILTRATION ON EQUIVARIANT BOREL–MOORE HOMOLOGY

2019 ◽  
Vol 7 ◽  
Author(s):  
ARAM BINGHAM ◽  
MAHIR BILEN CAN ◽  
YILDIRAY OZAN
Keyword(s):  
Fixed Point ◽  
Maximal Torus ◽  
Flag Varieties ◽  
Direct Sum ◽  
Equivariant Embedding ◽  
Graded Module ◽  
Homogeneous Variety

Let $G/H$ be a homogeneous variety and let $X$ be a $G$ -equivariant embedding of $G/H$ such that the number of $G$ -orbits in $X$ is finite. We show that the equivariant Borel–Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the $G$ -orbits. If $T$ is a maximal torus of $G$ such that each $G$ -orbit has a $T$ -fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel–Moore homology of $X$ . We apply our findings to certain wonderful compactifications as well as to double flag varieties.

scholarly journals Degenerate Affine Flag Varieties and Quiver Grassmannians

2020 ◽  
Author(s):  
Alexander Pütz
Keyword(s):  
Flag Varieties ◽  
Rational Singularities ◽  
Quiver Grassmannians ◽  
Finite Dimensional ◽  
Irreducible Components ◽  
Affine Flag

AbstractWe study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

scholarly journals Typical representations via fixed point sets in Bruhat–Tits buildings

2021 ◽  
Vol 25 (36) ◽  
pp. 1021-1048
Author(s):  
Peter Latham ◽  
Monica Nevins
Keyword(s):  
Fixed Point ◽  
Maximal Compact Subgroup ◽  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Supercuspidal Representation ◽  
Content Type ◽  
Branching Rules ◽  
Irreducible Components ◽  
Tits Building ◽  
Fixed Point Sets

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

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.

scholarly journals Goresky–MacPherson Calculus for the Affine Flag Varieties

2010 ◽  
Vol 62 (2) ◽  
pp. 473-480 ◽  
Author(s):  
Zhiwei Yun
Keyword(s):  
Fixed Point ◽  
Equivariant Cohomology ◽  
Flag Variety ◽  
Moment Map ◽  
Flag Varieties ◽  
The Moment ◽  
Affine Flag Variety ◽  
Affine Flag

AbstractWe use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety ℱ ℓ G generated by degree 2. We use this result to show that the vertices of the moment map image of ℱ ℓ G lie on a paraboloid.

Restriction properties for the Krull–Schmidt decomposition of vector bundles

2018 ◽  
Vol 29 (03) ◽  
pp. 1850022
Author(s):  
Mihai Halic
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
The Other ◽  
Flag Varieties ◽  
Schmidt Decomposition ◽  
Ample Vector Bundle ◽  
Projective Varieties ◽  
Splitting Criterion ◽  
Irreducible Components ◽  
Smooth Projective Varieties

We obtain decomposability criteria for vector bundles on smooth projective varieties [Formula: see text] by comparing the Krull–Schmidt decomposition on [Formula: see text], on one hand, and along the vanishing locus of a section in an ample vector bundle over [Formula: see text], on the other hand. We determine effective bounds for the amplitude of and also genericity conditions for its sections which ensure that the irreducible components of and those of its restriction correspond bijectively. Moreover, we get a simple splitting criterion for vector bundles on partial flag varieties.

scholarly journals Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

2021 ◽  
Vol 25 (32) ◽  
pp. 903-934
Author(s):  
Yiqiang Li
Keyword(s):  
Fixed Point ◽  
Geometric Realization ◽  
Classical Type ◽  
Symmetric Pair ◽  
Flag Varieties ◽  
Projective System ◽  
Content Type ◽  
Lagrangian Construction

We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.

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