The flag-varieties tag has no wiki summary.

**22**

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**3**answers

780 views

### Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...

**12**

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**2**answers

1k views

### What is the significance that the Springer resolution is a moment map?

Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution
$$
\mu: T^*\mathcal{B}\rightarrow \mathcal{N}
$$
is the moment ...

**9**

votes

**2**answers

344 views

### Partial (or complete) flag varieties as GIT quotients of affine spaces

I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...

**9**

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**2**answers

395 views

### Cohomology ring of a flag variety and representation theory

I'm interested in the cohomology ring $H^*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to ...

**8**

votes

**4**answers

782 views

### Riemannian metric on a flag variety

$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = ...

**6**

votes

**3**answers

267 views

### Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...

**6**

votes

**1**answer

208 views

### Quotienting $SU(3)$ by $U(1)$?

As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...

**6**

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**2**answers

635 views

### Hecke algebra and $H^*(G/B)$

Given a complex reductive group, with Weyl group $W$, one can associate to it lots of "algebras of size $|W|$". For example $B$ equivariant functions on $G/B$ with convolution, grothendieck groups of ...

**6**

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**2**answers

731 views

### The anticanonical bundle on a flag variety is ample

Hello,
I would like to get references or answers, for the following. How do I show that the anti-canonical line bundle (i.e. dual to top wedge power of cotangent bundle) on a flag variety (of a ...

**6**

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**0**answers

94 views

### Cominuscule property of nilpotent orbits

Let $G$ be a complex reductive Lie group, $G/P$ a flag manifold,
and $\Phi: T^* G/P \to {\mathfrak g}^*$ the moment map. So $\Phi(T^* G/P)$ is the closure of a nilpotent orbit.
Lots of classes of ...

**6**

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**0**answers

176 views

### Purity and six operations?

The six operations $f_!,f^!,f_*,f^*,\otimes,\mathcal Hom$ have the property that they preserve estimates on weights in one direction.
For $f_!,f^!,f_*,f^*$ I can see, that they don't preserve purity ...

**6**

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**0**answers

478 views

### Is the Springer resolution a blow-up?

Let's consider the Springer resolution of the nilpotent cone $\mathcal{N}$ of a complex semisimple Lie algebra $\mathfrak{g}$, which is
$$
\widetilde{\mathcal{N}}=T^*\mathcal{B}\rightarrow ...

**5**

votes

**1**answer

380 views

### What kind of algebra has geometric realization as in “Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups”

In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra ...

**5**

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**0**answers

191 views

### Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)

For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...

**4**

votes

**1**answer

235 views

### Moduli of flag varieties

I work over an algebraically closed field $k$ of characteristic zero.
Recall that a flag variety is a projective variety which is a homogeneous space for some semisimple algebraic group. Every flag ...

**4**

votes

**1**answer

216 views

### Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$:
$0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...

**4**

votes

**1**answer

317 views

### Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...

**4**

votes

**1**answer

127 views

### Stalks of intersection cohomology complexes of Schubert varieties and Bruhat order

All varieties are over $\mathbb{C}$.
Let $G$ be a connected reductive group, $B\subseteq G$ a Borel subgroup.
Let $O_w$ be a $B$-orbit in $G/B$. I.e., $O_w$ is a Bruhat cell. In particular, it is ...

**3**

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**2**answers

353 views

### Defining Equations of a Flag Variety

I've been reading Fulton's Young Tableaux, and I'm trying to understand flag varieties. I want to understand the defining equations of a Flag Variety, but the coordinates in Fulton's Plucker ...

**3**

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**2**answers

225 views

### Kahler Metric Fundamental Forms and Cohomology Ring Generators

For the projective line $CP^1$, its cohomology ring has a single generator. Moreover, this generator is given by the cohomology class of the fundamental form associated associated to the Fubini--Study ...

**3**

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**2**answers

125 views

### Why is $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the space of complete flags $GL_3/B$?

In one the the answers to this thread " Can one embedd the projectivezed tangent space of CP^2 in a projective space? " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety ...

**3**

votes

**1**answer

188 views

### Full exceptional set on flag variety

Define a full exceptional set in a triangulated category to be a partially ordered set of objects $\Delta_i$ which generate the category and such that $\text{Ext}^\bullet(\Delta_i, \Delta_j) = 0$ ...

**3**

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**1**answer

127 views

### How does the grading on the cohomology of a flag variety break up the regular representation of W?

The complex cohomology H^* of the manifold of flags in C^n is a quotient of C[x1,...,xn] by the ideal generated by symmetric polynomials with no constant term. In particular it has an action of the ...

**3**

votes

**1**answer

216 views

### What is known about formality of flag varieties?

Let $G$ be a connected, complex reductive algebraic group and $X=G/P$ a (partial) flag variety. For example by "Real homotopy theory of Kähler manifolds", we know that its cohomology with real ...

**3**

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**0**answers

81 views

### (semi-)Small resolutions of Peterson varieties

Peterson varieties (in type A) can be described as the subvarieties of the full flag variety
$$\{(F_{i})\;|\; F_{i}\subset \mathbb{C}^{n}, \; \dim F_{i} =i,\; N(F_{i})\subset F_{i+1}\}$$
where $N$ ...

**3**

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**0**answers

99 views

### How to deduce (8.1) in Lusztig's “Equivariant K-theory and representations of Hecke Algebras”

Let G be a connected complex algebraic group G and X=G/B, where B is the Borel subgroup. $ M=G\times C^{*}$, where $C^*$ acts on X trivially. Let $K_M(X)$ be the equivariant K-theory. Let $s\in S$ be ...

**3**

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168 views

### The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:
Let $j: ...

**2**

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**2**answers

167 views

### Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?

As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 ...

**2**

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**1**answer

180 views

### The real group orbits on the flag variety always contains the holomorphic directions?

Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of ...

**2**

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**0**answers

113 views

### Flags of varieties

I was wondering if there is a generalization of flags in the following way: Suppose you have a series of inclusions of affine varieties $V_1\hookrightarrow V_2\hookrightarrow\cdots\hookrightarrow V_n$ ...

**2**

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121 views

### Flag Varieties via Quiver Varieties

In type $A$ it is possible to realise the flag variety $\mathcal{F}$ of $\text{SL}_{n}(\mathbb{C})$ via Nakajima's quiver varieties: consider the vectors $v=(1,2,\ldots, n-1), w=(0,\ldots,0,n)$. Then, ...

**1**

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**1**answer

87 views

### Non-equivariant vector bundles over complex projective $N$-space

From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist ...

**1**

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**1**answer

296 views

### Computing relative Lie algebra cohomology (as appears in Borel-Weil-Bott theorem)

Suppose $G$ is a complex Lie group, $P$ a Borel subgroup, $E$ a representation of $P$ that induces a vector bundle ${\cal E}$ over $G/P$. The general version of Borel-Weil-Bott theorem, as stated in ...

**1**

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**1**answer

389 views

### moduli problem for flag varieties?

Hi,
Suppose $G$ is a reductive group over an algebraiclly closed field $k$
(suppose $k$ of char zero if you want at first). Let $X$ be its flag variety.
Question: What is the moduli problem that $X$ ...

**1**

vote

**1**answer

155 views

### When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and
$\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...

**1**

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**0**answers

191 views

### Pullback of a sheaf associated to a divisor

I am reading a paper Desingularisation des varietes de Schubert generalisees by Demazure. I am interested in Lemma 3 on page 58. In particular, I would like to know whether the lemma is true and how ...

**1**

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**1**answer

121 views

### Variety of factorizations of differential operator

Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...

**1**

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147 views

### Equivariant $K$-theory, singular vectors, and flag manifolds

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...

**1**

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207 views

### Weight decomposition and eigenspaces Euler vector field

Let $V$ be a finite dimensional vector space over $\mathbb{C}$, denote $V^{\times} = V -0$ and let $\pi : V^{\times} \rightarrow \mathbb{P}(V)$ be the quotient by the natural action of ...