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6
votes
0answers
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
votes
0answers
181 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
votes
0answers
481 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
0answers
196 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 ...
3
votes
0answers
88 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
votes
0answers
101 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
votes
0answers
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
votes
0answers
114 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
votes
0answers
127 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
vote
0answers
61 views

On the universal property of certain representable functors and rational sections

Let $P_1,P_2$ be two Hilbert polynomials of subschemes in $\mathbb{P}^n$. Denote by $H_{P_1,P_2}$ the corresponding flag Hilbert scheme (parametrizing pairs $(X\subset Y)$ where $X$ has Hilbert ...
1
vote
0answers
196 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
vote
0answers
148 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
vote
0answers
220 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 ...