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### What is the Hirzebruch-Riemann-Roch formula for the flag variety of a Lie algebra?

If we have a finite dimensional Lie algebra g, then the flag variety of g is a projective scheme.
My question is what is Hirzebruch-Riemann-Roch formula for this projective scheme? Are there any ...

**14**

votes

**1**answer

482 views

### Morphisms between Verma modules

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ ...

**13**

votes

**4**answers

1k views

### motivating geometric representation theory

I am wondering if there is a good motivation for geometric representation theory from within the questions of classical representation theory.
In other words, I'd be curious to see something using ...

**13**

votes

**1**answer

950 views

### How many ways are there to globalize Harish Chandra modules?

Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...

**12**

votes

**3**answers

656 views

### Introductory References for Geometric Representation Theory

Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...

**12**

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**2**answers

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### 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 ...

**11**

votes

**1**answer

1k views

### Is there Grothendieck Riemann Roch for abelian category?

From the answers in noncommutative algebraic geometry, one can take abelian category as a scheme(commutative or noncommutative). So I wonder whether anyone ever developed the Grothendieck Riemann Roch ...

**10**

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**2**answers

816 views

### Can the Quantum Torus be realized as a Hall Algebra?

Background
The Quantum Torus
Let $q$ be an arbitrary complex number, and define (the algebra of) the quantum torus to be
$$T_q:=\mathbb{C}\langle x^{\pm 1},y^{\pm 1}\rangle/xy-qyx$$
For $q=1$, this ...

**8**

votes

**1**answer

426 views

### How nice are representation varieties of Fuchsian groups?

Background
Let $S_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the non-hyperbolic cases:
$g=0$, $n=0,1,2$.
$g=1$, $n=0$.
Let $\Gamma$ be the fundamental ...

**8**

votes

**1**answer

240 views

### Kazhdan-Lusztig Polynomials and Intersection Cohomology

I hope this question has not been asked before.
I would like to know which Ideas led (Deligne), Kazhdan and Lusztig believe, that Kazhdanâ€“Lusztig polynomials can be expressed via intersection ...

**8**

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**3**answers

722 views

### Is there a good account of D-affinity and localization theorem for partial flag varieties?

Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated ...

**7**

votes

**2**answers

435 views

### Weil's theorem about maps from a discrete group to a Lie group.

Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...

**7**

votes

**1**answer

222 views

### Restriction to Levi Subgroups and the Affine Grassmannian

Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.
My Question:
What is the geometric analogue of the restriction ...

**7**

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**0**answers

208 views

### Characteristic Classes in Geometric Representation Theory

Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used.
I wonder if there are concrete applications of the ...

**6**

votes

**4**answers

799 views

### Why are the holomorphic line bundle sections finite dimensional?

I'm trying to understand the Borel--Weil theorem at the moment (not the whole Bott--Borel--Weil theorem as has been asked elsewhere). However, I am having a little difficulty finding a direct proof, ...

**6**

votes

**3**answers

2k views

### Beilinson-Bernstein and Koszul duality

For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed ...

**6**

votes

**1**answer

1k views

### How to understand character sheaves

There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247
How to understand these ...

**6**

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**0**answers

470 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 ...

**6**

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**0**answers

484 views

### Hirzebruch-Riemann-Roch for quiver varieties?

These days, I attended a workshop at North Carolina State University. The key lecturer is Professor Nakajima. He introduced two types of quiver variety. One of them is affine, another one is ...

**5**

votes

**3**answers

345 views

### Reference request: Affine Grassmannian and G-bundles

Let $G$ be an affine algebraic group over an algebraically closed field $k$ of zero characteristic. The set of cosets $X_G=G(k((t))/G(k[[t]])$ is called the Affine Grassmannian of $G$ and can be given ...

**5**

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**1**answer

363 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

209 views

### A question about equivariant derived categories and [BBD]

Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of ...

**5**

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189 views

### Action of $F_4$ on generalized flag manifolds

Does there exist (as elementary as possible) a generalized flag manifold (e.g. sphere, projective space, Grassmanian, etc.) of a simple classical Lie group (SL, SO or Sp over real or complex field), ...

**4**

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**1**answer

193 views

### About the pro-algebraic group structure of $G(\mathbb{C}[[t]])$

I hope this is not too elementary!
Let $G$ be a algebraic reductive group over $\mathbb{C}$.
The group $G(\mathbb{C}[[t]])$ can be given the structure of a pro algebraic group as follows.
Let $l\in ...

**4**

votes

**1**answer

624 views

### Explanation for Satake correspondence

Some time ago I was told there's an interesting classical Satake correspondence which I will write as
$$[\mathop{\mathrm{disk}} \Rightarrow G] \\,\backslash\\, [\mathop{\mathrm{disk}^\times} ...

**4**

votes

**1**answer

551 views

### Quiver varieties and the affine Grassmannian

Recently I was watching a talk: http://media.cit.utexas.edu/math-grasp/Ben_Webster.html and at the end the lecturer gave a correspondence (I was having trouble with subscripts so changed the notation ...

**4**

votes

**1**answer

116 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 ...

**4**

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**0**answers

339 views

### Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?

So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...

**3**

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**1**answer

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### Dimension of orbits in affine Grassmanian

Let $\mathcal{K} = \mathbb{C}((t)), \mathcal{O}=\mathbb{C}[[t]]$, $G$ be a reductive group, and $\text{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$; there is a left action of $G(\mathcal{O})$ on ...

**3**

votes

**1**answer

250 views

### A question on algebraic loop groops

Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...

**2**

votes

**2**answers

320 views

### Around the socle filtration of a Verma module

Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$.
Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of ...

**2**

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**2**answers

381 views

### Smoothness properties of the Springer fiber

The Springer fiber, recall, is defined (briefly) with reference to a chosen unipotent matrix $U \in \mathrm{GL}_n$, and consists of all flags $0 = F_0 \subset F_1 \subset \dots \subset F_n = ...

**2**

votes

**1**answer

252 views

### Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors

I hope this question is not too vague.
Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$.
Denote by $\pi:G/B\to G/P$ the canonical map. Consider ...

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vote

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198 views

### When representation of two different coadjoint orbits are equivalent?

Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where ...

**0**

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101 views

### Equivariant J-function of Laumon space

Let $G=SL(n)$ and $B$ be the Borel subgroup of $G$. $G/B$ is the complete flags variety $0\subset V_1 \subset \cdots \subset V_n=\mathbb{C}^n$. The Laumon space is the space ...