The geometric-rep-theory tag has no usage guidance.

**6**

votes

**1**answer

138 views

### Confusion about Subcategories of Category $\mathcal{O}$

So, in learning about category $\mathcal{O}$ representations of a semisimple Lie algebra $\mathfrak{g}$, I've come across two natural kinds of subcategories, and I think I'm confused about their ...

**3**

votes

**0**answers

66 views

### Double loop groups and cohomology

Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$.
What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?

**1**

vote

**1**answer

122 views

### Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?

I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}_n$. ...

**4**

votes

**0**answers

191 views

### Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in ...

**5**

votes

**0**answers

85 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**2**

votes

**0**answers

108 views

### What's the relation of the Hecke algebra of a pair and the flag variety?

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively.
Then the Hecke algebra ...

**7**

votes

**2**answers

435 views

### Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...

**1**

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**0**answers

126 views

### References for crystal bases and Demazure modules in representation theory

I was wondering what are some standard general references/books/survey articles about: (1) crystal bases, and string parameterizations and (2) Demazure modules, and Schubert varieties (containing ...

**4**

votes

**1**answer

194 views

### Are Strata of the affine Grassmannian total spaces of equivariant vector bundles over flag varieties

This question is closely related to Peter Crooks question.
Strata of the Affine Grassmannian
Let $G$ be a complex reductive group, $\mathcal{K}:= \mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and ...

**13**

votes

**3**answers

1k 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 ...

**9**

votes

**1**answer

472 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 ...

**4**

votes

**1**answer

306 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 ...

**1**

vote

**2**answers

226 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 ...

**3**

votes

**1**answer

275 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 ...

**4**

votes

**1**answer

186 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 ...

**7**

votes

**1**answer

245 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 ...

**2**

votes

**1**answer

309 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 ...

**8**

votes

**1**answer

313 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 Kazhdan–Lusztig polynomials can be expressed via intersection ...

**3**

votes

**1**answer

91 views

### 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 ...

**5**

votes

**0**answers

243 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 ...

**7**

votes

**3**answers

603 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 ...

**14**

votes

**4**answers

2k 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 ...

**10**

votes

**4**answers

2k 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, ...

**2**

votes

**2**answers

425 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 ...

**14**

votes

**1**answer

609 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 $\Delta_w$ ...

**4**

votes

**2**answers

584 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 = ...

**12**

votes

**2**answers

2k 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 ...

**7**

votes

**0**answers

524 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

433 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

197 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), ...

**8**

votes

**1**answer

455 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 ...

**4**

votes

**1**answer

623 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 ...

**6**

votes

**0**answers

522 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 ...

**8**

votes

**3**answers

759 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 ...

**17**

votes

**1**answer

1k 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 ...

**10**

votes

**2**answers

970 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

**2**answers

476 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 ...

**12**

votes

**1**answer

2k 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 ...

**20**

votes

**1**answer

1k views

### 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 ...

**4**

votes

**0**answers

369 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 ...

**4**

votes

**1**answer

726 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} ...

**7**

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

2k 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 ...