12
votes
3answers
660 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 ...
7
votes
0answers
210 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
1answer
195 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
2answers
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 ...
3
votes
1answer
251 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 ...
7
votes
1answer
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 ...
2
votes
1answer
253 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 ...
5
votes
0answers
211 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
votes
3answers
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 ...
13
votes
4answers
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 ...
2
votes
2answers
384 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 = ...
6
votes
0answers
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 ...
8
votes
1answer
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 ...
4
votes
1answer
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 ...
6
votes
0answers
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 ...
8
votes
3answers
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 ...
11
votes
1answer
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 ...
19
votes
1answer
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
0answers
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 ...