13
votes
3answers
761 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
228 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 ...
1
vote
2answers
210 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
258 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
227 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
263 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
1answer
260 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 ...
5
votes
0answers
215 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 ...
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 ...
8
votes
4answers
1k 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
2answers
335 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
1answer
501 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 $\Delta_w$ ...
2
votes
2answers
407 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 = ...
8
votes
3answers
727 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 ...
13
votes
1answer
992 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 ...
7
votes
2answers
443 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 ...
4
votes
1answer
636 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} ...
6
votes
3answers
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
1answer
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 ...