12
votes
4answers
814 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 se …
5
votes
4answers
508 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 find …
13
votes
1answer
372 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 $W$ …
2
votes
2answers
243 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 t …
5
votes
1answer
302 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", t …
1
vote
2answers
284 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 \su …
10
votes
2answers
1k 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
$$
\mu: T^*\mathcal{B}\rightarrow \mathcal{ …
5
votes
0answers
384 views
Is the Springer resolution a blow-up?
Let's consider the Springer resolution of the nilpotent cone $\mathcal{N}$ of a complex semisimple Lie algebra $\mathfrak{g}$, which is
$$
\widetilde{\mathcal{N}}=T^*\mathcal{B}\r …
7
votes
1answer
391 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 …
5
votes
0answers
178 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 …
4
votes
1answer
477 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 c …
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 …
10
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 Groth …
8
votes
3answers
685 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 fin …
5
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 …

