All Questions
Tagged with perverse-sheaves geometric-representation-theory
13 questions
8
votes
0
answers
259
views
What is an example of Beilinson's theorem on $D^b\mathrm{Perv}$ failing for non-field coefficients?
In Beilinson's paper "On the derived category of perverse sheaves", he proves that the realization functor $D^b\mathrm{Perv}(X,R)\to D^b_c(X, R)$ is an equivalence when $R$ is a field. Here $...
- 495
6
votes
0
answers
225
views
What advantages do perverse sheaves provide over D-modules? (or vice versa)
My question is as in the title: What advantages do perverse sheaves provide over D-modules? (or vice versa)
As a specific example: could something like the modular generalized Springer correspondence ...
- 495
3
votes
0
answers
413
views
Understanding the proof of the Springer correspondence
Let $G$ be a connected reductive group over an algebraically closed field $k$ with Weyl group $W$.
Let
$$
\mathcal{S} = R\pi_*\mathbb{Q}_\ell[\dim \mathcal{N}]
$$
be the Springer sheaf, where $\...
- 31
12
votes
0
answers
388
views
Perverse sheaves and representation theory
At my university we are having a working group on perverse sheaves, with the aim of applying them to representation theory (Lusztig canonical bases for quivers/quantum groups etc). We are still ...
- 1,061
6
votes
0
answers
275
views
The geometric "hands-on" vs. algebraic approach to nearby cycles
Feel free to skip to the question below; the following is just context and discussion:
An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...
- 272
15
votes
0
answers
541
views
Applications of character sheaves
There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain ...
- 2,964
7
votes
0
answers
168
views
Explicit computation for perverse cohomology
To construct the convolution product for two ($G(O)$-equivariant) perverse sheaves $\mathcal{F}, \mathcal{G}$ on affine grassmanian, the first thing we need to compute is $^PH^0(\mathcal{F} \boxtimes^...
- 677
4
votes
1
answer
296
views
Computation of multiplicity of irreducible representation in some representation via geometric Satake correspondence
Let $G$ be a reductive algebraic group over $\mathbb{C}$. Let $\operatorname{Gr}_{G}$ be the corresponding affine Grassmannian ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$...
- 103
3
votes
1
answer
304
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 $\...
- 2,554
4
votes
1
answer
379
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 ...
- 1,595
8
votes
1
answer
530
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 ...
- 2,554
2
votes
1
answer
687
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 ...
- 2,554
14
votes
1
answer
1k
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 $\Delta_w$ ...
- 1,595