All Questions
17 questions with no upvoted or accepted answers
15
votes
0
answers
542
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
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
193
views
Bundles equivariant with respect to a transitive Lie algebra action
Suppose an algebraic group $G$ transitively acts on a variety $X$. Then it is well known that $G$-equivariant vector bundles on $X$ are in correspondence with representations of the stabilizer of a ...
- 151
6
votes
0
answers
304
views
Geometric interpretation of a formula for the induced character (fix point localization?)
Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
- 7,789
5
votes
0
answers
223
views
Making Virasoro uniformization explicit for elliptic curves
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...
user108998
5
votes
0
answers
244
views
Borel–Weil–Bott theorem and tensor product
Let $G=\operatorname{SL}(2)$ and $V_n$ be the n+1 dimensional irreducible representation of $G$. This gives a representation for $G(O)=\operatorname{Map}(\operatorname{Spec} k[[t]], G)$ and hence an ($...
- 189
5
votes
0
answers
324
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 $\mathbb{C}...
- 2,554
4
votes
0
answers
115
views
Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$
It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
- 1,343
4
votes
0
answers
486
views
Plucker coordinates of flag varieties
I am interested in understanding Lemma A.2 in the paper "Moduli spaces of principal F-bundles" by varshavsky which you can find here. It uses so called "Plücker" coordinates of the flag variety for ...
- 41
4
votes
0
answers
597
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 $Rep(G)$...
- 1,982
3
votes
0
answers
414
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
3
votes
0
answers
229
views
Spherical perverse sheaves on the affine Grassmannian and critically twisted $D$-modules
Let $G$ be a reductive algebraic group and let $Gr_G=G((z))/G[[z]]$ be its affine Grassmannian. Define $\mathcal{D}(Gr_G)_{crit}-mod$ to be the category of right $D$-modules on $Gr_G$ twisted by the ...
- 3,019
3
votes
0
answers
144
views
Computation of nearby cycles, monodromy action and action of $sl_{2}$ on $\operatorname{gr}(Ψ)$ for Picard-Lefshetz family
Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{1}$ be a map that sends $(x,y)$ to $xy$. Let $U \hookrightarrow \mathbb{A}^{2}$ be the preimage $f^{-1}(\mathbb{A}^{2} \setminus \{0\})$ and $X:=f^{−1}(0)...
- 103
2
votes
0
answers
169
views
Counting points of parabolic Springer fibers
Let $G$ be a reductive group over an (algebraically closed ) field. To each parabolic subgroup $P \subseteq G$ and $x \in G$ we can consider two types of partial Springer fibers associated to it :
$$1)...
- 1,061
2
votes
0
answers
640
views
Areas of algebraic geometry useful for geometric representation theory
What topics/areas of algebraic geometry (aside from perverse sheaves/D-modules, etale cohomology, and possibly derived algebraic geometry) is it useful to learn/master if one is interested in doing ...
- 2,964
1
vote
0
answers
146
views
Factoriality of schubert cells in affine flag variety
Take for simplicity $G=SL_n$ and consider the affine flag variety $Fl=G(\mathbb{C}((t)))/I$ for $I$ the Iwahori corresponding to the Borel of upper triangular matrices of determinant one.
For each $...
- 3,472
0
votes
0
answers
82
views
The closure of the orbits of $\mathcal{F} \times \mathcal{F}$
Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$, and $\dim V=d$. Let $G=GL(V)$, $P$ is the parabolic subgroup of $G$. Let $\mathcal{F}$ be the set of all partial ...
