Questions tagged [geometric-representation-theory]
The geometric-representation-theory tag has no usage guidance.
208 questions
-5
votes
0
answers
124
views
Is a quiver variety a moduli stack of quiver representations?
As the title, I was just wondering is a quiver variety a moduli stack of quiver representations? I am familiar with quiver varieties but not that familiar with moduli stacks, so I was just wondering ...
- 133
2
votes
0
answers
97
views
An injective map in equivariant algebraic K-theory
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
- 653
5
votes
1
answer
300
views
Operations on filtered / graded vector spaces via $\mathbb{A}^1 / \mathbb{G}_m$
A well known fact (e.g. Moulinos - The geometry of filtrations) is that one can describe the category of filtered vector spaces as $\operatorname{FilVect} \simeq \operatorname{QCoh}(\mathbb{A}^1/\...
- 834
3
votes
0
answers
122
views
Canonical basis in equivariant K-theory of the Springer resolution
In Definition 15.0.2 of the notes from a course by Bezrukavnikov there is a characterization of canonical basis in K-theory of a Springer fiber which is due to Lusztig. This characterization is in ...
- 2,964
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
8
votes
1
answer
204
views
Relative de Rham cohomology of flag varieties
Let $B \subset P \subset G$ be a parabolic, Borel, and reductive (split) group over the complex numbers. Consider the projection $\pi: G/B \to G/P$, I am interested in computing $R\pi_{*} \Omega^{\...
- 461
13
votes
0
answers
188
views
Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
- 3,772
6
votes
0
answers
116
views
Properties of a functor from Soergel bimodules to Soergel modules
I am looking for an extension of a result of Riche-Soergel about a functor which maps Soergel bimodules to Soergel modules.
Fix a given Coxeter system $(W,S)$, together with a (reflection faithful) ...
- 211
5
votes
0
answers
113
views
Smoothness of some varieties related to the Slodowy slice
Let $G$ be a complex algebraic group with simple Lie algebra $\mathfrak{g} = \operatorname{Lie} G$. Let $\mathcal{B}$ be the flag variety consisting of all the Borel subalgebras of $\mathfrak{g}$.
Let ...
- 51
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
1
answer
170
views
Factoring out an element of a root subgroup to make a conjugation integral
Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix
$$\begin{pmatrix} a & \varpi b \\ c & d \...
- 503
1
vote
0
answers
79
views
Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
- 1,677
2
votes
0
answers
129
views
Flag variety type Beilinson resolution
The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
- 653
3
votes
0
answers
119
views
Different definitions of the thick affine flag variety
I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same.
Some ...
- 403
3
votes
1
answer
392
views
Representation theory and topology of Teichmüller space
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\char{char}$I am reading a note on Teichmüller space, and I come across a somewhat algebraic problem in the picture below,...
- 77
1
vote
0
answers
162
views
Definition of nearby cycle over an affine line
In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...
- 291
1
vote
0
answers
133
views
What is Pic of the torus global affine Grassmannian?
Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$.
What is $\text{Pic}(\text{Gr}_{T,X^n})$?
Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $...
- 5,701
8
votes
1
answer
372
views
Two versions of parabolic category O
Let $ \mathfrak{g} $ be a semisimple Lie algebra with corresponding complex semisimple group $ G$. Let $ P \subset G $ be a parabolic subgroup. Let $ W^P $ be the set of shortest coset ...
- 4,612
2
votes
0
answers
101
views
Multiplicities of components of a Springer fibre
Given a Springer fibre of type A, are multiplicities of its irreducible components known in general, or at least in the special cases of two-row/hook types?
By multiplicities I mean considering a ...
- 1,677
0
votes
1
answer
205
views
What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
- 275
2
votes
0
answers
123
views
Lie Algebra representations outside of generalized central characters
For a simple Lie algebra $\mathfrak{g}$, we can view its category of representations as fibered over $\operatorname{Spec}Z(\mathfrak{g})$ (a representation will lie over a point if the center's action ...
- 834
3
votes
0
answers
110
views
additive vs multiplicative quiver/hypertoric varieties - properties
It is a standard fact that a smooth Nakajima quiver variety / hypertoric variety $X$ has the following properties:
It is holomorphic symplectic $(X,\omega)$, in fact 1'. hyperkahler
It has a ...
- 1,677
2
votes
0
answers
180
views
Are parabolic Springer fibers equal dimensional?
Let $G$ be a simple algrbraic group （ of type BCDEFG ） over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \...
- 653
2
votes
1
answer
227
views
Component group of stabilizer group of a nilpotent element
Let $G$ be a semisimple complex algebraic group, and $\mathfrak{g}$ be its Lie algebra. $G$ acts on $\mathfrak{g}$ by adjoint action. Let $x$ be an nilpotent element in $\mathfrak{g}$, and $G(x)\...
- 653
3
votes
0
answers
58
views
Locally finite positive energy modules generated by singular vectors at positive levels?
This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background.
Backround on affine Lie algebras. Let $\...
user108998
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
14
votes
1
answer
697
views
What is the Zhu algebra of a vertex algebra "really"?
Given any vertex algebra $V$, you can give a particular quotient $\DeclareMathOperator\Zhu{Zhu}\Zhu V=V/\cdots$ an algebra structure using (a small amount of) the vertex algebra structure. As far as ...
- 5,701
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
5
votes
0
answers
212
views
Relations between Whittaker functions/W algebras and Stokes data/resurgence
Skippable background: A Whittaker function is more or less a function on a flag manifold which is twisted-invariant for the action of a unipotent subgroup. E.g. consider functions $f$ on $\mathbf{P}^1$...
- 5,701
8
votes
1
answer
295
views
What are twisted Verma modules? Basic properties?
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...
- 5,701
7
votes
0
answers
223
views
Twisted D-module structure on pushfoward of structure sheaf of Bruhat cell
Apologies for the basic question, but my experience on math.stackexchange tells me that this will go unanswered there.
Background: In the principal block, the dual Verma modules (with highest weight $...
- 311
5
votes
1
answer
279
views
Exterior products of irreducible representations of sl_2(C)
It is well-known that $\mathfrak{sl}_2(\mathbb{C})$ admits exactly one irreducible representation $V_n$ of dimension $n+1$ for all $n\geq 0$. It is explicitly given by the action on homogeneous ...
- 617
12
votes
0
answers
2k
views
Roadmap to geometric Langlands for a mathematical physics student
I am a student of both mathematics and physics, who has recently been studying string theory. My mathematics background is mostly differential geometry (principal bundles, Lie groups, etc.), although ...
- 349
2
votes
0
answers
124
views
Levi quotients of parahorics in loop group
I am looking for some references on Levi quotients of parahorics in $LG = G(\mathbb{C}((t)))$, $G$ being an algebraic group with Weyl group $W$.
I have read that parahoric subgroups of $LG$ are in ...
- 57
4
votes
0
answers
153
views
Verlinde formula via chiral/vertex algebras
The Verlinde formula gives an explicit formula for any finite dimensional simple Lie algebra $\mathfrak{g}$ an explicit formula for the dimension of conformal blocks for the associated WZW conformal ...
- 5,701
5
votes
0
answers
127
views
Is there a derived version of affine Schur-Weyl duality?
One version of affine Schur-Weyl duality states that there is a fully faithful functor from representation of $A_r$ affine Hecke algebra to the representation of $A_n$ affine Lie group assuming $r<...
- 189
1
vote
1
answer
90
views
Dimension of maximal split subtorus and fixed point subspace of Lie algebra
Let $F = \mathbb{C}((t))$. Let $G$ be a complex semisimple algebraic group. Then conjugacy classes of maximal tori in $G(F)$ are in bijection with conjugacy classes in $W$, the Weyl group of $G$ with ...
- 11
4
votes
1
answer
348
views
Verma modules and Borel–Weil
Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
- 181
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 ...
1
vote
0
answers
216
views
Visualizing the affine Bruhat decomposition for $\operatorname{SL}_2$
$
\newcommand\Fl{\mathcal{F}\!\ell}
\newcommand\numC{\mathbb{C}}
\newcommand\numZ{\mathbb{Z}}
\newcommand\ringO{\mathbb{O}}
\newcommand\ringK{\mathbb{K}}
\newcommand\power{\...
- 295
0
votes
0
answers
231
views
Relationship between vector bundles and modules
THE GROTHENDIECK RING IN GEOMETRY AND TOPOLOGY - M.F. ATIYAH
§1. The Grothendieck ring in homotopy theory
I am going to be talking about vector bundles, i.e. fibre bundles with
fibre a vector space ...
- 61
4
votes
0
answers
172
views
Frobenius kernel by Greenberg functor
I was not able to find much on representation theory (with a geometric perspective) of algebraic groups over local artinian rings. In particular, studying on the classic book of Jantzen, I was asking ...
- 41
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 :
$$1)...
- 1,061
4
votes
1
answer
189
views
Equivalence of categories of modules over $U(\mathfrak{g})$ on which $Z(\mathfrak{g})$ acts by central characters
$\newcommand{\g}{\mathfrak{g}}$Setting: $\mathfrak{g}$ is a semisimple complex Lie algebra. Here $\chi_\lambda$ denotes the central character corresponding to the action of $Z(\g)$ on a highest weight ...
- 311
5
votes
0
answers
152
views
Concrete computation of pullback of the $D$-module $\mathbb{C}[t,t^{-1}]$
I have some problems in calculating some example explicitly. Consider
$$ \{0\} \overset{i}{\rightarrow} \mathbb{C} \overset{j}{\leftarrow} \mathbb{C}^*.$$
Then $Rj_+\mathbb{C}[t,t^{-1}] = \mathbb{C}[t,...
- 1,168
6
votes
1
answer
543
views
Chebotarev density theorem and pure weight local systems
How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper.
Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...
- 677
5
votes
1
answer
187
views
Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation.
One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan ...
- 5,701
20
votes
0
answers
597
views
Your favourite alternative proof of Borel–Weil–Bott
There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple ...
5
votes
1
answer
266
views
Two identities involving Ext functors in the context of D-modules
I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety ...
- 541
7
votes
0
answers
160
views
comparison of polynomial loop group and smooth loop group
I have a question about Section 8.6 of Pressley-Segal's Loop groups book. Let $G$ be a compact, connected Lie group. Proposition 8.6.6 concerns the comparison of homotopy type between its polynomial ...
- 959