Questions tagged [geometric-representation-theory]
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208 questions
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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 ...
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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 ...
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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/\...
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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 ...
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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) ...
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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 $...
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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
$$\begin{pmatrix} a & \varpi b \\ c & d \...
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8
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1
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Relative de Rham cohomology of flag varieties
Let $B \subset P \subset G$ be a parabolic, Borel, and reductive (split) group over the complex numbers. Consider the projection $\pi: G/B \to G/P$, I am interested in computing $R\pi_{*} \Omega^{\...
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Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
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Smoothness of some varieties related to the Slodowy slice
Let $G$ be a complex algebraic group with simple Lie algebra $\mathfrak{g} = \operatorname{Lie} G$. Let $\mathcal{B}$ be the flag variety consisting of all the Borel subalgebras of $\mathfrak{g}$.
Let ...
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What advantages do perverse sheaves provide over D-modules? (or vice versa)
My question is as in the title: What advantages do perverse sheaves provide over D-modules? (or vice versa)
As a specific example: could something like the modular generalized Springer correspondence ...
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Different definitions of the thick affine flag variety
I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same.
Some ...
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Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
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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 ...
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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? Are there any ...
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Branching rule of $S_n$ and Springer theory
Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
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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 ...
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3
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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,...
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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", ...
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1
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Coulomb branch varieties and symplectic singularities
I was recently looking at the survey article of Fu on symplectic resolutions which has a number of open questions and conjectures at the end. (I think one of these was existence of a classification ...
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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 $...
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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 ...
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2
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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 ...
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1
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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 ...
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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 ...
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180
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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 \...
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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)\...
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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 $\...
CommunityBot
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1
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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 ...
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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 ...
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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 $\mathbf{P}^1$...
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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 ...
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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 $\...
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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 $...
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1
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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 ...
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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 ...
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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 ...
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What do Hecke eigensheaves actually look like?
Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
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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 ...
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3
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3
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Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
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Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay?
Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the ...
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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<...
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1
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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 $\...
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1
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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 ...
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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 ...
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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{\...
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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 ...
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0
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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
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0
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169
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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
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Symplectic resolutions amongst cotangent bundles
It is known that a generalized flag variety $X=T^*(G/P)$ is a (symplectic) resolution of singularities of its affinization $X^\text{aff}\mathrel{:=}\operatorname{Spec}(H^0(X,\mathcal{O}_X))$. In type ...
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