Questions tagged [geometric-representation-theory]

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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)...
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4 votes
1 answer
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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 ...
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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,...
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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 $\...
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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 ...
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13 votes
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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 ...
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1 answer
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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 ...
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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 ...
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4 votes
1 answer
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A problem on Kazhdan–Lusztig theorem

I am reading Chriss, Ginzburg's book Representation theory and complex geometry. In theorem 7.2.16 it says that the convolution action of the Steinberg variety $St=\tilde{\mathcal{N}}\times_\mathfrak{...
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Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group

What is the geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group? Could Springer theory of Weyl group representations be used to obtain such a geometric meaning?
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Error in Proposition 8.7.1 of Pressley–Segal

Let $G$ be a connected, compact Lie group, $T$ a maximal torus. Let $LG$ be the group of smooth (or polynomial) loops and $X=LG/T$ the affine flag variety ($T$ acts say by right multiplication). In ...
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12 votes
1 answer
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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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Sign error in Chriss-Ginzburg?

On page 118, Theorem 2.7.26 (iii) in Chriss-Ginzburg "Representation Theory and Complex Geometry" there is a formula for the convolution of the classes of conormal bundles of $Y_{12}\subset ...
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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 ...
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1 answer
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Nakajima quiver varieties for ADE quiver with one dimensional framing

Let $Q$ be a quiver of type $ADE$, $I$ is the set of vertices of $Q$. Let $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ be a Nakajima quiver variety for such quiver (here ${\mathbf{v}}=(v_i)_{i \in I}$ is ...
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Reference for combinatorics with view towards representation theory/algebraic geometry

I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
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Poincare polynomials for Borel Moore homology and fibrations

For an algebraic variety $X$ over $\mathbb{C}$, we denote $H_k(X)$ as its Borel-Moore homology of degree $k$. Let us define the Poincare polynomial associated it by $$P(X)=\sum_{k\in \mathbb{N}}dim ...
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Stratified fibration property of the "Ran" affine Grassmannian

Let us consider the so-called Ran Grassmannian $Gr_{Ran}$, i.e. the geometric object defined e.g. in [Zhu, An Introduction to the affine Grassmannian and the Geometric Satake equivalence, Definition 3....
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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 (...
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  • 273
8 votes
1 answer
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Why are VOA characters modular forms (geometrically)?

In Zhu's seminal paper, he proves (5.3.2) that if $V$ is a vertex algebra the character of all of its modules are modular forms! (This is not literally true- there are conditions). I have always found ...
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5 votes
1 answer
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Statement of local geometric Langlands

A precise statement of the global geometric Langlands conjecture is well-known. However, I am unable to find a statement of the local Langlands conjecture. Does anyone have a modern statement or a ...
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1 vote
1 answer
157 views

Irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$

I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ of finite dimension (in the following by "representation" I mean a representation of ...
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1 vote
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Characteristic functions of character sheaves on tori

I am currently reading a set of lecture notes by V. Ostrik and G. Williamson, Character sheaves, tensor categories and non-Abelian Fourier transform. In Theorem 1.1, they make the assertion that the ...
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8 votes
3 answers
441 views

Intuitive reason that the regular representation is a uniform function

Corollary 12.14 of Digne-Michel's book Representations of finite groups of Lie type gives various decompositions of the regular representation $\operatorname{reg}_G$ in terms of the Deligne-Lusztig ...
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6 votes
1 answer
273 views

Irreducible representations of product of profinite groups

It is a standard fact in the representation theory of finite groups that for $G,H$ finite groups, all of the irreducible representations of $G \times H$ are the external tensor product of irreps of $G$...
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3 votes
2 answers
228 views

Frobenius reciprocity for Deligne-Lusztig induction/restriction

I am currently trying to understand the properties of Deligne-Lusztig induction, following Carter's Finite groups of Lie type and Digne-Michel's Representations of finite groups of Lie type. I am ...
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9 votes
1 answer
233 views

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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4 votes
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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 ...
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4 votes
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On the order of the head of product of two simple modules over Quiver Hecke Algebras

My question is: We assume the underlying quiver is a Dynkin quiver. Let $L(\lambda)$ and $L(\mu)$ be two simple modules over Quiver Hecke algebra $R$ where $\lambda$ and $\mu$ are two Konstant ...
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3 votes
1 answer
230 views

Iterating specialization of sheaves?

This is a question about the operation of taking the specialization of sheaves along a subspace. I'll recall the settings in which I've encountered a notion of specialization of sheaves: The real, ...
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3 votes
1 answer
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Geometric explanation for the $\widehat{\mathfrak{sl}}_2$ free field realisation (alias $L_1(\mathfrak{sl}_2)\to V(\sqrt{2}\mathbf{Z})$)

Let $\sqrt{n}\mathbf{Z}$ be the one dimensional lattice, whose generator has length $2$. Associated to this is a lattice vertex algebra $$V(\sqrt{2}\mathbf{Z}).$$ We also have the simple quotient of ...
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2 votes
1 answer
208 views

Hyperkahler and symplectic complex geometry: reference?

I would need some references regarding symplectic and hyperkahler (complex) geometry. My background is mostly from algebraic geometry and I know a little bit the basics on Kahler manifolds. I would be ...
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1 vote
0 answers
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Coincidence of notation in the classification of representations of affine Hecke algebras

This is spurred by a short discussion I had in the comments of this MO question. In Ginzburg's 1998 paper, https://arxiv.org/abs/math/9802004v3, or equivalently in the book by Chriss and Ginzburg, &...
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6 votes
1 answer
260 views

Are there cases in which the Weyl group _does_ act on the flag variety/springer fiber?

In nearly every reference on the classical springer correspondence (for example Chriss/Ginzburg's book on Complex Geometry) it is stated that the action of the Weyl Group on the homology of the ...
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3 votes
0 answers
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Fundamental representation bases and generalized minors

Let $G$ be a simple simpy connected complex algebraic group. I was wondering if there is a clear relationship between the generalized minors (defined by Berenstein, Fomin and Zelevinsky) and bases of ...
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5 votes
0 answers
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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 ($...
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3 votes
2 answers
287 views

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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4 votes
0 answers
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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 $\...
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6 votes
1 answer
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Modern proofs of the Verlinde formula?

Let $G$ be a semisimple algebraic group and $\Sigma$ a smooth proper curve. Then $\text{Bun}_G(\Sigma)$ comes equipped with a line bundle $\mathcal{L}$ which generates the torsion free part of $\text{...
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  • 4,004
7 votes
2 answers
367 views

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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5 votes
1 answer
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Drinfeld Sokolov and the semiinfinite flag variety

For a long time I've been confused about Drinfeld Sokolov/BRST reduction/semiinfinite cohomology for affine Lie algebras. Most treatments define it in what to me feels like a fairly ad-hoc way, by ...
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8 votes
0 answers
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Representation theory of Chevalley groups as a categorical trace

Dennis Gaitsgory's 2016 preprint, From Geometric to Function-Theoretic Langlands (or How to Invent Shtukas) includes in the third section a very compressed but suggestive discussion of the ...
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3 votes
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Are there six functors for twisted D modules?

Is there a notion of holonomic D module which admits the six functor formalism in the world of twisted D modules? Recall that twisted D modules on $X$ are well-defined for any $T$ torsor $\tilde{X}\...
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4 votes
0 answers
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Confusion about twisted Vermas in Feigin-Frenkel

Let $G$ be a finite dimensional semisimple algebraic group, and for $s\in W$ write $i_s: F_s=B^+sB^-/B^-\to F$ for the $s$th Bruhat cell in the flag variety $F$. Then (in Affine Kac-Moody Algebras and ...
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  • 4,004
7 votes
0 answers
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Conceptual proof of braid group actions on quantum groups

Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $U_q(\mathfrak{g})$ is the proof which is least conceptual. The original paper ...
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9 votes
0 answers
518 views

Geometric meaning of twist

It is sometimes the case that a Galois representation or a motive acquires a desirable property only after a twist by a character, usually a Tate twist. The latest example of this I have come across ...
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4 votes
0 answers
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Computing $\mathcal D$-module direct image along group action map

Say everything is over $\mathbb C$, and I have an action $act: N \times X \to X$ of an affine algebraic group $N$ on a smooth variety, say with finitely many orbits. I'm trying to compute the $\...
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1 vote
1 answer
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Two different formulations of the Bott–Samelson resolution

There seem to be two formulations of the Bott–Samelson resolution flowing around. For concreteness, let $ G = \mathrm{GL}_{n} ( \mathbb{C} ) $ with the Borel subgroup $ B \subset G $ of upper ...
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2 votes
1 answer
96 views

Representation variety in $\mathrm{SU}(p,q)$

$\DeclareMathOperator\SU{SU}$Let $\Gamma$ be a cocompact oriented Fuchsian group, and consider the representation variety $\textrm{Hom}(\Gamma, \SU(p,q))$. Consider a point $\rho : \Gamma \to \SU(p,q)$...
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  • 543
6 votes
0 answers
164 views

Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules

Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...
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