Questions tagged [geometric-representation-theory]
The geometric-representation-theory tag has no usage guidance.
197
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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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1
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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
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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 ...
- 13
1
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0
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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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3
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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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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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2
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284
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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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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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0
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208
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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 ...
user108998
4
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69
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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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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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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 ...
- 5,565
2
votes
1
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358
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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 ...
1
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0
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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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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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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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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
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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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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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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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2
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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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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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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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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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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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356
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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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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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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
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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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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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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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Beilinson-Bernstein for nonintegral levels
If one wants to understand representations of $\mathfrak{g}$ (a finite dimensional semisimple Lie algebra) of weight $\lambda$, the happiest you could be is if $\lambda+\rho$ is (integral) regular ...
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Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian
The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ ...
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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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Nakajima reflection functors and the flavour/framing group action
Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$
$$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
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Coefficient ring of Satake isomorphism
Let $G$ be a split reductive group over local field $F$, $G^L$ be the (complex) Langlands dual group of $G$. Denote $H$ to be the $\mathbb{Z}$-Hecke algebra of $G$, that is the ring of $G(\mathcal{O}...
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Freeness of completed homology over universal deformation ring
In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...
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3
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261
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Nontrivial Poisson relations for affine Poisson algebras
Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, ...
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Relations between double coinvariants and affine Springer fibers
Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme.
There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv....
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Presentation of $H^2(\overline{M}_{0,n},\mathbb{Z})$ as an $S_n$-module?
Let $\overline{M}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points. Let $I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2,
|S^c|\geq2\}$ be the set of partitions of $\{1,\dots n\}$ ...
user39380
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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 ...
- 2,490
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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 ...
- 2,490
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Hochschild cohomology of (generalizations) of Khovanov's arc algebra
Backgroud: In his seminal paper A functor-valued invariant of tangles, Khovanov (among many other things) introduced the arc algebra $H^{n}$ and several functors between $H^{n}$ and $H^{m}$ related to ...
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Projection of conormal bundle of Schubert variety under Springer resolution
Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ ,
$\mu:T^*(G/B)\to \mathcal{N}$ ...
- 849
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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 $...
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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 ...
- 2,969
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Convolution in K-Theory via an Example (From StackExchange)
I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still ...
- 841
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214
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Non-Archimedean Soergel
If you are studying finite groups, and you have classified all finite simple groups, your job is not done yet since potentially there might be some interesting phenomena in how two finite simple ...
user141500
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121
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Geometric Arthur-Selberg
What should be the analogue of the Arthur-Selberg trace formula in the geometric Langlands theory?
user140765
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Intersection of components in Springer fibre of type A
From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...
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