The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
1answer
49 views

A question related to the semisimplification of a Weil-Deligne representation

I have been trying to find the answer to this question, I think it must not be hard but I don't get it. I have a Weil-Deligne representation ($\rho,N$) of the Weil group $W$ of $Q_p$, that is $\rho$ ...
15
votes
1answer
380 views

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 ...
10
votes
1answer
163 views

Meaning of topological tensor products in Frenkel-Gaitsgory

The appendix to http://arxiv.org/abs/math/0508382 by Frenkel & Gaitsgory (following an earlier work of Beilinson) describes three different monoidal structures, denoted by $\otimes^!,\otimes^*,$ ...
22
votes
6answers
2k views

Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...
9
votes
3answers
845 views

Why is the simple trace formula a weaker tool than the Arthur trace formula?

What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker? (So I do not mean weaker in the sense ...
9
votes
0answers
195 views

Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...
7
votes
0answers
110 views

Homological contractibility of a prestack

This question is in reference to Gaitsgory's preprint Contractibility of the space of rational maps. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine ...
4
votes
1answer
449 views

What are local spaces and what are they good for?

Factorization structures have been popular in the past decade. Recently a variant of this structure has been suggested by Ivan Mirkovic (and possibly collaborators). This variant, which goes under the ...
13
votes
2answers
439 views

Langlands duality and multiplying cocharacters

Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation) of the Langlands dual group ...
3
votes
1answer
340 views

Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...
11
votes
1answer
651 views

Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.  Background : I recently chanced ...
23
votes
2answers
2k views

Current Status on Langlands Program

The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where ...
1
vote
0answers
127 views

Twists in “Eisenstein property” in Geometric Langlands

I am trying to read and understand (parts of) Gaitsgory's “Outline of the proof of the Geometric Langlands conjecture for GL(2)” [arXiv link]. In Section 6.4.8 he states "Property Ei", which basically ...
7
votes
1answer
995 views

On Geometric Langlands Correspondence

The Geometric Langlands correspondence introduced by Drinfeld and Laumon conjectures a 1 to 1 correspondence between (A) local systems on a projective smooth curve over a field and (B) (Hecke ...
35
votes
1answer
6k views

Consequences of Geometric Langlands

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...
5
votes
0answers
269 views

Real representations of G = those of Langlands dual and maps of a cylinder

There is a result about the real representations of a simple Lie group $G$ which is known as Soergel conjecture or Vogan duality. We'll focus on the formulation that for a fixed character of $G$ ...
2
votes
0answers
168 views

Langlands correspondence for reducible representations

The Langlands correspondence over a function field matches irreducible $n$-dimensional Galois representations with cuspidal irreducible automorphic representations. My question is: Is there any idea ...
10
votes
7answers
3k views

Langlands Dual Groups

Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does ...
0
votes
1answer
344 views

Reference on Casselman-Shalika formula for GL(n) and PGL(n)?

I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.
2
votes
0answers
404 views

Complex Finite Dimensional Representation of GL(N,C)

What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$? We already know all the complex finite dimensional linear representation of SU(N).
1
vote
2answers
242 views

Symmetric and Exterior products of sl(n,C)-module

Let M be the $sl(n,C)$-representation of the inclusion $sl(n,C)\hookrightarrow gl(n,C)$. Let q be a symbol. $f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$ $g(q)=\sum_{i=0}^\infty Sym^iM \; q^i$ ...
8
votes
2answers
591 views

Examples of Eigensheaves outside of langlands

In geometric Langlands, one looks at correspondences of the form $$ Bun_n(X) \leftarrow Hecke \rightarrow X\times Bun_n(X)$$ and calls a sheaf on the lefthand space Hecke eigensheaf, if pulling ...
6
votes
0answers
562 views

The De Rham Stack and $\text{LocSys}$

Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$): $Maps(S, LocSys_G(X)) := ...
1
vote
0answers
260 views

Orbit stratification of semi infinite flag manifold?

Denote semi infinite flag manifold by $Fl_{\infty/2}=G((t))/N_-((t))H[[t]]$, denote $B_-((t))=N_-((t))H[[t]]$ from the book of Frenkel and Benzvi" Vertex algebras and algebraic curves", They take ...
2
votes
0answers
232 views

Local counterpart of the NON-Hitchin Hecke eigen-sheaves ?

Insight of Beilinson and Drinfeld at early 90-ies - that Hitchin's D-modules are Hecke eigen-D-modules. However they are NOT all Hecke-eigensmodules and actually they are only the half-dimensional ...
13
votes
1answer
1k views

What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ?

What is the relation between Lafforgue's result on Langlands and Frenkel-Gaitsgory-Vilonen ? ( http://arxiv.org/abs/math/0012255 , http://arxiv.org/abs/math/0204081 ) Does one imply other ? If not ...
17
votes
3answers
2k views

A good example of a curve for geometric Langlands

I'm currently working through Frenkel's beautiful paper: http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf. I'm looking for a good example of a projective curve to get my hands dirty, and go ...
10
votes
3answers
2k views

ubiquitous quantum cohomology

Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...
10
votes
2answers
816 views

The affine Grassmannian and the Bogomolny equations

In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more generally, in the ...
1
vote
2answers
619 views

Opers, connections

My questions here are from my attempt at trying to understand the definition on pg 15 in [FG2]-"Local Geometric Langlands Correspondence & Affine Kac-Moody Algebras" ...
4
votes
1answer
1k views

Why is the Arthur trace formula so powerful?

Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all automorphic ...
6
votes
0answers
357 views

Generalizations of Drinfeld Symmetric Space? (Drinfeld homogeneous space, Drinfeld flag variety?)

Are there natural generalizations of the Drinfeld symmetric space? For $\mathbb{K}$, a non-Archimedean local field, the Drinfeld symmetric space can be defined as the complement of all ...
5
votes
1answer
673 views

Fiber functor of category of D-module on affine Grassmannian.

Geometric Satake correspondence allows us to construct Langlands dual group in a canonical way. In Mirkovic-Vilonen's paper, they prove that category of spherical perverse sheaves is an commutative ...
43
votes
1answer
3k views

Double affine Hecke algebras and mainstream mathematics

This is something of a followup to the question "Kapranov's analogies", where a connection between Cherednik's double affine Hecke algebras (DAHA's) and Geometric Langlands program was mentioned. I ...
7
votes
2answers
868 views

A question on group action on categories

Let $Gr$ be the affine Grassmannian of $G=G((t))/G[[t]]$, and let $Perv(Gr)$ be the category of perverse sheaves on $Gr$. We have action of $G((t))$ on the left-hand side of $Perv(Gr)$, also we have ...
28
votes
3answers
2k views

Topological Langlands?

In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
2
votes
1answer
549 views

Understanding formula in Frenkel-Witten

I'm not the person to understand everything in Geometric Endoscopy and Mirror Symmetry, but some parts of it are reasonably clear to me. In particular, one of the main objects, mathematically ...
4
votes
1answer
780 views

Explanation for Satake correspondence

Some time ago I was told there's an interesting classical Satake correspondence which I will write as $$[\mathop{\mathrm{disk}} \Rightarrow G] \\,\backslash\\, [\mathop{\mathrm{disk}^\times} ...
11
votes
3answers
2k views

What is Eisenstein series?

There are several related questions here, the latter being especially interesting. We know the classical Eisenstein series. What are the Eisenstein series on a group G and why they are interesting? ...
5
votes
2answers
408 views

Sheaves on Bun_G

What's the background I need to know to understand the conjectural D (Bun_G) =?= O(LocSys) from this question. I know the LHS is about the derived category of ...
7
votes
2answers
2k views

What is an Oper?

Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems. I keep reading that there's a substack of "opers" but am having trouble locating a ...
4
votes
1answer
254 views

Reverse Langlands transform

What os the meaning of a reverse Langlands transform to which Drinfeld seems to refer?