# Questions tagged [geometric-langlands]

The geometric-langlands tag has no usage guidance.

65
questions

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### On a remark of Langlands

I'm been wondering about this for a while and hope someone can enlighten me.
In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I ...

**9**

votes

**1**answer

217 views

### Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.
Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.
Are ...

**3**

votes

**1**answer

216 views

### Understanding moduli of shtukas of non-minuscule cocharacter

I have kind of a soft question. I've studied the basics of L. Lafforgue's proof of function field Langlands for GLn, and its use of the moduli of shtukas with two legs, and the cocharacters $[1,0,\...

**8**

votes

**2**answers

459 views

### Geometric Langlands: From D-mod to Fukaya

This post is rather wordy and speculative, but I promise there is a concrete question embedded within. For experts, I'll open with a question:
Question: Given a compact Riemann surface $X$, why ...

**2**

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**0**answers

84 views

### Integral kernels for geometric langlands

My apologies for the imprecise question(s), it should be clear enough that I´m a complete beginner in this subject.
The (de Rham) Geometric Langlands Conjecture over $\mathbb{C}$ takes as input a ...

**4**

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179 views

### Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?

In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...

**7**

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663 views

### Ramified Geometric Langlands

Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...

**1**

vote

**1**answer

174 views

### What is the relationship between the sheaf-function dictionary and cohomology of moduli spaces of shtukas?

I'm a newcomer to the geometric Langlands setting, and have mostly consulted surveys like Laumon's overview of L. Lafforgue's proof or Frenkel's recent advances survey, so apologies if this is ...

**4**

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285 views

### What is the analogy between the moduli of shtukas and Shimura varieties?

I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...

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963 views

### Number Theory and Gravity

Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands at IAS (1967, 1970), it seeks to relate Galois ...

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361 views

### Analog of Ramanujan-Petersson conjecture in Geometric Langlands

The Ramanujan conjecture asserts that
\begin{align}
|\tau(p)|\leq 2p^{11/2}
\end{align}
where $\tau(p)$ is the $p^{th}$ Fourier coeffecient in the q-expansion of the weight 12 cusp form $\Delta(z)$. ...

**7**

votes

**1**answer

325 views

### Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...

**7**

votes

**1**answer

336 views

### Beilinson-Drinfeld quantization and stable bundles

To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. ...

**7**

votes

**1**answer

404 views

### Implications of gauge symmetry breaking on the spectral side of geometric Langlands?

Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^*[2]/W$ (...

**3**

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85 views

### Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for ...

**4**

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123 views

### Langlands dual and integrable representations

Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...

**6**

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**1**answer

338 views

### Examples of function fields Langlands for small genus (<= 2)

See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...

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255 views

### Bi-Whittaker functions and local Langlands compatibility

I'm trying to figure out the arithmetic analogue of a key conjecture in the geometric local Langlands correspondence. Briefly, one expects for $K=\mathbb{C}((t))$ an equivalence of dg categories $$\...

**16**

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**1**answer

712 views

### References for Langlands classification

I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.
My ...

**15**

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**1**answer

983 views

### LMS Lectures on Geometric Langlands

Everybody knows how insightful are David Ben-Zvi talks (and comments/answers here on mathoverflow). I was trying to watch the LMS 2007 Lecture Series on Geometric Langlands by David, supposedly made ...

**3**

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**1**answer

628 views

### Global Langlands function fields

Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields?
What is the current status, more generally?
Related ...

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537 views

### Any progress on Strominger-Yau and Zaslow conjecture?

In 2002 Hausel - Thaddeus interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it
Let $\pi : E \to Σ$ a complex vector bundle of rank $r$ and ...

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183 views

### Feigin-Frenkel centre and opers for reductive Lie algebras

Edward Frenkel (together with Boris Feigin and others) has proven many interesting results connecting the representation theory of an affine Kac-Moody algebra at the critical level with the geometry ...

**1**

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**1**answer

120 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 $\rho$ ...

**17**

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**1**answer

1k 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 ...

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369 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

**1**answer

207 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 $\mathbb{C}$-...

**10**

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**1**answer

327 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 $\otimes^!,\otimes^*,$ ...

**6**

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**1**answer

735 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 ...

**35**

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### 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 ...

**13**

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**2**answers

534 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 $^...

**5**

votes

**1**answer

780 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 ...

**14**

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**1**answer

1k 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 ...

**35**

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**2**answers

5k 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 ...

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156 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 ...

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184 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 ...

**9**

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**1**answer

2k 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 eigen-)...

**0**

votes

**1**answer

576 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.

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518 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).

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**2**answers

388 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.
$f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$
$g(q)=\sum_{i=0}^\infty Sym^iM \; q^i$
...

**10**

votes

**2**answers

903 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 ...

**7**

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854 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)) := \text{...

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305 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 ...

**15**

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**1**answer

2k 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 ...

**2**

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**0**answers

342 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 ...

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**2**answers

908 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" (http://arxiv.org/PS_cache/...

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1k 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 ...

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2k 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 ...

**9**

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**3**answers

1k 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 ...

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### 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 $\mathbb{K}$-...