All Questions
Tagged with geometric-langlands langlands-conjectures
25 questions
40
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Implications and consequences of the recent proof of the geometric Langlands conjecture
I am a beginner in mathematical physics and geometric Langlands, having very limited knowledge in both fields so far.
The proof of geometric Langlands conjecture is published a few months ago. What ...
9
votes
0
answers
963
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Geometrization of the global Langlands correspondence?
Fargues-Scholze famously describe arithmetic local Langlands via global geometric Langlands on the Fargues-Fontaine (FF) curve.
The FF curve acts like an algebraic curve over $\mathbb{C}_p$ (its ...
2
votes
0
answers
177
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Why are they called reductive groups? [duplicate]
The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?
1
vote
1
answer
375
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From Galois representations to automorphic forms for $\mathfrak{sl}_2$ (via Drinfeld's shtukas)
Drinfeld-Lafforgue have proven function fields Langlands conjectures in type A: see https://arxiv.org/pdf/math/0212417.pdf (Laumon's survey in English), https://arxiv.org/pdf/math/0212399.pdf (...
16
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2
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2k
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Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these notes or this lecture) some important aspects of the Langlands correspondence are stated in the language of topological ...
5
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1
answer
830
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Categorical-geometric Langlands for tori
Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?...
8
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0
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454
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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
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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
607
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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,\...
18
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0
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1k
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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 ...
7
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1
answer
555
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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 &...
6
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0
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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 $$\...
17
votes
1
answer
1k
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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 ...
3
votes
1
answer
988
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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 ...
24
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1
answer
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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 ...
43
votes
7
answers
13k
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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 ...
6
votes
1
answer
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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 ...
42
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2
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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 ...
2
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0
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191
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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 ...
2
votes
1
answer
753
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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
0
answers
562
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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).
3
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0
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334
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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
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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 ...
8
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1
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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 ...
9
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3
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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 ...