Skip to main content

All Questions

Filter by
Sorted by
Tagged with
40 votes
1 answer
2k views

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 ...
Qichang Huangfu's user avatar
9 votes
0 answers
963 views

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 ...
David Corwin's user avatar
  • 15.4k
2 votes
0 answers
177 views

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?
Ola Sande's user avatar
  • 705
1 vote
1 answer
375 views

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 (...
Puraṭci Vinnani's user avatar
16 votes
2 answers
2k views

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 ...
Anton Hilado's user avatar
  • 3,309
5 votes
1 answer
830 views

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?...
Dat Minh Ha's user avatar
  • 1,516
8 votes
0 answers
454 views

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 ...
Waleed Qaisar's user avatar
9 votes
1 answer
435 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 ...
annie marie cœur's user avatar
3 votes
1 answer
607 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,\...
xir's user avatar
  • 2,044
18 votes
0 answers
1k 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 ...
wonderich's user avatar
  • 10.5k
7 votes
1 answer
555 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 &...
Puraṭci Vinnani's user avatar
6 votes
0 answers
328 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 $$\...
dhy's user avatar
  • 5,958
17 votes
1 answer
1k 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 ...
Malkoun's user avatar
  • 5,215
3 votes
1 answer
988 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 ...
user avatar
24 votes
1 answer
3k 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 ...
Will Sawin's user avatar
  • 148k
43 votes
7 answers
13k 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 ...
Ofra's user avatar
  • 1,613
6 votes
1 answer
1k 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 ...
Aswin's user avatar
  • 1,073
42 votes
2 answers
8k 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 ...
Tian An's user avatar
  • 3,799
2 votes
0 answers
191 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 ...
user avatar
2 votes
1 answer
753 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.
7-adic's user avatar
  • 3,804
2 votes
0 answers
562 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).
7-adic's user avatar
  • 3,804
3 votes
0 answers
334 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 ...
Alexander Chervov's user avatar
15 votes
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 ...
Alexander Chervov's user avatar
8 votes
1 answer
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 ...
Marc Palm's user avatar
  • 11.2k
9 votes
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 ...
Marc Palm's user avatar
  • 11.2k