# Questions tagged [geometric-langlands]

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### 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 ...
0answers
247 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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827 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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319 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 $\Delta(z)$. ...
1answer
255 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 ...
1answer
223 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. ...
1answer
370 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 $\mathfrak{h}^*[2]/W$ (...
0answers
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### 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 ...
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112 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 ...
1answer
292 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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230 views