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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 because there isn't really a "right" answer: what does GLC imply? Lots of big conjectures have well known consequences (Riemann Hypothesis and distribution of primes) but what about GLC? Are there any nice things that are known to follow from this equivalence of derived categories?

EDIT: The Geometric Langlands Conjecture says the following: Let C be an algebraic curve (any field, though I think the formulation I know is only good in characteristic 0), G a reductive algebraic group, its Langlands dual (the characters of G are cocharacters of , if I recall correctly). Then there's a natural equivalence of categories from the derived category of coherent sheaves on the stack of local G-local systems to the derived category of coherent D-modules on the moduli stack of principal -bundles, such that the structure sheaf of a point is sent to a Hecke Eigensheaf (and I'm not going to sit down and define that on top of the rest here...the idea is that local G-local systems on the curve are equivalent to eigensheaves for some collection of operators, but actually making it precise and having a hope of being true gets technical)

Edit 2: This paper states one version of the conjecture (for GL(n) only) as 1.3, after defining the Hecke operators.

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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 because there isn't really a "right" answer: what does GLC imply? Lots of big conjectures have well known consequences (Riemann Hypothesis and distribution of primes) but what about GLC? Are there any nice things that are known to follow from this equivalence of derived categories?

EDIT: The Geometric Langlands Conjecture says the following: Let C be an algebraic curve (any field, though I think the formulation I know is only good in characteristic 0), G a reductive algebraic group, its Langlands dual (the characters of G are cocharacters of , if I recall correctly). Then there's a natural equivalence of categories from the derived category of coherent sheaves on the -gerbe stack of local systems to the derived category of coherent D-modules on the moduli stack of principal -bundles, such that the structure sheaf of a point is sent to a Hecke Eigensheaf (and I'm not going to sit down and define that on top of the rest here...the idea is that local systems on the curve are equivalent to eigensheaves for some collection of operators, but actually making it precise and having a hope of being true gets technical)

Edit 2: This paper states one version of the conjecture (for GL(n) only) as 1.3, after defining the Hecke operators.

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