Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does one go about writing down what group it is?
[Math] Langlands Dual Groups
ag.algebraic-geometryalgebraic-groupsgeometric-langlandslie-groups
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OK, this is a very broad question so I'll be telegraphic. There is a sequence of increasingly detailed conjectures going by the name GL -- it's really a "program" (harmonic analysis of $\mathcal{D}$-modules on moduli of bundles) rather than a conjecture -- and only the first of this sequence has been proved (and only for $GL_n$), but I don't want to get into this.
There are several kinds of reasons you might want to study geometric Langlands:
direct consequences. One application is Gaitsgory's proof of de Jong's conjecture (arXiv:math/0402184). If you prove the ramified geometric Langlands for $GL_n$, you will recover L. Lafforgue's results (Langlands for function fields), which have lots of consequences (enumerated eg I think in his Fields medal description), which I won't enumerate. (well really you'd need to prove them "well" to get the motivic consequences..) In fact you'll recover much more (like independence of $l$ results). To me though this is the least convincing motivation..
Original motivation: by understanding the function field version of Langlands you can hope to learn a lot about the Langlands program, working in a much easier setting where you have a chance to go much further. In particular the GLP (the version over $\mathbb{C}$) has a LOT more structure than the Langlands program -- ie things are MUCH nicer, there are much stronger and cleaner results you can hope to prove, and hope to use this to gain insight into underlying patterns.
By far the greatest example of this is Ngo's proof of the Fundamental Lemma --- he doesn't use GLP per se, but rather the geometry of the Hitchin system, which is one of the key geometric ingredients discovered through the GLP. To me this already makes the whole endeavor worthwhile..
- Relations with physics. Once you're over $\mathbb{C}$, you (by which I mean Beilinson-Drinfeld and Kapustin-Witten) discover lots of deep relations with (at least seemingly) different problems in physics.
a. The first is the theory of integrable systems -- many classical integrable systems fit into the Hitchin system framework, and geometric Langlands gives you a very powerful tool to study the corresponding quantum integrable systems. In fact you (namely BD) can motivate the entire GLP as a way to fully solve a collection of quantum integrable systems. This has has lots of applications in the subject (eg see Frenkel's reviews on the Gaudin system, papers on Calogero-Moser systems etc).
b. The second is conformal field theory (again BD) --- they develop CFT (conformal, not class, field theory!) very far towards the goal of understanding GLP, leading to deep insights in both directions (and a strategy now by Gaitsgory-Lurie to solve the strongest form of GLP).
c. The third is four-dimensional gauge theory (KW). To me the best way to motivate geometric Langlands is as an aspect of electric-magnetic duality in 4d SUSY gauge theory. This ties in GLP to many of the hottest current topics in string theory/gauge theory (including Dijkgraaf-Vafa theory, wall crossing/Donaldson-Thomas theory, study of M5 branes, yadda yadda yadda)...
- Finally GLP is deeply tied to a host of questions in representation theory, of loop algebras, quantum groups, algebraic groups over finite fields etc. The amazing work of Bezrukavnikov proving a host of fundamental conjectures of Lusztig is based on GLP ideas (and can be thought of as part of the local GLP). (my personal research program with Nadler is to use the same ideas to understand reps of real semisimple Lie groups). This kind of motivation is secretly behind much of the work of BD --- the starting point for all of it is the Beilinson-Bernstein description of reps as $\mathcal{D}$-modules.
There's more but this is already turning into a blog post so I should stop.
To answer [a paraphrase of] your second question first: yes the Kapustin-Witten perspective on geometric Langlands has I think been taken very seriously by a segment of the math community. I find it very misleading though to say (as is often done) that "geometric Langlands is mirror symmetry for the Hitchin space" -- mirror symmetry is a statement about 2d TFTs, while geometric Langlands is one about 4d TFTs which implies a vast amount more structure -- specifically the most important structure for the Langlands story, the action of Hecke operators, is part of the 4d story but not of the mirror symmetry statement.
In any case as Will mentions the Betti Geometric Langlands conjecture formulated in https://arxiv.org/abs/1606.08523 is a direct response to your question - in particular to have a version of geometric Langlands which as you ask should depend only topologically on the Riemann surface (so eg have a mapping class group symmetry). However it is not formulated in Fukaya category language directly. I'm fairly ignorant of Fukaya categories but my impression is that the vast technical difficulties in the subject prevent them currently being rigorously defined on the kind of spaces we are talking about here -- namely both singular and stacky. So the Fukaya-theoretic conjecture you discuss is still more of a guiding principle than a precise question.
Also since the Hitchin space is noncompact you have to decide what KIND of Fukaya category you'd mean (assume say we are dealing with a smooth manifold), i.e. what conditions to put at infinity as Will says - infinitesimal, wrapped, or partially wrapped ("with stops"). The Betti conjecture, which Nadler and I felt captured the spirit of Kapustin-Witten, is morally taking the Fukaya category with stops in the direction of the Hitchin base - i.e. your prototypical Lagrangians allowed are Hitchin fibers, not sections. [By the way for one of probably quite a few papers I can't remember this instant which does treat aspects of GL in a Fukaya perspective there's Nadler's paper on Springer theory https://arxiv.org/abs/0806.4566]
So what to do instead of Fukaya categories? by the microlocal perspective of Nadler, Zaslow, Kontsevich,.... we expect to replace Fukaya categories with categories of microlocal sheaves, eg for cotangent bundles, with constructible sheaves on the base (and you can impose singular support conditions for the "stops" or growth conditions on the Lagrangians). This actually gets you very close to the original characteristic p origin of the geometric Langlands correspondence, which dealt with l-adic sheaves -- via the Grothendieck function-sheaf dictionary those are natural "categorified" substitutes for functions, eg automorphic functions. There are no D-modules in this story.
The beautiful D-module version developed by Beilinson-Drinfeld and Arinkin-Gaitsgory in particular -- the de Rham geometric Langlands correspondence -- has a quite different flavor in many respects, and I would claim is one step further from both the arithmetic (l-adic) origins and from the mirror symmetry story. It is motivated by two (closely related) stories -- the Beilinson-Bernstein realization of representations of Lie algebras as D-modules, which gives it a very close relation to representation theory of affine Kac-Moody algebras; and conformal field theory (eg theory of vertex algebras). This allowed Beilinson-Drinfeld to leverage a crucial result of Feigin-Frenkel to prove a "big chunk" (half-dimensional slice) of the de Rham conjecture, and Gaitsgory and collaborators to develop an amazing program to understand and solve the conjecture in general.
[As a side note I think it's overly dramatic to say the "best hope" is "dashed" -- rather the distance from the original dream to the Arinkin-Gatisgory formulation is technical and not very large and not terribly unexpected - though led to some beautiful math - and is purely about understanding how to match growth conditions on one side with singularity conditions on the other, just as in studying the Fourier transform in different function spaces.]
The de Rham story also has deep relations to physics. The physics I would say is somewhat insensitive to the de Rham vs Betti distinction, which is about different algebraic structures underlying an analytic equivalence, but many of the mathematical questions require you to pick your setting more precisely (except the "core" ones that live in the intersection of the two conjectures). The de Rham story comes up naturally in relation to CFT, to things like gauge theories of Class S and the AGT conjecture, and a whole world that is part of.
OK this is now way too long.
Best Answer
You can construct the dual group in a combinatorial manner as follows: Reductive groups are classified by their root datum. There is an obvious duality on the set of all root data, and the dual group is the reductive group with the dual root datum.
You can see Wikipedia for the notion of the root datum.