abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
What is called the Langlands correspondence in number theory (Langlands 67) is first of all a conjectural correspondence (a bijection subject to various conditions) between
$n$-dimensional complex linear representations of the Galois group $Gal(\bar F/F)$ of a given number field $F$, and
certain representations – called automorphic representations – of the $n$-dimensional general linear group $GL_n(\mathbb{A}_F)$ with coefficients in the ring of adeles of $F$, arising within the representations given by functions on the double coset space $GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O})$ (where $\mathcal{O} = \prod_v \mathcal{O}_p$ is the ring of integers of all formal completions of $F$).
This is motivated from the abelian case ($n=1$), which is fully understood: For $n = 1$ then an $n$-dimensional representation of the Galois group factors through $GL_1$ and hence through an abelian group. Therefore, by adjunction, it is equivalently a representation of the abelianization of the Galois group. The Kronecker-Weber theorem says that for $F = \mathbb{Q}$, then the abelianized Galois group is the idele class group $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})$, and hence 1-dimensional representations of the Galois group are equivalently representations of this. Moreover, one finds that for any prime number $p$, then those representations which are “unramified at $p$” are invariant under the subring of p-adic integers, hence are representations of the double quotient group $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})/GL_1(\mathbb{Z}_p)$. More generally, the Artin reciprocity law says that for number fields there is an isomorphism between $GL_1(K) \backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O}_K)$ and the abelianized Galois group.
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Moreover:
Conjecture 1 To each such automorphic representation $\pi$ is associated an L-function – the automorphic L-function $L_\pi$ – and in generalization of Artin reciprocity the conjecture of Langlands is that the Artin L-function $L_\sigma$ associated with the given Galois representation $\sigma$ is equal to this: $L_\sigma = L_\pi$ (Gelbhart 84, conjecture 1 (page 27 (204))).
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More generally, analogous statements are supposed to hold for general reductive algebraic groups $G$ other than $GL_n$. Here now an L-function is assigned to data which in addition to the Galois representation consists of a linear representation ${}^L G \longrightarrow GL_n$ of the Langlands dual group of $G$.
First of all:
Conjecture 2 This more general L-function is conjectured to indeed behave like a decent L-function in that it has meromorphic analytic continuation to the complex plane and satisfies the “functional equation”-invariance under sending its parameter $s$ to $1-s$ (Gelbhart 84, conjecture 2’ (page 29 (205))).
Second:
Conjecture 3 This construction is supposed to behave well with respect to an analytic homomorphism ${}^L G \to {}^L G^{'}$ in that when changing the representation of ${}^L G^{'}$ by precomposition with this homomorphism one may find an accompanying change of Galois representation/automorphic representation from $G$ to $G^{'}$ such that the associated L-function remains invariant under these joint changes. This statement is what Robert Langlands calls functoriality (Gelbhart 84, conjecture 3 (page 31 (207)))
In fact this last conjecture implies the previous two (Gelbhart 84, (page 32 (208))).
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Various versions and refinements of this conjecture have since been considered, for some perspective see (Taylor 02, Langlands 14, Harris 14). On the one hand the “localization” of the program to local fields leads to the conjecture of local Langlands correspondences (Gelbhart 84, (page 34 (210))). On the other hand, the interpretation of the above story dually in arithmetic geometry in view of the function field analogy motivates the conjectural geometric Langlands correspondence, based on the following analogy:
the Galois group is essentially the fundamental group of an algebraic curve;
hence a representation of it is a principal/vector bundle with flat connection on this curves;
while that curious double quotient $GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O}_F)$ is, in view of the Weil uniformization theorem, analogous to the canonical Cech-realization of the moduli stack of bundles (see there for details) on that curve.
analogies in the Langlands program:
arithmetic Langlands correspondence | geometric Langlands correspondence |
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ring of integers of global field | structure sheaf on complex curve $\Sigma$ |
Galois group | fundamental group of $\Sigma$ |
Galois representation | flat connection/local system on $\Sigma$ |
idele class group mod integral adeles | moduli stack of line bundles on $\Sigma$ |
nonabelian $\;$ “ | moduli stack of vector bundles on $\Sigma$ |
automorphic representation | Hitchin connection D-module on bundle of conformal blocks over the moduli stack |
From this arithmetic geometry point of view the Langlands conjecture seems to speak of a correspondence that sends Dirac distributions on the moduli space of flat connections over an algebraic curve to certain “automorphic” functions on the moduli stack of bundles on the same curve. This suggests that the Langlands correspondence should be understood as a nonabelian version of a Fourier-Mukai-type integral transform. This version of the conjecture is known as the geometric Langlands correspondence. See there for more details.
The original conjecture is due to
Surveys of the state of the program include
Richard Taylor, Galois Representations, Proceedings of the ICM 2002 (long version pdf).
Robert Langlands, Problems in the theory of automorphic forms -- 45 years later, Oxford 2014
Michael Harris, Automorphic Galois representations and Shimura varieties, Proceedings of the ICM 2014 (pdf).
Introductions and expository surveys include
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)
Edward Frenkel, Commentary on “An elementary introduction to the Langlands Program” by Steven Gelbart, Bull. Amer. Math. Soc. 48 (2011), 513-515, (pdf)
Mark Goresky, Langlands’ conjectures for physicists (pdf)
James W. Cogdell, L-functions and non-abelian class eld theory, from Artin to Langlands, 2012 (pdf)
Minhyong Kim, A superficial introduction to Langlands functoriality, 2014 (pdf)
Discussion with an eye towards geometric class field theory and geometric Langlands duality and quantum field theory is in
Mikhail Kapranov, Analogies between the Langlands Correspondence and Topological Quantum Field Theory, in Functional Analysis on the Eve of the 21st Century Progress in Mathematics Volume 131/132, 1995, pp 119-151
Peter Toth, Geometric abelian class field theory, 2011 (web)
Edward Frenkel, Gauge Theory and Langlands Duality (arXiv:0906.2747)
More resources are at
the digital archive of Robert Langlands’ articles at sunsite.ubc.ca, not currently being maintained, see:
The Work of Robert Langlands at the Institute of Advanced Study.