2 DENNIS GAITSGORY, 1 1 2 Then came Drinfeld s ground breaking paper Dr In a sense it was the extension of. Deligne s construction to the vastly more complicated case when instead of grossen characters. we consider unramified automorphic functions for the group GL2 Here again we inter. pret the unramified automorphic space as the groupoid of Fq points of the moduli space. Bun2 BunGL2 classifying rank 2 vector bundles on X Drinfeld s idea is to attach to a. 2 dimensional Galois representation an adic sheaf by which we actually mean an object of. the corresponding derived category F on Bun2 and then obtain the sought for function by. taking the traces of the Frobenius, The main difference from the commutative case considered by Deligne which corresponds to. the case of the group Gm GL1 is that the construction of F starting from is much more. involved The intermediate player i e E is now interpreted as the adic sheaf that records. the Whittaker a k a Fourier coefficients of F So our task is to reconstruct an automorphic. object from its Fourier coefficients This is again done via appealing to geometry ultimately. the simply connectedness of fibers of some map, 1 1 3 After Drinfeld s paper came one by Laumon Lau1 which gave a conjectural extension. of Drinfeld s construction from GL2 to GLn To the best of our knowledge the title of Laumon s. paper was the first place where the combination of words geometric Langlands appeared. While the stated goal of Drinfeld s paper was to construct an automorphic function Laumon s. paper had the effect of shifting the goal people became interested in automorphic sheaves. adic sheaves on Bunn X for their own sake, Following the appearance of Laumon s paper it became clear that one should also try to. attack BunG X for an arbitrary reductive G even though it was not clear how to do this. because the Whittaker model does not work as nicely outside the case of G GLn. 1 1 4 The next paradigm shift came in the work of Beilinson and Drinfeld BD They con. sidered the same BunG X but now over a ground field k of characteristic zero and instead of. adic sheaves they proposed to consider D modules, In this case a new method for constructing objects becomes available by generators and rela. tions A fancy version of generators and relations principle the localization functor pioneered. in BB lies in the core of the manuscript BD which produces automorphic D modules using. representations of the Kac Moody Lie algebra thought of as the Lie algebra of infinitesimal. symmetries of a G bundle on the formal punctured disk. 1 1 5 In an independent development in Lau2 Laumon showed that if we take G to be a. torus T a generalized version of the Fourier Mukai transform identifies the derived category. of D modules on the stack BunT X with the derived category of quasi coherent sheaves on. the stack LocSysT X of de Rham local systems on X with respect to the Langlands dual torus. I e Laumon s paper extends the poinwtise Langlands correspondence i e construction of. F corresponding to a fixed local system to a statement about the universal family of local. 1 1 6 Finally combining Laumon s equivalence for the torus and accumulated evidence for. the general G Beilinson and Drinfeld came up with the idea of categorical geometric Langlands. equivalence,RECENT PROGRESS IN GEOMETRIC LANGLANDS THEORY 3. In its crude form this should be an equivalence between the derived category D BunG X. of D modules on the stack BunG X and the derived category QCoh LocSysG X of quasi. coherent sheaves on the stack LocSysG X Such as equivalence is what Beilinson and Drinfeld. called the best hope but they never stated it explicitly because it is and was known that it. cannot hold as is beyond the case of a torus the reason reason for this will be indicated in. Sect 3 1 2, 1 2 What do we mean by geometric Langlands nowadays There are several meta. problems that comprise what one can call the geometric Langlands theory we shall list some. of them below the order in which they will appear reflects our perception of the historical. development and the increasing level of technical complexity rather than how the complete. picture should ultimately look like e g we do think that the quantum case is more fundamental. than the usual one, We will only consider the categorical geometric Langlands theory in particular we will assume. that the ground field k is of characteristic zero and on the automorphic side we will work with. D modules rather than adic sheaves, We should remark that whatever conjectures and meta conjectures we mention below they. are all theorems when the group G is a torus thanks to the various generalizations of the. Deligne Fourier Mukai Laumon transform, 1 2 1 First we have the categorical1 global unramified geometric Langlands This is an attempt. to formulate and prove a version of the best hope by Beilinson and Drinfeld mentioned above. I e we want a category that it a close cousin or identical twin of D BunG X to be equivalent. to a category that is a close cousin of QCoh LocSysG X. This is the aspect of the geometric Langlands theory that has been developed the most It. will be discussed in Sects 2 and 3, 1 2 2 Next there is the local ramified geometric Langlands theory Unlike the global case in the. local version we are interested in an equivalence of 2 categories rather than 1 categories i e. just categories For a long time it was not even clear how to formulate our wish specifically. what 2 category to consider on the Galois side However recently a breakthrough has been. achieved in the work of S Raskin Ras We will discuss this in Sect 4. We should also mention that the tamely ramified case of the local ramified geometric Lang. lands had been settled by R Bezrukavnikov in Bez even before the general program was. formulated, 1 2 3 Next there is the global ramified Langlands theory Its tamely ramified case has not. been explicitly studied in detail but the current state of knowledge should allow to bring it to. the same status as the unramified case, The general ramified case is wide open and there are formidable technical difficulties that. one needs to surmount in order to start investigating it One of the difficulties is that we do not. know whether the category of D modules on the automorphic side i e the derived category. of D modules on the moduli space BunG X k x of G bundles on X equipped with structure of. level k 1 at a point x is compactly generated2, 1From now on we will drop the adjective categorical because everything will be categorical. 2If one surveys the literature in most of the statements that involve an equivalences of two triangulated DG. categories the categories of question are compactly generated The reason is that we do not know very well. how to compute things outside the compactly generated case. 4 DENNIS GAITSGORY, 1 2 4 Finally all of the above three aspects unramified global ramified local and ramified. global admit quantum versions The quantum parameter in the quantum geometric Langlands. theory is a non degenerate W invariant symmetric bilinear form on the Cartan Lie algebra h of. g whose inverse is a similar kind of datum for g, At the risk of making a controversial statement the author speaker has to admit that he. came to regard the quantum version as the ultimate reason of why something like the Langlands. theory takes place with the usual geometric Langlands being its degeneration letting the. quantum parameter tend to zero and the classical i e function theoretic Langlands theory. as some sort of residual phenomenon, We will not say anything about the quantum case in this talk but rather refer the reader. to Ga2 where the dream of quantum geometric Langlands is discussed Here we will only. mention the following two facts, One is that in the quantum theory restores the symmetry between G and G we no longer. have the Galois side but rather both sides are automorphic but twisted by the quantum. The other is that the guiding principle of the quantum theory is that Whittaker is dual. to Kac Moody which is striking because Whittaker has a classical i e number theoretic. meaning while Kac Moody does not,1 3 Terminology and notation. 1 3 1 The global unramified geometric Langlands conjecture can be formulated as an equiva. lence of triangulated categories But if one wants to dig a tiny bit deeper into attempts of. its proof one needs to work with categories in this case with k linear DG categories We. refer the reader to GR1 Sect 10 for the definition of the latter. For the reader not familiar with categories we recommend the following approach On. the first pass pretend that there no difference between categories and ordinary categories. On the second pass pretend that you already know what categories are and stay tuned for. the language used when working with them a survey of the syntax of categories can be. found in Lu Sect 1 or from a somewhat different perspective in GR1 Sect 1 On the. third pass learn the theory properly, 1 3 2 Another piece of bad news is that when working on the Galois side of the geometric. Langlands theory we cannot stay within the realm of classical algebraic geometry and one. needs to plunge oneself into the world of DAG derived algebraic geometry For example the. stack LocSysG X has a non trivial derived structure for G T being a torus The reader is. referred to GR2 for an introduction to DAG, In what follows when we say scheme or algebraic stack we will tacitly mean the corre. sponding derived notions, 1 3 3 To a scheme or algebraic stack Y one attaches the DG category QCoh Y we will. somewhat abusively refer to it as the derived category of quasi coherent sheaves on 3 Y We. refer the reader to GR3 for the definition We note however that the definition of QCoh Y. is much more general it makes sense for Y which is an arbitrary prestack 4. 3When Y is a classical as opposed to derived scheme or a sufficiently nice algebraic stack this category is. the derived category of its heart with respect to a naturally defined t structure. 4But in this more general setting QCoh Y is not at all the derived category of any abelian category even. if Y itself is a classical prestack,RECENT PROGRESS IN GEOMETRIC LANGLANDS THEORY 5. 1 3 4 If Y is a scheme algebraic stack prestack locally of finite type over k one can attach to. it the DG category D Y we will also somewhat abusively refer to it as the derived category. of D modules on 5 Y We refer the reader to GR4 for the definition. The good news is that when discussing D Y the derived structure on Y plays no role So. on the automorphic side of the geometric Langlands theory we can stay within the realm of. classical algebraic geometry, 1 3 5 Whenever we talk about functors between derived categories of sheaves D modules on. various spaces and direct and inverse images we will always mean the corresponding. derived functors I e abelian categories will not appear unless explicitly stated otherwise. 1 3 6 For the motivational parts of the talk i e analogies with the function theoretic situa. tion we will assume the reader s familiarity with the basics of algebraic number theory ade les. ramification Frobenius elements etc, 1 4 Acknowledgements The author speaker wishes to thank D Arinkin A Beilinson. J Bernstein V Drinfeld E Frenkel D Kazhdan S Raskin and E Witten discussions with. whom has informed his perception of the Langlands theory. 2 Hecke action, The classical Langlands correspondence and historically also the geometric one were char. acterized by relating the spectrum of the action of the Hecke operators resp functors on the. automorphic side to a Galois datum We begin by discussing this aspect of the theory. 2 1 Hecke action on automorphic functions Let K be the field of rational functions on. our curve X let A be the ring of ade les and O A the subring of integral ade les For a place. x X we let Ox Kx denote the corresponding local ring and local field respectively. 2 1 1 The automorphic space is by definition the quotient G A G K It is acted on by left. translations by the group G A The unramified automorphic space is the set but properly. speaking groupoid,G O G A G K, Our object of study is the space Autom X of unramified Q valued automorphic functions. i e functions on G O G A G K or which is the same the space of G O invariant functions. on G A G K, 2 1 2 Since the subgroup G O G A is not normal we do not have an action of G A on. Autom X Instead the action of G A on G A G K induces an action on Autom X of the. spherical Hecke algebra H G X By definition as a vector space H G X consists of compactly. supported G O biinvariant functions on G A and it is endowed with a structure of associative. algebra via the operation of convolution, The datum of the action of H G X is equivalent to that of a family of pairwise commuting. actions of the local Hecke algebras H G x for every place x of X where each H G x is the. algebra with respect to convolution of G Ox biinvariant compactlt supported functions on. Our interest is to find the spectrum of H G X i e the joint spectrum of the algebras H G x. acting on Autom X, 5If Y is a scheme this is the derived category of its heart but this is no longer true for algebraic stacks. 6 DENNIS GAITSGORY, 2 1 3 Fix x X The first basic fact about the associative algebra H G x is that it is actually. commutative But in addition to this we can actually describe it very explicitly

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