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The $$\mathrm {SU}(r)$$ Vafa–Witten partition function, which virtually counts Higgs pairs on a projective surface S, was mathematically defined by Tanaka–Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $$\mu _r$$ -gerbes. In this paper, we instead use Yoshioka’s moduli spaces of twisted sheaves. Using Chern character twisted by rational B-field, we give a new mathematical definition of the $$\mathrm {SU}(r) / {\mathbb {Z}}_r$$ Vafa-Witten partition function when r is prime. Our definition uses the period-index theorem of de Jong. S-duality, a concept from physics, predicts that the $$\mathrm {SU}(r)$$ and $$\mathrm {SU}(r) / {\mathbb {Z}}_r$$ partition functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all K3 surfaces and prime numbers r.

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface. We give a geometric interpretation of these intersection pairings in the case that $G=SL_n$. In particular, we show that they can be computed as the length of minimal weighted networks in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [LO] The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, Konig's theorem, and the Kuhn-Munkres algorithm., Comment: 15 pages. We prove some conjectures from arXiv:1511.00165

We observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$-theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse., Comment: 36 pages, 2 figures; published version

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirkovi\'c-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^+ G \backslash LG / L^+ G$ and representations of Deligne's modification of the Langlands dual group $\hat G$., Comment: final published version (v2 contains a new section on sheaves vs. functions, v3 has an overhauled appendix and many more minor edits)

This paper serves two purposes, by adopting the classical Casselman–Tadi c ´ \acute {c} ’s Jacquet module machine and the profound Langlands–Shahidi theory, we first determine the explicit Langlands classification for quasi-split groups P G S O 8 E PGSO^E_8 which provides a concrete example to guess the internal structures of parabolic inductions. Based on the classification, we further sort out the unitary dual of P G S O 8 E PGSO^E_8 and compute the Aubert duality which could shed light on the final answer of Arthur’s conjecture for P G S O 8 E PGSO_8^E . As an essential input to obtain a complete unitary dual, we also need to determine the local poles of triple product L-functions which is done in the appendix. As a byproduct of the explicit unitary dual, we verified Clozel’s finiteness conjecture of special exponents and Bernstein’s unitarity conjecture concerning AZSS duality for P G S O 8 E PGSO_8^E .

We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual., Comment: Finalized published version. Introduction has been rewritten to reflect recent progress and references added. Some typos fixed

In this article we give a new proof of Ng\^o's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa}$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.

In this paper we discuss various $N=3$ SCFTs in 4 dimensions and in particular those which can be obtained as a discrete gauging of an $N=4$ SYM theories with non-simply laced groups. The main goal of the project was to compute the Coulomb branch superconformal index and Higgs branch Hilbert series for the $N=3$ SCFTs that are obtained from gauging a discrete subgroup of the global symmetry group of $N=4$ Super Yang-Mills theory. The discrete subgroup contains elements of both $SU(4)$ R-symmetry group and the S-duality group of $N=4$ SYM. This computation was done for the simply laced groups (where the S-duality groups is $SL(2, \mathbb{Z})$ and Langlands dual of the the algebra $L[\mathfrak{g}]$ is simply $\mathfrak{g}$) by Bourton et al. arXiv:1804.05396, and we extended it to the non-simply laced groups. We also considered the orbifolding groups of the Coulomb branch for the cases when Coulomb branch is relatively simple; in particular, we compared them with the results of Argyres et al. arXiv:1904.10969, who classified all $N\geq 3$ moduli space orbifold geometries at rank 2 and with the results of Bonetti et al. arXiv:1810.03612, who listed all possible orbifolding groups for the freely generated Coulomb branches of $N\geq 3$ SCFTs. Finally, we have considered sporadic complex crystallographic reflection groups with rank greater than 2 and analyzed, which of them can correspond to an $N=3$ SCFT., 12 pages

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group $$G^\vee $$ , and its Poisson–Lie dual group $$G^*$$ , respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell $$G^{\vee ; w_0, e} \subset G^\vee $$ is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of $$K^* \subset G^*$$ (the Poisson–Lie dual of the compact form $$K \subset G$$ ). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of $$K^*$$ are equal to symplectic volumes of the corresponding coadjoint orbits in $${{\,\mathrm{Lie}\,}}(K)^*$$ . To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Super (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells $$G^{w_0, e} \subset G$$ and $$G^{\vee ; w_0, e} \subset G^\vee $$ .

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

In this addendum, we remove the ambiguity for roots of higher order of denominator formulas in our paper. These refinements state that there are roots of order 4, 5, 6, which is the first such observation of a root of order strictly larger than 3 to the best knowledge of the authors.

We introduce a procedure to fold the structure of a crystal B of simply-laced Cartan type ${\mathscr{C}}$ by the action of an automorphism σ. This produces a crystal Bσ for the folded Langlands dual datum ${\mathscr{C}}^{\sigma \vee }$ which properly contains the well-studied ${\mathscr{C}}^{\sigma \vee }$ crystal of σ-invariant points. Our construction preserves normality and the Weyl group action, and is compatible with Kashiwara’s tensor product rule. Combinatorial properties of $B(\infty )_{\sigma }$ reflect the structure of a subalgebra of $U_{q}^{-}({\mathscr{C}})$ , which is naturally a module over the graded-σ-fixed-point subalgebra of $U_{q}^{-}({\mathscr{C}})$ via Berenstein and Greenstein’s quantum folding procedure. We find that $B(\infty )_{\sigma }$ is generated by a set of highest-weight elements over the monoid of root operators. Through the Kashiwara-Nakashima-Zelevinsky polyhedral realization, the highest-weight set identifies with a commutative monoid which admits a Hilbert basis in finite type. A subset of the Weyl group called the balanced parabolic quotient is in one-to-one correspondence with the Hilbert basis for the pair $\left ({\mathscr{C}}, {\mathscr{C}}^{\sigma \vee } \right ) = \left (D_{r}, C_{r-1} \right )$ , and identifies with a proper subset of the Hilbert basis in other finite types. We obtain an explicit combinatorial description of the highest-weight set of $B(\infty )_{\sigma }$ by establishing a connection between the action of root operators on $B(\infty )$ and the semigroup structure in the polyhedral realization.

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, $\mathscr{C}_{\mathscr{Q}}^{(1)}$ and $\mathscr{C}_{\mathfrak{Q}}^{(1)}$. These results give Dorey's rule for all exceptional affine types, prove the conjectures of Kashiwara-Kang-Kim and Kashiwara-Oh, and provides the partial answers of Frenkel-Hernandez on Langlands duality for finite dimensional representations of quantum affine algebras of exceptional types., 67 pages, 1 figure; v2 incorporated changes from referee report; v3 incorporated an addendum that removes ambiguities

Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is motivated by 3D mirror symmetry, and it is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.

We introduce a C∗-algebra \({\mathcal A}_V\) of a variety V over the number field K and a C∗-algebra \({\mathcal A}_G\) of a reductive group G over the ring of adeles of K. Using Pimsner’s Theorem, we construct an embedding \({\mathcal A}_V\hookrightarrow {\mathcal A}_G\), where V is a G-coherent variety, e.g. the Shimura variety of G. The embedding is an analog of the Langlands reciprocity for C∗-algebras. It follows from the K-theory of the inclusion \({\mathcal A}_V\subset {\mathcal A}_G\) that the Hasse-Weil L-function of V is a product of the automorphic L-functions corresponding to irreducible representations of the group G.

In this expository note, we review the recent developments about Maulik and Okounkov’s stable bases for the Springer resolution \(T^*(G/B)\). In the cohomology case, we compute the action of the graded affine Hecke algebra on the stable basis, which is used to obtain the localization formulae. We further identify the stable bases with the Chern–Schwartz–MacPherson classes of the Schubert cells. This relation is used to prove the positivity conjecture of Aluffi and Mihalcea. For the K theory stable basis, we first compute the action of the affine Hecke algebra on it, which is used to deduce the localization formulae via root polynomial method. Similar as the cohomology case, they are also identified with the motivic Chern classes of the Schubert cells. This identification is used to prove the Bump–Nakasuji–Naruse conjecture about the unramified principal series of the Langlands dual group over non-Archimedean local fields. In the end, we study the wall R-matrices, which relate stable bases for different alcoves. As an application, we give a categorification of the stable bases via the localization of Lie algebras over positive characteristic fields.

We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well as the mirror dual Landau-Ginzburg model $(\check{X}^\circ, W_q:\check{X}^\circ \to \mathbb C)$, where $\check{X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{X} = Gr_k((\mathbb C^n)^*)$, and the superpotential W_q has a simple expression in terms of Pl\"ucker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a $\mathcal X$-cluster chart $\Phi_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal A$-cluster chart $\Phi_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. To each $\mathcal X$-cluster chart $\Phi_G$ and ample `boundary divisor' $D$ in $X\setminus X^{\circ}$, we associate a Newton-Okounkov body $\Delta_G(D)$ in $\mathbb R^{k(n-k)}$, which is defined as the convex hull of rational points. On the other hand using the $\mathcal A$-cluster chart $\Phi_G^{\vee}$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential $W_q$ in the $\mathcal A$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies $\Delta_G(D)$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when $G$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus., Comment: 55 pages, many figures; to appear in Duke

Let g be an affine Lie algebra with index set I = { 0 , 1 , 2 , ⋯ , n } and g L be its Langlands dual. It is conjectured in [16] that for each k ∈ I ∖ { 0 } the affine Lie algebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for g L . Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra g = A n ( 1 ) for each Dynkin index k ∈ I ∖ { 0 } and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for A n ( 1 ) given in [24] . In the process we develop and use some lattice-path combinatorics.

We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking the spectra of quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine algebra. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz equations for a large class of quantum integrable models associated to arbitrary untwisted quantum affine algebras, under a mild genericity condition. We also conjecture analogues of these results for the twisted quantum affine algebras and elucidate the notion of opers for twisted affine algebras, making a connection to twisted opers introduced in [FG]., 54 pages (v3: some examples added; opers for twisted affine algebras elucidated). Accepted for publication in Communications in Mathematical Physics

We prove that the Grothendieck rings of category $$\mathcal {C}^{(t)}_Q$$ over quantum affine algebras $$U_q'(\mathfrak {g}^{(t)})$$ $$(t=1,2)$$ associated with each Dynkin quiver Q of finite type $$A_{2n-1}$$ (resp. $$D_{n+1}$$ ) are isomorphic to one of the categories $$\mathcal {C}_{\mathscr {Q}}$$ over the Langlands dual $$U_q'({^L}\mathfrak {g}^{(2)})$$ of $$U_q'(\mathfrak {g}^{(2)})$$ associated with any twisted adapted class $$[\mathscr {Q}]$$ of $$A_{2n-1}$$ (resp. $$D_{n+1}$$ ). This results provide simplicity-preserving correspondences on Langlands duality for finite-dimensional representation of quantum affine algebras, suggested by Frenkel–Hernandez.

We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $\mathcal{S}$-groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered $L$-packets. This provides a formula of the multiplicity for $p$-adic groups by means of dimensions of irreducible representations of their $\mathcal{S}$-groups., This present version supersedes arXiv:1306.6118v1-v5 -- changed the title; modified considerable parts of the manuscript; removed some sections. Accepted for publication in Mathematische Zeitschrift

Let G be a connected reductive algebraic group over ℂ, and let Λ + be the monoid of dominant weights of G. We construct integrable crystals BG(λ), λ ∈ Λ + , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of G. We also construct tensor product maps $$P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} $$ in terms of multiplication in generalized transversal slices. Let L ⊂ G be a Levi subgroup of G. We describe the functor Res : Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.

In this note, we study conjectures of Artin and Langlands and derive the automorphy of all solvable groups of order at most 200, three groups excepted.

Let $X$ be a spherical variety for a connected reductive group $G$. Work of Gaitsgory-Nadler strongly suggests that the Langlands dual group $G^\vee$ of $G$ has a subgroup whose Weyl group is the little Weyl group of $X$. Sakellaridis-Venkatesh defined a refined dual group $G^\vee_X$ and verified in many cases that there exists an isogeny $\phi$ from $G^\vee_X$ to $G^\vee$. In this paper, we establish the existence of $\phi$ in full generality. Our approach is purely combinatorial and works (despite the title) for arbitrary $G$-varieties., Comment: v1: 30 pages; v2: 30 pages, Lemma 10.4 (pertaining to Galois group actions) has been fixed, v3: 31 pages, revised according to referee's report

This paper begins a series studying D-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory. Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certain finite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groups realize the nilradical of a Borel subalgebra for the Langlands dual group in a precise sense, following earlier work of Feigin-Finkelberg-Kuznetsov-Mirkovic and Braverman-Gaitsgory. Moreover, we compare this geometric realization of the Langlands dual group to the standard one provided by (factorizable) geometric Satake.

Motivated by quantum field theoretic partition functions that can be expressed as products of tau functions of the KP hierarchy we attach several types of local geometric Langlands parameters to quivers in the Sato Grassmannian. We study related questions of Virasoro constraints, of moduli spaces of relevant quivers, and of classical limits of the Langlands parameters.

In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O 2 n \mathrm {O}_{2n} established by Arthur. We give two applications: Prasad’s conjecture and Gross–Prasad conjecture for O n \mathrm {O}_n . Also, we discuss the Arthur conjecture for O 2 n \mathrm {O}_{2n} , and establish the Arthur multiplicity formula for O 2 n \mathrm {O}_{2n} .

In this note, we study Artin's conjecture via group theory and derive Langlands reciprocity for certain solvable Galois extensions of number fields, which extends the previous work of Arthur and Clozel. In particular, we show that all nearly nilpotent groups and all groups of order less than 60 are of automorphic type.

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions as appear in recent advances in the Geometric Langlands program. The key technical tools in our arguments are: a new descent theory for coherent sheaves or D-modules with prescribed singular support; and the theory of integral transforms for coherent sheaves developed in the companion paper [BNP]., Comment: 28 pages

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve G$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$. Denote by $\Bun_G$ the moduli stack of $G$-bundles on $C$ and $ \Loc_{\breve G}$ the moduli stack of $\breve G$-local systems on $C$. Let $D_{\Bun_G}$ be the sheaf of crystalline differential operators on $\Bun_G$. In this paper we construct an equivalence between the bounded derived category $D^b(\on{QCoh}(\Loc_{\breve G}^0))$ of quasi-coherent sheaves on some open subset $\Loc_{\breve G}^0\subset\Loc_{\breve G}$ and bounded derived category $D^b(D_{\Bun_G}^0\on{-mod})$ of modules over some localization $D_{\Bun_G}^0$ of $D_{\Bun_G}$. This generalizes the work of Bezrukavnikov-Braverman in the $\GL_n$ case., Comment: 57 pages, corrected some arguments in section 3.6 and 3.7, to appear in Compositio Math

We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac--Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra should be encoded by affine opers associated to the Langlands dual affine algebra. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding affine opers may be expressed as Schroedinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this and other recently found examples of the correspondence between quantum integrals of motion and differential operators may be viewed as special cases of the Langlands duality., Comment: 70 pages. Final version, to appear in Proceedings of the conference in honor of A. Tsuchiya (Nagoya, March 2007), published in the series Advanced Studies of Pure Mathematics

Open access at https://arxiv.org/pdf/1707.00392.pdf., Given a compact connected Riemann surface X equipped with an anti-holomorphic involution and a complex semisimple Lie group G equipped with a real structure, we define anti-holomorphic involutions on the moduli space of G-Higgs bundles over X. We describe how the various components of the fixed point locus match up, as one passes from G to its Langlands dual G. As an example, the case of G=SL(2,C) and LG=PGL(2,C) is investigated in detail., We thank International Centre for Theoretical Sciences for hospitality while a part of the work was carried out. The first author is partially supported by a J. C. Bose Fellowship.

In this paper, we present a refinement of the global Langlands correspondence for discrete symplectic motives of rank 2n over Q. To such a motive Langlands conjecturally associates a generic, automorphic representation of the split orthogonal group SO2n+1 over Q, which appears with multiplicity one in the cuspidal spectrum. Using the local theory of generic representations of odd orthogonal groups, we define a new vector F in this representation, which is the tensor product of local test vectors for the Whittaker functionals [9]. I hope that the defining properties ofF will make it easier to investigate the Langlands correspondence computationally, especially for the cohomology of algebraic curves. Our refinement is similar to the refinement that Weil [24] proposed for the conjecture that elliptic curves over Q are modular. Namely, Weil proposed that such a curve should be associated with a homomorphic newform F = P anq n of weight 2 on 0(N), where N is equal to the conductor of the curve. This paper expands on a letter that I wrote to Serre in 2010. It was motivated by a question Serre posed at my 60th birthday conference, and a suggestion Brumer made of a family of discrete subgroups generalizing 0(N). I would like to thank them, and to thank Deligne for his comments.

Let G be a split reductive p-adic group with connected centre. We show that each Bernstein block in the principal series of G admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form T//W where T is a maximal torus in the Langlands dual group of G and W is the Weyl group of G.

The central result of this paper is an identification of the shifted Cartier dual of the moduli stack M g ( C ) of G ˜ -Higgs bundles on C of arbitrary degree (modulo shifts by Z ( G ˜ ) ) with a quotient of the Langlands dual stack M g L ( C ) . Via hyperkahler rotation, this may equivalently be viewed as the identification of an SYZ fibration relating Hitchin systems for arbitrary Langlands dual semisimple groups, coupled to nontrivial finite B-fields. As a corollary certain self-dual stacks M g ( C ) Γ are observed to exist, which I conjecture to be the Coulomb branches for the 3d reduction of the 4d N = 2 theories of class S .

We consider the Grassmannian X = G r n − k ( C n ) and describe a ‘mirror dual’ Landau-Ginzburg model ( X ˇ ∘ , W q : X ˇ ∘ → C ) , where X ˇ ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian X ˇ , and we express W succinctly in terms of Plucker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to W q a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten.

This paper reviews the moment graph technique that allows to translate certain representation theoretic problems into geometric ones. For simplicity we restrict ourselves to the case of semisimple complex Lie algebras. In particular, we show how the original Kazhdan-Lusztig conjecture on the characters of irreducible highest weight representations can be translated into a multiplicity problem for parity sheaves on the (Langlands dual) flag variety., 18 pages, references added

The Langlands classification theorem describes all admissible representations of a reductive group $G$ in terms of the tempered representations of Levi subgroups of $G$. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of $G$ in terms of certain very special unitary representations of Levi subgroups.

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of Langlands' constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker vector associated to unramified automorphic representations of G(Q_p). Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore introduce also some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics where automorphic forms occur in string theory.

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory $T[\Sigma,G]$ on $L(k,1) \times S^1$, the other is the $^LG$ "equivariant Verlinde formula", or equivalently partition function of $^LG_{\mathbb{C}}$ complex Chern-Simons theory on $\Sigma\times S^1$. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally $G$ and its Langlands dual $^LG$. When $G$ is not simply-connected, we provide a recipe of computing the index of $T[\Sigma,G]$ as summation over indices of $T[\Sigma,\tilde{G}]$ with non-trivial background 't Hooft fluxes, where $\tilde{G}$ is the simply-connected group with the same Lie algebra. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for $G=SU(2)$ or $SO(3)$. In the end, as an application of this newly found relation, we consider the more general case where $G$ is $SU(N)$ or $PSU(N)$ and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the $SU(3)$ equivariant Verlinde coefficients., Comment: 40 pages, 7 figures, Mathematica Notebook attached; v2: misprints corrected; v3: Acknowledgement added, corrections made based on the journal version

Given a semisimple group G, Kostant and Kumar defined a nil-Hecke algebra that may be viewed as a degenerate version of the double affine nil-Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical subalgebra of the nil-Hecke algebra with a Whittaker type quantum Hamiltonian reduction of the algebra of differential operators on G. This result has an interpretation in terms of geometric Satake and the Langlands dual group. Specifically, the isomorphism provides a bridge between very differently looking descriptions of equivariant Borel-Moore homology of the affine flag variety (due to Kostant and Kumar) and of the affine Grassmannian (due to Bezrukavnikov and Finkelberg), respectively.

We conjecture that quantum Gaudin models in affine types admit families of local higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals. We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle $\Omega$. Each fibre is isomorphic to the direct product of the space of sections of the square of $\Omega$ with the direct product, over the exponents $j$ not equal to 1, of the twisted cohomology of the $j^{\rm th}$ tensor power of $\Omega$., Comment: 53 pages; v2: minor edits; version to appear in Advances in Mathematics; v3: references clarified

Let H be a split reductive group over a local non-Archimedean field, and let \(\check {H}\) denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of “Satake transform” and perform the analogous calculation for a large class of spherical varieties.

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence., Comment: v1: 66 pages; v2: 67 pages, minor corrections; v3: 65 pages, final version, to appear in JEMS