Search

Deligne constructed a remarkable local system on $\bP^1-\{0,\infty\}$ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct l-adic Kloosterman sheaves for any reductive group in a uniform way, and describe the local and global monodromy of these Kloosterman sheaves. In particular, they give motivic Galois representations with exceptional monodromy groups G_2,F_4,E_7 and E_8. This also gives an example of the geometric Langlands correspondence with wild ramifications for any reductive group., 58 pages

Let \(\mathfrak{g}\) be an affine Lie algebra with index set I={0,1,2,…,n} and \(\mathfrak{g}^{L}\) be its Langlands dual. It is conjectured in Kashiwara et al. (Trans. Am. Math. Soc. 360(7):3645–3686, 2008) that for each i∈I∖{0} the affine Lie algebra \(\mathfrak{g}\) has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for \(\mathfrak{g}^{L}\). We prove this conjecture for i=2 and \(\mathfrak{g} = A_{n}^{(1)}\).

We establish the Langlands classification for graded Hecke algebras. The proof is analogous to the proof of the classification of highest weight modules for semisimple Lie algebras.

When a group G G is in the Harish-Chandra class, the goal of classifying its tempered representations and the goal of decomposing the Langlands subrepresentation for any of its standard representations are equivalent. The main result of this work is given in Theorem (5.3.5) that consists of a formula for decomposing any Langlands subrepresentation for the group G G . The classification of tempered representations is a consequence of this theorem (Corollary (5.3.6)).

We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials., Title changed. Fixed typos in the representations of C_n and F_4. Fixed typos in the matrix at (4.48) and signs of lambda. Add a remark on the representation of type G2

Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by Moeglin and for the exceptional Chevalley group G_2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E_6, E_7, E_8, and F_4. In particular we prove Arthur's conjecture that the spherical constituent of an unramified principal series of a Chevalley group over any local field of characteristic zero is unitarizable if its Langlands parameter coincides with half the marking of a coadjoint nilpotent orbit of the Langlands dual Lie algebra., 8 pages. The computational data and programs can be found at http://www.math.rutgers.edu/~sdmiller/L2

Interpolating Langlands Quantum Groups of Rank 1 -- We study a certain family, parameterized by an positive integer g, of double deformations of the envelopping algebra U(sl2), in the spirit of arXiv:0809.4453. We prove that each of these double deformations simultaneously deforms two rank 1 quantum groups. We show this interpolating property explains the Langlands duality for the representations of the quantum groups in rank 1. Hence we prove a conjecture of arXiv:0809.4453 in this case : we prove for all g the existence of representations which simultaneously deform two Langlands dual representations. We also study more generaly the finite rank representation theory of this family of double deformations. ----- On \'etudie une certaine famille, param\'etr\'ee par un entier g strictement positif, de doubles d\'eformations de l'alg\'ebre enveloppante U(sl2), dans l'esprit de arXiv:0809.4453. On prouve que chacune de ces doubles d\'eformations d\'eforme simultan\'ement deux groupes quantiques de rang 1. On montre que cette propri\'et\'e d'interpolation explique la dualit\'e de Langlands pour les repr\'esentations des groupes quantiques en rang 1. On r\'esout ainsi une conjecture de arXiv:0809.4453 dans ce cas : on prouve pour tout g l'existence de repr\'esentations qui d\'eforment simultan\'ement deux repr\'esentations Langlands duales. On \'etudie aussi plus g\'en\'eralement la th\'eorie des repr\'esentations de rang fini de de cette famille de doubles d\'eformations., Comment: 44 pages, in French

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over $R$. We also consider the related notion of Hermitian dual, in the case of GL$(n,{\Bbb R})$.

This article addresses the question of whether Langlands duality for complex reductive Lie groups may be implemented by T-dualization. We prove that for reductive groups whose simple factors are of Dynkin type A, D, or E, the answer is yes.

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call ‘admissible’ but we call them ‘of Harish-Chandra type’. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, Ν) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations UV,Ν with imaginaryΝ and withV from the ‘quasi-discrete series’ of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.

We study a duality of 5d maximally supersymmetric Yang-Mills on S^1, which exchanges the tower of Kaluza-Klein W-bosons and the tower of instantonic monopoles. This duality maps a non-simply-laced gauge theory to a simply-laced gauge theory twisted by an outer automorphism around S^1, and is closely related to the Langlands dual of affine Lie algebras. We also discuss how this S-duality is implemented in terms of 6d N=(2,0) theory. This is straightforward except for the 6d theory of type SU(2n+1) with Z_2 outer-automorphism twist, for which a few new properties are deduced. For example, this 6d theory, when reduced on an S^1 with Z_2 twist, gives 5d USp(2n) theory with nontrivial discrete 5d theta angle., 30 pages, 7 figures; v2: more references. Published version

In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of its Langlands dual reductive group. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians., 62 pages

The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of $p$ -adic groups and their Langlands parameters, which are homomorphisms from the local Weil-Deligne group to the $L$ -group. In this article, we refine a conjecture of Hiraga, Ichino, and Ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter. We attach a rational function in $x$ with rational coefficients to each discrete parameter, which specializes at $x=q$ , the cardinality of the residue field, to the quotient of this local gamma factor by the gamma factor of the Steinberg parameter. The order of this rational function at $x=0$ is also an important invariant of the parameter—it leads to a conjectural inequality for the Swan conductor of a discrete parameter acting on the adjoint representation of the $L$ -group. We verify this conjecture in many cases. When we impose equality, we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations, both of which are found and described in this article.

This is an introduction to the work of Beilinson and Drinfeld [6]_on the Langlands program.

We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby cycles in certain regular situations., Comment: 37 pages

In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra g^sigma along a Dynkin diagram automorphism sigma of g For each quantum folding we replace g^sigma by its Langlands dual g^sigma^v and construct a nilpotent Lie algebra n which interpolates between the nilpotnent parts of g and (g^sigma)^v, together with its quantized enveloping algebra U_q(n) and a Poisson structure on S(n). Remarkably, for the pair (g, (g^sigma)^v)=(so_{2n+2},sp_{2n}), the algebra U_q(n) admits an action of the Artin braid group Br_n and contains a new algebra of quantum n x n matrices with an adjoint action of U_q(sl_n), which generalizes the algebras constructed by K. Goodearl and M. Yakimov in [12]. The hardest case of quantum folding is, quite expectably, the pair (so_8,G_2) for which the PBW presentation of U_q(n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each., Comment: 45 pages, AMSLaTeX; journal version: referees suggestions incorporated; some misprints corrected

We review the approach to the geometric Langlands program for algebraic curves via S-duality of an N = 4 supersymmetric four-dimensional gauge theory, initiated by Kapustin and Witten in 2006. We sketch some of the central further developments. Placing this four-dimensional gauge theory into a six-dimensional framework, as advocated by Witten, holds the promise to lead to a formulation which makes geometric Langlands duality a manifest symmetry (like covariance in differential geometry). Furthermore, it leads to an approach toward geometric Langlands duality for algebraic surfaces, reproducing and extending the recent results of Braverman and Finkelberg.

Let F be a totally real Galois number field. We prove the existence of base change relative to the extension F/Q for every classical newform of odd level, under simple local assumptions on the field F., 27 pages. Main result improved: now there is no condition on the primes in the level of the newform (they are allowed to ramify in F)

Let K be a connected compact Lie group, and G be its complexification. The homology of the based loop group \Omega K with integer coefficients is naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify H_*(\Omega K,\ZZ) with the Hopf algebra of algebraic functions on B^\vee_e, where B^\vee is a Borel subgroup of the Langlands dual group scheme G^\vee of G and B^\vee_e is the centralizer in B^\vee of a regular nilpotent element e\in\Lie B^\vee. We also give a similar interpretation for the equivariant homology of \Omega K under the maximal torus action., Comment: 23 pages

The Langlands Program was conceived initially as a bridge between Number Theory and Automorphic Representations, but it has now expanded into such areas as Geometry and Quantum Field Theory. Remarkably and surprisingly, the same salient features, such as the Langlands dual group, appear in these different contexts. Understanding the hidden patterns of the Langlands duality may thus hold the key to many important problems in modern mathematics and physics. One of the most exciting developments in this area in the last few years has been the work of Edward Witten and collaborators unifying the Langlands Program with dualities in quantum field theory and string theory. Specifically, they have related the so-called S-duality of the four-dimensional gauge theory to the geometric Langlands correspondence via the dimensional reduction from four to two dimensions. My work on this project continued this line of research as well as other topics on Langlands duality in Representation Theory, Geometry and Quantum Physics.

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of $G_{\mathrm{cpt}}$-instantons on $\R^4/\Z_r$ correspond to weight spaces of representations of the Langlands dual group $G_\aff^\vee$ at level $r$. When $G = \SL(l)$, the Uhlenbeck compactification is the quiver variety of type $\algsl(r)_\aff$, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for $G=\SL(l)$., 37 pages

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual., Comment: 40 pages; several results and comments added. Accepted for publication in Letters in Mathematical Physics

We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.

In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of ``surface operators,'' which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves. We describe the relevant surface operators in N=4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters. Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification. The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space., Comment: 160 pp

A recent attempt to extend the geometric Langlands duality to affine Kac-Moody groups, has led Braverman and Finkelberg [arXiv:0711.2083] to conjecture a mathematical relation between the intersection cohomology of the moduli space of G-bundles on certain singular complex surfaces, and the integrable representations of the Langlands dual of an associated affine G-algebra, where G is any simply-connected semisimple group. For the A-type groups, where the conjecture has been mathematically verified to a large extent, we show that the relation has a natural physical interpretation in terms of six-dimensional compactifications of M-theory with coincident five-branes wrapping certain hyperkahler four-manifolds; in particular, it can be understood as an expected invariance in the resulting spacetime BPS spectrum under string dualities. By replacing the singular complex surface with a smooth multi-Taub-NUT manifold, we find agreement with a closely related result demonstrated earlier via purely field-theoretic considerations by Witten. By adding OM five-planes to the original analysis, we argue that an analogous relation involving the non-simply-connected D-type groups, ought to hold as well. This is the first example of a string-theoretic interpretation of such a two-dimensional extension to complex surfaces of the geometric Langlands duality for the A-D groups., Comment: 43 pages. Technical refinement of section 4.3. Minor correction of sections 5.2 and 5.3: level 2k changed to k

The global geometric Langlands correspondence relates Hecke eigensheaves on the moduli stack of G-bundles on a smooth projective algebraic curve X and holomorphic G'-bundles with connection on X, where G' is the Langlands dual group of G. This correspondence is relatively well understood when the connection has no singularities, i.e., is unramified. But what should happen if the connection has singularities, or ramification, at finitely many points of X? In this case we expect to attach to it a category of Hecke eigensheaves on a suitable moduli stack of G-bundles on X with parabolic (or level) structures at the ramification points. In this paper I review an approach to constructing these categories developed by D.Gaitsgory and myself. It uses representations of affine Kac-Moody algebras of critical level and localization functors from these representations to D-modules on the moduli spaces of bundles with parabolic (or level) structures. As explained in hep-th/0512172, these D-modules may be viewed as sheaves of conformal blocks naturally arising in the framework of Conformal Field Theory.

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck space of instantons on the quotient of the affine plane by a finite cyclic subgroup of SL(2)., Section 7 has been rewritten to correct previous mistakes

We study the two-dimensional twisted (0,2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence - at the level of the holomorphic chiral algebra - between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical W-algebras and a level relation which underlie a geometric Langlands correspondence for G=SL(N,C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G=SL(N,C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G=SL(N,C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on., 80 pages + appendix. Minor changes, further clarifications and improvements. Published version

We discuss several applications of the recent developments in the Langlands functoriality conjecture such as the automorphy of the symmetric powers of 2-dimensional complex representations of Galois groups of number fields, lattice point problems, Ramanujan– Selberg and Sato–Tate conjectures.We conclude by explaining how these recent developments are established.

Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group ^LG in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian Gr_G=G((t))/G[[t]] of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to ^LG in terms of the geometry of Gr_G. The idea of the construction belongs to Jacob Lurie.

Let G be a quasi-split p-adic group. Under the assumption that the local coefficients C ψ defined with respect to ψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.

We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m+1). We then show that duality for G2 is implemented by an involution on the base space which takes one fibre to its dual. A formula for the natural cubic form is given and shown to be invariant under the involution., 33 pages

Let $X$ be a (smooth and complete) curve and $G$ a reductive group. In [BG] we introduced the object that we called "geometric Eisenstein series". This is a perverse sheaf $\bar{Eis}_E$ (or rather a complex of such) on the moduli stack $Bun_G(X)$ of principal $G$-bundles on $X$, which is attached to a local system $E$ on $X$ with respect to the torus $\check{T}$, Langlands dual to the Cartan subgroup $T\subset G$. In loc. cit. we showed that$\bar{Eis}_E$ corresponds to the $\check{G}$-local system induced from $E$, in the sense of the geometric Langlands correspondence. In the present paper we address the following question, suggested by V. Drinfeld: what is the perverse sheaf on $Bun_G(X)$ that corresponds to the universal deformation of $E$ as a local system with respect to the Borel subgroup $\check{B}\subset \check{G}$? We prove, following a conjecture of Drinfeld, that the resulting perverse sheaf if the classical, i.e., non-compactified Eisenstein series.

We show that the Hitchin integrable system for a simple complex Lie group $G$ is dual to the Hitchin system for the Langlands dual group $\lan{G}$. In particular, the general fiber of the connected component $\Higgs_0$ of the Hitchin system for $G$ is an abelian variety which is dual to the corresponding fiber of the connected component of the Hitchin system for $\lan{G}$. The non-neutral connected components $\Higgs_{\alpha}$ form torsors over $\Higgs_0$. We show that their duals are gerbes over $\Higgs_0$ which are induced by the gerbe of $G$-Higgs bundles $\gHiggs$. More generally, we establish a duality between the gerbe $\gHiggs$ of $G$-Higgs bundles and the gerbe $\lan{\gHiggs}$ of $\lan{G}$-Higgs bundles, which incorporates all the previous dualities. All these results extend immediately to an arbirtary connected complex reductive group $\mathbb{G}$., Comment: 75 pages, 1 figure, LaTeX. New version substantially expanded and revised for publication

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics., Comment: 225 pp; further clarifications

Smooth Representations.- Finite Fields.- Induced Representations of Linear Groups.- Cuspidal Representations.- Parametrization of Tame Cuspidals.- Functional Equation.- Representations of Weil Groups.- The Langlands Correspondence.- The Weil Representation.- Arithmetic of Dyadic Fields.- Ordinary Representations.- The Dyadic Langlands Correspondence.- The Jacquet-Langlands Correspondence.

Using a level-raising argument (and a result of Larsen on the image of Galois representations in compatible systems), we prove that for any automorphic representation $\pi$ for U(3), the l-adic Galois representation $\rho_l$ which is attached to $\pi$ by the work of Blasius and Rogawski is the one expected by local Langlands correspondance at every finite place (at least up to semi-simplification and for a density one set of primes l). We rely on the work of Harris and Taylor, who have proved the same results (for U(n)) assuming the base change of $\pi$ is square-integrable at one place. As a corollary, every automorphic representation which is tempered at an infinite number of places is tempered at all places

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ``Langlands dual'' group. We generalize this description to an arbitrary spherical variety X of G as follows: Irreducible unramified quotients of the space $C_c^\infty(X)$ are in natural ``almost bijection'' with a number of copies of $A_X^*/W_X$, the quotient of a complex torus by the ``little Weyl group'' of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ``distinguished'' by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by F. Knop, of the Weyl group on the set of Borel orbits., Comment: Final version, to appear in Compositio Mathematica

— By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups. Resume. — Par la theorie de Colmez et Fontaine, une representation de de Rham du groupe de Galois d’un corps local correspond essentiellement a une representation du groupe de Weil-Deligne dont l’espace sous-jacent est muni d’une filtration admissible. En modifiant la correspondance locale de Langlands, on associe a chaque couple forme d’une representation du groupe de Weil-Deligne et des poids d’une filtration (admissible ou pas) une representation localement algebrique particuliere d’un groupe lineaire general. On conjecture qu’un couple provient d’une representation de de Rham si et seulement si la representation localement algebrique correspondante possede une norme invariante. Dans le cas cristallin, la representation du groupe de Weil-Deligne est non-ramifiee et la representation localement algebrique associee peut s’etudier grâce a l’isomorphisme de Satake classique. En prolongeant ce dernier a une completion de l’algebre de Hecke, on montre que l’existence d’une norme invariante comme ci-dessus implique que le couple provient effectivement d’une representation cristalline. On montre aussi, en utilisant le formalisme des categories tannakiennes, que ce dernier fait est compatible avec la fonctorialite de Langlands non-ramifiee classique, et donc qu’il se generalise a tout groupe reductif deploye. 2 C. BREUIL & P. SCHNEIDER

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra $\mathfrak g$. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice are given by the zero set of the $\tau$-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the $\tau$-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra., Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics"