Local Langlands Correspondence Research Articles

Fonctions d'une variable p-adique et représentations de [formula omitted]

We extend the dictionary between Fontaine rings and p-adic functionnal analysis, and we give a refinement of the p-adic local Langlands correspondence for principal series representations of GL2(Qp).

L-packets over strong real forms

Langlands [On the classification of irreducible representations of real algebraic groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined L L -packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992] combined L-packets over all real forms belonging to an inner class. In the tempered setting, using different methods, Kaletha [Ann. of Math. (2) 184 (2016), pp. 559–632] also defines such combined L-packets with a refinement to the local Langlands correspondence. We prove that the tempered L-packets of Adams-Barbasch-Vogan and Kaletha are the same and are parameterized identically.

Local-global compatibility for regular algebraic cuspidal automorphic representations when

Abstract We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$ . We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$ , the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$ .

Bruhat-Tits buildings, representations of p-adic groups and Langlands correspondence

The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of p-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the case of supercuspidal representations, in particular regarding the local Langlands correspondence and the internal structure of the L-packets.We prove that the enhanced L-parameters with semisimple cuspidal support are those which are obtained via the (ordinary) Springer correspondence.Let G be a connected reductive group over a non-archimedean field F of residual characteristic p. In the case where G splits over a tamely ramified extension of F and p does not divide the order of the Weyl group of G, we show that the enhanced L-parameters with semisimple cuspidal support correspond to irreducible smooth representations of G(F) with non-singular supercuspidal support via the local Langlands correspondence constructed by Kaletha, under the assumption that the latter satisfies certain expected properties of functorial nature. As a consequence, we obtain that every compound L-packet of G(F) contains at least one representation with non-singular supercuspidal support.

Hecke algebras for p-adic reductive groups and Local Langlands Correspondences for Bernstein blocks

We study the endomorphism algebras attached to Bernstein components of reductive p-adic groups and construct a local Langlands correspondence with the appropriate set of enhanced L-parameters, using certain “desiderata” properties for the LLC for supercuspidal representations of proper Levi subgroups. We give several applications of our LLC to various reductive groups with Bernstein blocks cuspidally supported on general linear groups.In particular, for Levi subgroups of maximal parabolic subgroups of the split exceptional group G2, we compute the explicit weight functions for the corresponding Hecke algebras, and show that they satisfy a conjecture of Lusztig's. Some results from §4 are used by the same authors to construct a full local Langlands correspondence in [9]. Moreover, we prove a “reduction to depth-zero” result for regular Bernstein blocks (i.e., blocks for which the supercuspidal support of each irreducible representation is regular).

Coherent Springer theory and the categorical Deligne-Langlands correspondence

Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$K$\\end{document}-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$F$\\end{document} with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$K$\\end{document}-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu).In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of GLn(F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathrm{GL}_{n}(F)$\\end{document} into coherent sheaves on the stack of Langlands parameters.

On the étale cohomology of Hilbert modular varieties with torsion coefficients

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm {GL}_2(\mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.

Towards a mod-p Lubin–Tate theory for GL2 over totally real fields

In this paper, we show that the conjectural mod [Formula: see text] local Langlands correspondence can be realized in the mod [Formula: see text] cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field [Formula: see text] and of mod [Formula: see text] local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13].

On unipotent representations of ramified 𝑝-adic groups

Let G G be any connected reductive group over a non-archimedean local field. We analyse the unipotent representations of G G , in particular in the cases where G G is ramified. We establish a local Langlands correspondence for this class of representations, and we show that it satisfies all the desiderata of Borel as well as the conjecture of Hiraga, Ichino and Ikeda about formal degrees. This generalizes work of Lusztig and of Feng, Opdam and the author, to reductive groups that do not necessarily split over an unramified extension of the ground field.

Rigid inner forms over local function fields

We generalize the concept of rigid inner forms, defined by Kaletha in [12], to the setting of a local function field F in order state the local Langlands conjectures for arbitrary connected reductive groups over F. To do this, we define for a connected reductive group G over F a new cohomology set H1(E,Z→G)⊂Hfpqc1(E,G) for a gerbe E attached to a class in Hfppf2(F,u) for a certain canonically-defined profinite commutative group scheme u, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over F, and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and s˙-stable virtual characters for a semisimple s˙ associated to a tempered Langlands parameter.

An approach to the characterization of the local Langlands correspondence

Scholze and Shin [J. Amer. Math. Soc. 26 (2013), pp. 261–294] gave a conjectural formula relating the traces on the automorphic and Galois sides of a local Langlands correspondence. Their work generalized an earlier formula of Scholze, which he used to give a new proof of the local Langlands conjecture for G L n \mathrm {GL}_n . Unlike the case for G L n \mathrm {GL}_n , the existence of non-singleton L L -packets for more general reductive groups constitutes a serious representation-theoretic obstruction to proving that such a formula uniquely characterizes such a correspondence. We show how to overcome this problem, and demonstrate that the Scholze–Shin equation is enough, together with other standard desiderata, to uniquely characterize the local Langlands correspondence for discrete parameters.

Local exterior square and Asai L-functions for GL(n) in odd characteristic

Let $F$ be a non-archimedean local field of odd characteristic $p > 0$. In this paper, we consider local exterior square $L$-functions $L(s,\pi,\wedge^2)$, Bump-Friedberg $L$-functions $L(s,\pi,BF)$, and Asai $L$-functions $L(s,\pi,As)$ of an irreducible admissible representation $\pi$ of $GL_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,\wedge^2(\phi(\pi)))$, $L(s+1/2,\phi(\pi))L(s,\wedge^2(\phi(\pi)))$, and $L(s,As(\phi(\pi)))$ of the associated Langlands parameter $\phi(\pi)$ under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local to global argument due to Henniart and Lomeli.

Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields. The topics of the talks included foundational results on analytic spaces as well as applications to the local Langlands conjecture, birational geometry, p-adic cohomology theories, Shimura varieties and the non-Archimedean Simpson correspondence.

The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

Abstract In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p {\operatorname{\mathbb{Q}}_{p}} in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.

Local distinction, quadratic base change and automorphic induction for GL n

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual or conjugate-dual representations. When applied to semi-simple representations of the Weil–Deligne group W F ′ of a non Archimedean local field F, and further translated in terms of representations of GL n (F) via the local Langlands correspondence when F has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When F has residual characteristic different from 2, combining of one of the simple results that we obtain with the tiviality of conjugate-orthogonal root numbers ([8]), we recover without using the LLC a result of Serre on the parity of the Artin conductor of orthogonal representations of W F ′ ([23]). On the other hand we discuss its parity for symplectic representations using the LLC and the Prasad and Takloo-Bighash conjecture.

Local Langlands correspondence for the twisted exterior and symmetric square $$\epsilon $$-factors of $$\text {GL}_n$$

Local Langlands correspondence for the twisted exterior and symmetric square $$\epsilon $$-factors of $$\text {GL}_n$$

The Local Langlands Conjecture for

Abstract We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.

The local Langlands correspondence for GLn over function fields

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze, we give a new proof of the local Langlands correspondence for $\GL_n$ over $F$. More specifically, we construct $\ell$-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map $\DeclareMathOperator{\rec}{rec}\pi\mapsto\rec(\pi)$ from isomorphism classes of irreducible smooth representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$. Our map $\rec$ is characterized in terms of a local compatibility condition on traces of a certain test function $f_{\tau,h}$, and we prove that $\rec$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).

Compatibility of semisimple local Langlands parameters with parahoric Satake parameters

In this paper, we prove that there is at most one correspondence between parahoric-spherical representations and semisimple local Langlands parameters which satisfies certain natural properties. Our proof of this uniqueness statement is very formal. In particular, the semisimple local Langlands parameters constructed in Fargues and Scholze (Geometrization of the local Langlands correspondence, 2021. arXiv:2102.13459 ) yield the unique candidate for such representations. As a corollary, we prove the conjecture (Haines in IMRN 2015(20):10367–10398, 2015) [Conjecture 13.1] which posits the compatibility of semisimple local Langlands parameters with parahoric Satake parameters.

P-adic L-functions via local–global interpolation: the case of

AbstractLet F be a totally real field, and let $E/F$ be a CM quadratic extension. We construct a p-adic L-function attached to Hida families for the group $\mathrm {GL}_{2/F}\times \mathrm {Res}_{E/F}\mathrm {GL}_{1}$ . It is characterized by an exact interpolation property for critical Rankin–Selberg L-values, at classical points corresponding to representations $\pi \boxtimes \chi $ with the weights of $\chi $ smaller than the weights of $\pi $ .Our p-adic L-function agrees with previous results of Hida when $E/F$ splits above p or $F=\mathbf {Q}$ , and it is new otherwise. Exploring a method that should bear further fruits, we build it as a ratio of families of global and local Waldspurger zeta integrals, the latter constructed using the local Langlands correspondence in families.In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof of an identity concerning double factorials.