We prove that the global Jacquet--Langlands correspondence ${\rm JL}$ for ${\rm GL}(2)$ can be realized via tensor products over Hecke algebras. Let $G$ be a non-split inner form of ${\rm GL}(2)$ over a number field. Using the similitude theta correspondence, the space $L^2(D(\mathbb{A})\times \mathbb{A}^{\times})$ acquires the structure of a $G(\mathbb{A})$-$(G(\mathbb{A})\times {\rm GL}(2,\mathbb{A}))$ bimodule such that
$L^2(G(F)\backslash G(\mathbb{A}),χ)\otimes_{\mathcal{H}(G)}L^2(D(\mathbb{A})\times \mathbb{A}^{\times})~\cong~\oplus_{π\in {\mathcal{A}}(G,χ^{-1})}$ $π\otimes{\rm JL}(π).$
This decomposition into irreducible representations of $G(\mathbb{A})\times {\rm GL}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence.
We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of GL(n) over a non-archimedean local field by means of the local Jacquet-Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).
We extend the Jacquet-Langlands'correspondence between the Hecke-modules of usual and quaternionic modular forms, to overconvergent p-adic forms of finite slope. We show that this correspondence respects p-adic families and is induced by an isomorphism between some associated eigencurves.
We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet-Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes.
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $\text{\rm GL}_n(F)$. We give a transparent parametrization of the irreducible, totally ramified, cuspidal representations of $G$ of parametric degree $n$. We show that the parametrization is respected by the Jacquet-Langlands correspondence, relative to any other inner form. This expresses the Jacquet-Langlands correspondence for such representations within a single, compact formula.
We prove that Kaletha's local Langlands correspondence for regular supercuspidal representations gives the classical local Jacquet--Langlands correspondence due to Deligne--Kazhdan--Vigneras and Badulescu. As in a former joint paper with Oi, where a similar result is proved for the local Langlands correspondence for the general linear group, the key ingredients in our proof are the work of Bushnell--Henniart explicitly describing the local Jacquet--Langlands correspondence for essentially tame supercuspidal representations and its reinterpretation due to Tam in terms of Langlands--Shelstad's \…
Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).
Let $F$ be a a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^*=\mathrm{GL}_n$ over $F$. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^*$ can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for $G$ can be described using admissible embeddings of L-tori.
Let $F$ be a locally compact non-Archimedean field, and let $B/F$ be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations of $B^\times$ of dimension $>1$ and irreducible cuspidal representations of $GL_2(F)$. We present a new construction of this bijection in which the preservation of epsilon factors is automatic.
In this paper, we explicitly compute the semisimplifications of all Jacquet modules of irreducible representations with generic L-parameters of p-adic split odd special orthogonal groups or symplectic groups. Our computation represents them in terms of linear combinations of standard modules with rational coefficients. The main ingredient of this computation is to apply Moeglin's explicit construction of local A-packets to tempered L-packets.
Let $F$ be a non-archimedean local field. For any irreducible representation $π$ of an inner form $G'=\mathrm{GL}_{m}(D)$ of $G=\mathrm{GL}_{N}(F)$, there exists an irredubile representation of a maximal compact open subgroup in $G'$ which is also a type for $π$. Then we can consider the problem whether these types are unique or not in some sense. If such types for $π$ are unique, we say $π$ has the strong unicity property of types. On the other hand, there exists a correspondence connecting irreducible representations of $G'$ and $G$, called the Jacquet--Langland correspondence. In this paper…
We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from the set of automorphic square integrable representations of the multiplicative group of $D$ to the set of automorphic square integrable representations of GL_n(F), compatible at all places with the local Jacquet-Langlands correspondence for unitary representations. We characterize the image of the map. …
We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.
In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over $\bbR$, $\bbC$ or $\bbH$ of independent interest.
Let G be a general linear group over a p-adic field and let D^* be an anisotropic inner form of G. The Jacquet-Langlands correspondence between irreducible complex representations of D^* and discrete series of G does not behave well with respect to reduction modulo l\neq p. However we show that the Langlands-Jacquet transfer, from the Grothendieck group of admissible l-adic representations of G to that of D^* is compatible with congruences and reduces modulo l to a similar transfer for l-modular representations, which moreover can be characterized by some Brauer characters identities. Studying…
In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic techn…
A Jacquet-Langlands functor for $p$-adic locally analytic representations
Université de Clermont-Auvergne（克莱蒙奥弗涅大学）；Institute of Mathematics ”Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米恩·斯托伊洛瓦数学研究所）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）
Let F be a non-Archimedean local field and let E be an unramified extension of F of degree n>1. To each sufficiently generic multiplicative character of E (the details are explained in the body of the paper) one can associate an irreducible n-dimensional representation of the Weil group W_F of F, which corresponds to an irreducible supercuspidal representation π of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, π corresponds to an irreducible representation ρ of the multiplicative group of the central division algebra over F with invariant 1/…
Let K/F be a tamely ramified quadratic extension of non-archimedean locally compact fields. Let GL_m (D) be an inner form of GL_n (F) and GLp(R) = (M_m (D) \otimes K)^{\times} . Then GLp(R) is an inner form of GL_n (K). In this work, we determine conditions of GL_m (D)-distinction for level zero cuspidal representations of GL_p (R) which are the image of a level zero cuspidal representation by the Jacquet-Langlands correspondence, and we also prove that a level zero cuspidal representation of GL_n (K) is GL_n (F) distinguished if and only if its image by the Jacquet-Langlands correspondance is…
We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F). We consider the problem of describing explicitly the local Jacquet--Langlands correspondence between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet--Langlands corresponden…
We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the …
We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge-Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proo…
Let K/F be a quadratic extension of non archimedean local fields. Let V be an irreducible level zero discrete series representation of the group G = GL2(R), where R is a division algebra of center K and of index r. One assumes that V is not a supercuspidal representation. Let D be a division algebra of center F and index 2r. In the following work, we try to answer this question : Is the representation V D^{*}-distinguished if and only if its image by the Jacquet-Langlands correspondence is GL2(F)-distinguished. To answer this question, we use the coefficients system on the Bruhat-Tits building…
Given an induced representation of Langlands type $(π,V)$ of $\mathrm{GL}_n(F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $β$ and Schwartz-Bruhat function $Φ$ for which the Godement-Jacquet zeta integral $Z(s,β,Φ)$ attains the $L$-function $L(s,π)$.
Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the geometry and cohomology of the tamely ramified cover of Drinfeld's symmetric space over $K.$ We prove, in a purely local way, that the cohomology in middle degree realises the local Langlands correspondence and local Jacquet-Langlands correspondence for the depth zero supercuspidal representations.
In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups, analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let $X$ be the $p$-adic Deligne-Lusztig ind-scheme associated to a division algebra $D$ of invariant k/n over a non-Archimedean local field $K$. We study its homology groups by establishing a Deligne-Lusztig theory for families of finite unipotent groups that arise as subquotients of $D^\times$. The homology of $X$ induces a natural corresp…
Let F be a non-archimedean local field of characteristic zero with residual characteristic p. In this paper, we present a simple proof and construction of the local Langlands correspondence for simple supercuspidal representations of GL_n(F), when p does not divide n. As an application, we prove Jacquet's conjecture on the local converse problem for GL_n(F) in the case of simple supercuspidal representations, for arbitrary p.
For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformat…
We study the geometry and cohomology of the (generic fibres) of formal deformation schemes of one-dimensional formal modules of finite height. By the work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic etale cohomology of these spaces realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence for supercuspidal representations is realized by…
We generalize the rectifier of Bushnell and Henniart, which occurs in the local Langlands correspondence for $GL_n(K)$, to certain Langlands parameters for unramified connected reductive groups.
Let $K$ be a local non-Archimedean field of positive characteristic and let $L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and Jacquet-Langlands correspondences, to each sufficiently generic multiplicative character of $L$, one can associate an irreducible representation of the multiplicative group of the central division algebra $D$ of invariant $1/n$ over $K$. In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we prove th…
In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e., D=F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke-Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G' to statements on automorphic forms of its split form using the global Jacquet-Langlands correspondence developed by …
We prove the local Langlands correspondence for GSp_4(F), where F is a non-archimedean local field of positive characteristic with residue characteristic > 2.
In this paper, we continue the work of the first author and give a new construction of the tame local Langlands correspondence for PGSp(4,F), where F is a p-adic field, that is analogous to the construction of the local Langlands correspondence for real groups.
In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze's alternate proof of the local Langlands conjecture for $\mathrm{GL}_n$. More specifically, we show that if a local Langlands correspondence satisfies a Scholze--Shin equation, as in the paper of Scholze and Shin, in addition to the usual desiderata expected of such a correspondence then these properties uniquely characterize the correspondence.
Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.
We give an explicit description of the modified mod p local Langlands correspondence for GL_2(Q_{\ell}) of Emerton-Helm, where p is an odd prime different from \ell.
We prove a result which provides a link between the decomposition of parabolically induced representations and the Bushnell--Kutzko theory of typical representations. As an application, we show that there exists a well-defined inertial Langlands correspondence which respects the monodromy action of L-parameters, under some standard conjectures regarding the local Langlands correspondence. To allow for potential applications of this inertial Langlands correspondence, we also provide a complete construction of the set of typical representations, giving a parametrization of these in terms of the …
In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups (arXiv:1804.10901) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence established by Gan--Ichino. To combine them, we compare splittings for metaplectic covers of unitary groups constructed by Kudla with those constructed by Pan.
In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.
In this paper, we prove the coincidence of Kaletha's recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris--Taylor's one in the case of general linear groups. The keys are Bushnell--Henniart's essentially tame local Langlands correspondence and Tam's result on Bushnell--Henniart's rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha's and Tam's $χ$-data.
In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O_{2n} established by Arthur. We give two applications: Prasad's conjecture and Gross--Prasad conjecture for O_{n}. Also, we discuss the Arthur conjecture for O_{2n}, and establish the Arthur multiplicity formula for O_{2n}.
These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic groups, and the modular Langlands correspondence. These notes give an overview of the subject with several examples.
We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-étale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture.
Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.
