Christian Pierre
The Langlands functoriality conjecture envisaged in the bisemialgebra framework is proved to correspond to the nonorthogonal completely reducible cuspidal representations of the bilinear algebraic semigroups.
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We generalize a beautiful method of Blasius and Ramakrishnan, that in order to exhibit particular instances of the Langlands functorial correspondence, it is enough to show that the correspondence holds in the semistable case, provided the arithmetic data we are considering is closed with respect to cyclic base change and descent.
In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.
Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $η: \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible $G$-representations $π$ along $η$. Following Borel, Adler--Korman and Xu, we pose a conjecture on the decomposition of the pullback $η^* π$. It is formulated in terms of enhanced Langlands parameters and includes multiplicities. This can be regarded as a functoriality property of the local Langlands correspondence. We prove this conjecture for three classes: principal series repr…
Local Langlands functoriality for Yu's supercuspidals II: Fargues--Scholze's parametrization
本文通过计算Yu构造尖点表示的Tate上同调，结合模函子性，比较Fargues--Scholze与Kaletha参数化，证明前者具有有限纤维且满射到惯性L-参数。
Local Langlands functoriality for Yu's supercuspidals I: Kaletha's parametrization
本文为两篇系列论文之首，受Kaletha工作启发，为Yu构造的所有尖点表示构造任意系数的半单惯性L-参数，并基于函子性性质刻画对应，为后续计算Fargues–Scholze对应奠定基础。
Modular functoriality in the Local Langlands Correspondence
本文在Fargues-Scholze几何化框架中发展Smith-Treumann局部化理论，证明Treumann-Venkatesh关于模$\ell$局部Langlands函子性的猜想，并计算模$\ell$环面表示的参数。
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 …
The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey of my recent joint work with Robert Langlands and Ngo Bao Chau (arXiv:1003.4578 and arXiv:1004.5323) on a new approach to proving the Functoriality Conjecture using the trace formulas, and on the geometr…
We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical $G$-variety $X$, one associates a dual group ${^LG_X}$ and an $L$-value (encoded in a representation of ${^LG_X}$), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ${^LG_X}\to {^LG_Y}$ of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approac…
This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic $L$-functions.
A Functorial Refinement of the Franke Filtration and the Jacquet--Langlands Correspondence for Spaces of Automorphic Forms
本文通过扩展弗兰克滤波器的functorial细化，建立了自守形式空间中一般线性群及其内形式之间的雅各布-朗格兰斯对应关系，扩展了Badulescu的对应关系并包含完整functorial提升。
In this paper we first review the setting for the geometric Langlands functoriality and establish a result for the `backward' functoriality functor. We illustrate this by known examples of the geometric theta-lifting. We then apply the above result to obtain new Hecke eigen-sheaves. The most important application is a construction of the automorphic sheaf for G=GSp_4 attached to a G^L-local system on a curve X such that its standard representation is an irreducible local system of rank 4 on X.
Towards the Relative Langlands Duality for Orthosymplectic Pairs
本文证明了一个范畴等价猜想，即作用在张量积上的正交辛群的对偶性，作为Ben Zvi、Sakellaridis和Venkatesh的非极化相对朗兰兹对偶性的特例，并由此推出派生Satake同构的朗兰兹函子性由theta对应给出。
We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp(4) to GL(4) follows from our result as a special case.
We establish Langlands functoriality for the generic spectrum of GSp(4) and describe its transfer on GL(4). We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).
In this paper we propose a new way to realize cyclic base change (a special case of Langlands functoriality) for prime degree extensions of characteristic zero local fields. Let $F / E$ be a prime degree $l$ extension of local fields of residue characteristic $p \neq l$. Let $π$ be an irreducible cuspidal $l$-adic representation of $\mathrm{GL}_n(E)$ and $ρ$ be an irreducible cuspidal $l$-adic representation of $\mathrm{GL}_n(F)$ which is Galois-invariant. Under some minor technical conditions on $π$ and $ρ$ (for instance, we assume that both are level zero) we prove that the $\bmod l$-reducti…
Braverman and Kazhdan introduced influential conjectures generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands $L$-functions have meromorphic continuations and functional equations as predicted by Langlands' functoriality conjecture. As evidence for their conjectures, Braverman and Kazhdan considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. In this paper we consider a special case o…
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