scholarly journals THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

2021 ◽  
pp. 1-56
Author(s):  
Alexander Bertoloni Meli
Keyword(s):  
Levi Subgroup ◽  
Admissible Representation ◽  
Supercuspidal Representation ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
Grothendieck Groups ◽  
Local Langlands Correspondence

Abstract We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.

scholarly journals Transfer of Plancherel Measures for Unitary Supercuspidal Representations betweenp-adic Inner Forms

2014 ◽  
Vol 66 (3) ◽  
pp. 566-595 ◽  
Author(s):  
Kwangho Choiy
Keyword(s):  
Levi Subgroup ◽  
Plancherel Measure ◽  
Supercuspidal Representation ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
Working Hypothesis ◽  
Simply Connected ◽  
Positive Integers

AbstractLetFbe ap-adic field of characteristic 0, and letMbe anF-Levi subgroup of a connected reductiveF-split group such thatGLnifor positive integersrand ni. We prove that the Plancherel measure for any unitary supercuspidal representation ofM(F) is identically transferred underthe local Jacquet–Langlands type correspondencebetweenMand itsF-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups SO4nand Sp4nunderthe local Jacquet–Langlands correspondence. It can be applied to a simply connected simpleF-group of typeE6orE7, and a connected reductiveF-group of typeAn,Bn,CnorDn.

scholarly journals Local Langlands Correspondence for Regular Supercuspidal Representations of GL(n)

2020 ◽  
Author(s):  
Masao Oi ◽  
Kazuki Tokimoto
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
General Linear Groups ◽  
The Difference ◽  
Local Langlands Correspondence

Abstract 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 $\chi $-data.

scholarly journals Epipelagic Langlands parameters and L-packets for unitary groups

2020 ◽  
Vol 16 (07) ◽  
pp. 1449-1491
Author(s):  
Tony Feng ◽  
Niccolò Ronchetti ◽  
Cheng-Chiang Tsai
Keyword(s):  
Invariant Theory ◽  
Unitary Groups ◽  
General Construction ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
Langlands Parameters ◽  
New Construction ◽  
Local Langlands Correspondence ◽  
Invent Math

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations [M. Reeder and J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27(2) (2014) 437–477, MR 3164986]. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Epipelagic [Formula: see text]-packets and rectifying characters, Invent. Math. 202(1) (2015) 1–89, MR 3402796]. The interesting feature of our computation is that we find simplifications within [Formula: see text]-packets of the two novel invariants introduced in the above-mentioned paper of Kaletha, the toral invariant and the admissible L-embedding.

scholarly journals On the Notion of Conductor in the Local Geometric Langlands Correspondence

2017 ◽  
Vol 69 (1) ◽  
pp. 107-129
Author(s):  
Masoud Kamgarpour
Keyword(s):  
Irreducible Representation ◽  
Galois Representation ◽  
Loop Group ◽  
Langlands Correspondence ◽  
Categorical Representation ◽  
Swan Conductor ◽  
Geometric Langlands ◽  
Meromorphic Connection ◽  
Local Langlands Correspondence ◽  
Geometric Analogue

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

scholarly journals Sur les -blocs de niveau zéro des groupes -adiques

2018 ◽  
Vol 154 (7) ◽  
pp. 1473-1507
Author(s):  
Thomas Lanard
Keyword(s):  
Abelian Category ◽  
Langlands Correspondence ◽  
Parabolic Induction ◽  
Langlands Parameters ◽  
Tits Building ◽  
Local Langlands Correspondence

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

Globalization of Distinguished Supercuspidal Representations of GL(n)

2002 ◽  
Vol 45 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Jeffrey Hakim ◽  
Fiona Murnaghan
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Cuspidal Representation ◽  
Supercuspidal Representation ◽  
Local Component ◽  
Supercuspidal Representations

AbstractAn irreducible supercuspidal representation π of G = GL(n, F), where F is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup H of G and a quasicharacter χ of H if HomH(π, χ) ≠ 0. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to GL(n). Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

scholarly journals Local Langlands correspondence and ramification for Carayol representations

2019 ◽  
Vol 155 (10) ◽  
pp. 1959-2038
Author(s):  
Colin J. Bushnell ◽  
Guy Henniart
Keyword(s):  
Langlands Correspondence ◽  
Smooth Complex ◽  
Compact Field ◽  
Weil Group ◽  
Detailed Interpretation ◽  
Simple Character ◽  
Local Langlands Correspondence ◽  
Class Of Functions ◽  
Residual Characteristic

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group ${\mathcal{W}}_{F}$. Let $\unicode[STIX]{x1D70E}$ be an irreducible smooth complex representation of ${\mathcal{W}}_{F}$, realized as the Langlands parameter of an irreducible cuspidal representation $\unicode[STIX]{x1D70B}$ of a general linear group over $F$. In an earlier paper we showed that the ramification structure of $\unicode[STIX]{x1D70E}$ is determined by the fine structure of the endo-class $\unicode[STIX]{x1D6E9}$ of the simple character contained in $\unicode[STIX]{x1D70B}$, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$. In this paper we concentrate on the fundamental Carayol case in which $\unicode[STIX]{x1D70E}$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\unicode[STIX]{x1D70E}$, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$, as $\unicode[STIX]{x1D6EF}$ varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of $\unicode[STIX]{x1D70E}$ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.

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