Comparison of Bushnell-Kutzko and Yu's constructions of supercuspidal representations

The Langlands parameter of a simple supercuspidal representation: Symplectic groups

Let $π$ be a simple supercuspidal representation of the symplectic group $Sp_{2l}(F)$, over a $p$-adic field $F$. In this work, we explicitly compute the Rankin-Selberg $γ$-factor of rank-$1$ twists of $π$. We then completely determine the Langlands parameter of $π$, if $p \neq 2$. In the case that $F = \mathbb{Q}_2$, we give a conjectural description of the functorial lift of $π$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.

On the support of matrix coefficients of supercuspidal representations of the general linear group over a local non-archimedean field

We derive an upper bound on the support of matrix coefficients of suprecuspidal representations of the general linear group over a non-archimedean local field. The results are in par with those which can be obtained from the Bushnell--Kutzko classification of supercuspidal representations, but they are proved independently.

Simple characters and coefficient systems on the building

Let F be a non-archimedean local field and G be the group GL(N,F). Let πbe a smooth complex representation of G lying in the Bernstein block B(π) of some simple type in the sense of Bushnell and Kutzko. Refining the approach of the second author and U. Stuhler, we canonically attach to πa subset X_πof the Bruhat-Tits building X of G, as well as a G-equivariant coefficient system C[π] on X_π. Roughly speaking the coefficient system is obtained by taking isotypic components of πaccording to some representations constructed from the Bushnell and Kutzko type of π. We conjecture that when πhas cent…

Types and unitary representations of reductive p-adic groups

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

A geometric construction of types for the smooth representations of PGL(2) of a local field

We show that almost all (Bushnell and Kutzko) types of PGL(2,F), F a non-Archimedean locally compact field of odd residue characteristic, naturally appear in the cohomology of finite graphs.

Comparison of types for inner forms of $\mathrm{GL}_{N}$

Let $F$ be a non-archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. To construct types for supercuspidal representations of $G$, simple types by Sécherre--Stevens and Yu's construction are already known. In this paper, we compare these constructions. In particular, we show essentially tame supercuspidal representations of $G$ defined by Bushnell--Henniart are nothing but tame supercuspidal representations defined by Yu.

On finite symplectic modules arising from supercuspidal representations

Let $F$ be a non-Archimedean local field with finite residue field. Let $\mathcal{A}^{et}_n(F)$ be the collection of isomorphism classes of essentially tame irreducible supercuspidal representations of $\mathrm{GL}_n(F)$ studied by Bushnell-Henniart. It is known that we can parameterize $\mathcal{A}^{et}_n(F)$ by the collection $P_n(F)$ of equivalence classes of admissible pairs $(E, ξ)$ consisting of a tamely ramified extension $E/F$ of degree $n$ and an $F$-admissible character $ξ$ of $E^\times$. We are interested in a finite symplectic module $V = V(ξ)$ arising from the construction of the …

Local Jacquet-Langlands correspondence for regular supercuspidal representations

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 \…

Unicity of types for supercuspidal representations of p-adic SL(2)

We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of $\mathbf{SL}_2$ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the $\mathbf{SL}_2$-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in $\mathbf{SL}_2$ is restricted from one in $\mathbf{GL}_2$. From this it follows that any archetype must be induced from a Bushnell--Kutzko type. Given a supercuspidal representation $π$, we give an additional explicit description…

SL(2) 上非分裂覆盖的局部系数与 Gelfand-Graev 表示

Local coefficients and Gelfand-Graev representations for non-split covers on SL(2)

本文推广 Krishnamurthy-Kutzko 的局部方法至 SL(2) 非分裂覆盖，计算 Langlands-Shahidi 局部系数，并研究 Gelfand-Graev 表示的代数结构，推广至无特征限制的非阿基米德局部域。

On the realisation of maximal simple types and epsilon factors of pairs

Let $G$ be the group of rational points of a general linear group over a non-archimedean local field $F$. We show that certain representations of open, compact-mod-centre subgroups of $G$, (the maximal simple types of Bushnell and Kutzko) can be realized as concrete spaces. In the level zero case our result is essentially due to Gelfand. This allows us, for a supercuspidal representation $π$ of $G$, to compute a distinguished matrix coefficient of $π$. By integrating, we obtain an explicit Whittaker function for $π$. We use this to compute the epsilon factor of pairs, for supercuspidal represe…

从SO(5)到GL(4)的局部朗兰兹函子转移

On the Local Langlands Functorial Transfer from $\text{SO}(5)$ to $\text{GL}(4)$

针对特征零的非阿基米德局部域F，研究GL(4,F)超尖点表示从SO(5,F)到GL(4,F)的局部朗兰兹函子转移，给出其等价判定条件，推进了局部朗兰兹纲领相关研究。