Depth Zero Representations of Nonlinear Covers of $p$-adic Groups

A reducibility problem for even unitary groups: The depth zero case

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E/F$ a quadratic extension of p-adic fields the associated unitary group $G=\mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E)$. Let $π$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $ι_P^G π$ is reducible.

$p$-adic经典群与内形式的块分解

Block decompositions for $p$-adic classical groups and their inner forms

研究通过内参数分解$p$-adic经典群及其内形式的光滑表示范畴，证明诱导保分解并建立块与内参数的自然双射。

The unicity of types for depth-zero supercuspidal representations

We establish the unicity of types for depth-zero supercuspidal representations of an arbitrary $p$-adic group $G$, showing that each depth-zero supercuspidal representation of $G$ contains a unique conjugacy class of typical representations of maximal compact subgroups of $G$. As a corollary, we obtain an inertial Langlands correspondence for these representations, via the Langlands correspondence of DeBacker and Reeder.

通过Hecke代数同构实现驯顺p进群的深度零约化

Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms

该研究针对驯分歧p进连通约化群，证明Kim-Yu型对应的Hecke代数同构于某扭Levi子群的深度零型Hecke代数，在p不整除绝对Weyl群阶的条件下实现所有Bernstein块到深度零块的约化，且同构显式、系数域可推广。

Base change for Bernstein centers of depth zero principal series blocks

Let $G$ be an unramified group over a $p$-adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_1(p)$-level structure initiated by M. Rapoport and the author in [HR2].

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell--Kutzko's theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig.

p-进调和分析中的显式渐近展开 II

Explicit asymptotic expansions in p-adic harmonic analysis II

本文提出将深度为0的超 cuspidal 特征的渐近展开转化为任意正深度、 tame、超 cuspidal 特征的Kim-Murnaghan型渐近展开的方法，并探讨了相关傅里叶变换结果的独立价值。

Depth preserving property of the local Langlands correspondence for quasi-split classical groups in a large residual characteristic

For a quasi-split classical group over a p-adic field with sufficiently large residual characteristic, we prove that the maximum of depth of representations in each L-packet equals the depth of the corresponding L-parameter. Furthermore, for quasi-split unitary groups, we show that the depth is constant in each L-packet. The key is an analysis of the endoscopic character relation via harmonic analysis based on the Bruhat--Tits theory. These results are slight generalizations of a result of Ganapathy and Varma.

关于 $p$-adic 红uctive 组局部可积性结果的研究

On local integrability results for $p$-adic reductive groups

本文基于局部特征展开法，给出了哈里-查纳关于 $p$-adic 红uctive 组复特征局部可积性的著名定理的简短证明，展示了适用于某些非复系数的代数情形，并验证了文献未覆盖的局部可积性情况。

约化 $p$-进群覆盖群表示的纠缠算子

Intertwining operators for representations of covering groups of reductive $p$-adic groups

本文研究约化 $p$-进群覆盖群的抛物诱导表示间的纠缠算子，证明其伴随关系，并据此给出 Harish-Chandra $μ$-函数的显式公式及极点定位方法。

Patterns in Branching Rules for Irreducible Representations of SL_2(k), for k a p-adic field

Building on prior work, we analyze the decomposition of the restriction of an irreducible representation of SL_2(k), for k a p-adic field of odd residual characteristic, to a maximal compact subgroup K. The pattern of the decomposition varies between principal series and different supercuspidal representations, whereas the K-representations which occur in the "tail end" of these decompositions are precisely those occurring in the decomposition of depth-zero supercuspidal representations. Various applications are considered.

Geometrization of continuous characters of $\mathbb{Z}_p^\times$

We define the $p$-adic trace of certain rank-one local systems on the multiplicative group over $p$-adic numbers, using Sekiguchi and Suwa's unification of Kummer and Artin-Schrier-Witt theories. Our main observation is that, for every non-negative integer $n$, the $p$-adic trace defines an isomorphism of abelian groups between local systems whose order divides $(p-1)p^n$ and $\ell$-adic characters of the multiplicative group of $p$-adic integers of depth less than or equal to $n$.

Stability of character sums for positive-depth, supercuspidal representations

We re-write the character formulæ of Adler and the second-named author in a form amenable to explicit computations in $p$-adic harmonic analysis, and use them to prove the stability of character sums for a modification of Reeder's conjectural positive-depth, unramified, toral supercuspidal L-packets.

Deligne-Lusztig特征的Fourier-Jacobi模型与酉群的深度零局部下降

Fourier-Jacobi models of Deligne-Lusztig characters and depth zero local descent for unitary groups

本文推导了有限辛群、酉群和一般线性群的Deligne-Lusztig特征的Fourier-Jacobi模型重数公式，并应用于$p$-adic酉群的深度零局部下降，为非温Gan-Gross-Prasad纲领提供了具体实例。

Homological duality for covering groups of reductive $p$-adic groups

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…

Motivic proof of a character formula for SL(2)

We give a motivic proof of a character formula for depth zero supercuspidal representations of $p$-adic SL(2). We begin by finding the virtual Chow motives for the character values of all depth zero supercuspidal representations of $p$-adic SL(2), at topologically unipotent elements. Then we find the virtual Chow motives for the values of the Fourier transform of all regular elliptic orbital integrals with depth zero in their Cartan subalgebra, at topologically nilpotent elements. Finally, we prove a character formula for depth zero supercuspidal representations by showing that the formula cor…

Hecke Modules from Metaplectic Ice

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of $p$-adic groups and $R$-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on $p$-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of $R$-matri…

Whittaker models for depth zero representations of covering groups

We study the dimension of the space of Whittaker functionals for depth zero representations of covering groups. In particular, we determine such dimensions for arbitrary Brylinski-Deligne coverings of the general linear group. The results in the paper are motivated by and compatible with the work of Howard and the second author, and earlier work by Blondel.

奇酉群诱导表示的可约性：深度为零的情形

On reducibility of induced representations of odd unitary groups: the depth zero case

本文研究 $p$-adic 酉群中抛物诱导表示的可约性，针对深度为零的不可约超尖表示，利用 Hecke 代数方法给出判定条件。

$\overline{\mathbb{Z}}[\frac{1}{p}]$-块的 tame $p$-adic 群的参数化与降维到深度零

Parametrization and reduction to depth zero of $\overline{\mathbb{Z}}[\frac{1}{p}]$-blocks of tame $p$-adic groups

研究通过参数化和降维到深度零的方法，揭示了 tame $p$-adic 群的 $\overline{\mathbb{Z}}[\frac{1}{p}]$-块结构与局部朗兰兹对应的关系。

On the wavefront sets associated with theta representations

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over $p$-adic fields. In particular, it is shown that the formula agrees with the existing work in the literature for various families of groups. We also recapitulate the results of an analogous formula in the archimedean case, which motivated the conjectural formula in the $p$-adic setting.

Motivic nature of character values of depth-zero representations

It is shown that the values of Harish-Chandra distribution characters on definable compact subsets of the set of topologically unipotent elements of symplectic or special orthogonal p-adic groups can be expressed as the trace of Frobenius action on virtual Chow motives. The result is restricted to a class of depth-zero representations that can be obtained by inflation from Deligne-Lusztig representations. The proof relies on arithmetic motivic integration.

Endoscopic character identities for depth-zero supercuspidal L-packets

We prove the conjectural endoscopic transfer of L-packets for the local Langlands correspondence for pure inner forms of unramified p-adic groups and depth-zero parameters established by DeBacker and Reeder. More precisely, we show that under mild conditions on the residual characteristic, endoscopic induction identifies an unstable character of such an L-packet with the stable character of the corresponding endoscopic L-packet.

Supercuspidal characters of $\operatorname{SL}_2$ over a $p$-adic field

The character formulas of Sally and Shalika are an early triumph in $p$-adic harmonic analysis, but, to date, the calculations underlying the formulas have not been available. In this paper, which should be viewed as a precursor of the forthcoming volume by the authors and Alan Roche, we leverage modern technology (for example, the Moy-Prasad theory) to compute explicit character tables. An interesting highlight is the computation of the 'exceptional' supercuspidal characters, i.e., those depth-zero representations not arising by inflation-induction from a Deligne-Lusztig representation of fin…

$p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系

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

本文研究了 $p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系，通过构建增强的 $L$-参数集并应用到各种群上，证明了某些猜想并简化了深度零情况。

L'espace de Drinfeld et correspondance de Langlands locale II

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld's $p$-adic symmetric space. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $\ell$-adic cohomologies, as well as the torsion freeness of the integral cohomology. During the study, we obtain some results of independent interest on the coefficient systems over the Bruhat-Tits building associated to $GL_d(K).$

Supercuspidal characters of reductive p-adic groups

We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.

覆盖巴巴奇-沃根对偶性及真表示的前沿集

Covering Barbasch-Vogan duality and wavefront sets of genuine representations

本文提出并证明了覆盖巴巴奇-沃根对偶性下真表示前沿集的上界猜想，应用于一般线性群的Kazhdan-Patterson覆盖。

关于 $p$ 进群的 Hecke 代数的参数

On parameters of Hecke algebras for $p$-adic groups

本文证明了深度零类型的仿射 Hecke 代数与幂幺类型的同构，从而可显式计算其参数，并在特定假设下证明了 Lusztig 猜想关于任意 Bernstein 块与幂幺块参数一致的版本。

The local Langlands conjecture for GL(n) over a p-adic field, n < p

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs). It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representa…

尖点表示中的新形式

Newforms in Cuspidal Representations

本文将 $p$-adic 群尖点表示的 newform 理论推广至 $\ell$-modular 表示，证明了导子与模 $\ell$ 同余的相容性，并在复数与 modular 设定下给出了深度零及 minimax 尖点表示的显式公式。

Signs of self-dual depth-zero supercuspidal representations

Let $G$ be a quasi-split tamely ramified connected reductive group defined over a $p$-adic field $F$. We show that if $-1$ is in the $F$-points of the absolute Weyl group of $G$, then self-dual supercuspidal representations of $G(F)$ exist. Now assume further that $G$ is unramified and that the center of $G$ is connected. Let $π$ be a generic self-dual depth-zero regular supercuspidal representation of $G(F)$. We show that the Frobenius-Schur indicator of $π$ is given by the sign by which a certain distinguished element of the center of $G(F)$ of order two acts on $π$.

极大非分歧域扩张上的波前集

The wavefront set over a maximal unramified field extension

本文针对分裂于非分歧扩张的p-进约化群的深度0容许光滑复表示，发展了极大非分歧域扩张上波前集的研究理论，并将其用于计算球面Arthur表示的几何波前集，揭示了其与朗兰兹对偶群上反常层的关联。

量子 wreath 产品与 $p$-adic 一般线性群

Quantum wreath products and $p$-adic general linear group

本文提出量子 wreath 产品理论，用于研究 $p$-adic 一般线性群的 Hecke 代数及其模结构，并给出了相关 Schur 代数的结构理论和多个 $p$-adic 重要结果的代数证明。

On Branching Rules of Depth-Zero Representations

Using Bruhat-Tits theory, we analyse the restriction of depth-zero representations of a semisimple simply connected $p$-adic group $G$ to a maximal compact subgroup $K$. We prove the coincidence of branching rules within classes of Deligne-Lusztig supercuspidal representations. Furthermore, we show that under obvious compatibility conditions, the restriction to $K$ of a Deligne-Lusztig supercuspidal representation of $G$ intertwines with the restriction of a depth-zero principal series representation in infinitely many distinct components of arbitrarily large depth. Several qualitative and qua…

On the Langlands parameter of a simple supercuspidal representation: odd orthogonal groups

In this work, we explicitly compute a certain family of twisted gamma factors of a simple supercuspidal representation $π$ of a $p$-adic odd orthogonal group. These computations, together with analogous computations for general linear groups carried out in previous work with Liu, allow us to give a prediction for the Langlands parameter of $π$. If we assume the "depth-preserving conjecture", we prove that our prediction is correct if $p$ is sufficiently large.

仿射抛物型德利涅-卢斯蒂格归纳的标量积公式

The scalar product formula for parahoric Deligne--Lusztig induction

本文针对p进群仿射抛物子群正深度表示的仿射抛物型德利涅-卢斯蒂格归纳，解决了其是否满足标量积公式的基础猜想，证明所有豪可分解分裂泛型对及特定椭圆型子群与非挠素数情形下该公式成立。

Arc-descent for the perfect loop functor and $p$-adic Deligne--Lusztig spaces

We prove that the perfect loop functor $LX$ of a quasi-projective scheme $X$ over a local non-archimedean field $k$ satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group $G$, the map $LG \rightarrow L(G/B)$ is a $v$-surjection. This gives a mixed characteristic version (for $v$-topology) of an equal characteristic result (in étale topology) of Bouthier--Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne--Lusztig spaces $X_w(b)$ attached to unramified $p$-adic reductive …

单幂零 $\ell$-块与连通且单连通的 $p$-进群

Unipotent $\ell$-blocks for simply-connected $p$-adic groups

本文研究连通且单连通约化群的 $F$-点集的单幂零 $\ell$-块，通过引入基于 Harish-Chandra 理论的 $(d,1)$-级数和 Bruhat-Tits 建筑上的一致幂等系来构造 $\ell$-块，并描述了非分歧经典群深度零范畴的稳定 $\ell$-块分解。

A categorical approach to the stable center conjecture

The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.

真正的表示的波前集：Kazhdan-Patterson或Savin类型的GL覆盖的波前集

Wavefront sets for genuine representations of $\rm GL$-covers of Kazhdan--Patterson or Savin types

本文研究了Kazhdan-Patterson和Savin类型的GL覆盖的不可约真实表示的波前集，通过迭代的最高Bernstein-Zelevinsky导数来表达，并利用局部Langlands对应和覆盖Barbasch-Vogan对偶性重新解释结果。

Character Sheaves and Depth-zero representations

In this paper we provide a geometric framework for the study of characters of depth-zero representations of unramified groups over local fields with finite residue fields which is built directly on Lusztig's theory of character sheaves for groups over finite fields and uses ideas due to Schneider-Stuhler. Specifically, we introduce a class of coefficient systems on Bruhat-Tits buildings of perverse sheaves sheaves on affine algebraic groups over an algebraic closure of a finite field and to each supercuspidal depth-zero representation of an unramified $p$-adic group we associate a formal sum o…

经典群的 BD-覆盖中的分解 Levi 子群

Decomposed Levi subgroups in BD-covers of classical groups

本文研究经典群 BD-覆盖中的分解 Levi 子群，证明局部域上辛群和特殊正交群（除分裂 SO(4)）在 Brylinski–Deligne 中心扩张下所有 Levi 子群均分解，并讨论酉群等情形。

On the computability of some positive-depth supercuspidal characters near the identity

This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space $B$ (that depends on the group and a real number $r>0$) for the set of equivalence classes of the representations of minimal depth $r$…