Representations of unramified U(2,2) over a p-adic field I: representations of non-integral level

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p--adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p--adic type.

Weakly unramified representations, finite morphisms, and Knapp-Stein $R$-groups

We transfer Knapp-Stein $R$-groups for unitary weakly unramified characters between a $p$-adic quasi-split group and its non-quasi-split inner forms, and provide the structure of those $R$-groups for a general connected reductive group over a $p$-adic field. This work supports previous studies on the behavior of $R$-groups between inner forms, and extends Keys' classification for unitary unramified cases of simply-connected, almost simple, semi-simple groups.

Maximal Unramified Tori in Symplectic Groups

For a reductive group over a $p$-adic field, DeBacker gives a paramaterization of the conjugacy classes of maximal unramified tori using Bruhat-Tits theory. On the other hand, for unramified classical groups, Waldspurger gives a parameterization in terms of triples of partitions by constructing a regular semisimple element whose structure is governed by the parts of the partitions. In this paper, we compare these two parameterizations in the case of the symplectic group Sp$_{2n}$.

On the unramified spectrum of spherical varieties over p-adic fields

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ``Langlands dual'' group. We generalize this description to an arbitrary spherical variety X of G as follows: Irreducible unramified quotients of the space $C_c^\infty(X)$ are in natur…

On the signature character of representations of p-adic general linear groups

In this article we calculate the signature character of certain Hermitian representations of $GL_N(F)$ for a $p$-adic field $F$. We further give a conjectural description for the signature character of unramified representations in terms of Kostka numbers.

Weyl groups, the Dirac inequality, and isolated unitary unramified representations

I present several applications of the Dirac inequality to the determination of isolated unitary representations and associated "spectral gaps" in the case of unramified principal series. The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. The results could be viewed as a $p$-adic analogue of Salamanca-Riba's classification of irreducible unitary $(\mathfrak g,K)$-modules with strongly regular infinitesimal character.

On Representations of Reductive $p$--adic Groups over $\mathbb Q$--algebras

In this paper we study certain category of smooth modules for reductive $p$--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb Q$--algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations proving by reduction to the complex case that if the space of $K$--invariants is finite dimensional in an irreducible smooth unramified representation that the representation is admissible.

Conductors and newforms for non-supercuspidal representations of unramified U(2,1)

Let G be the unramified unitary group in three variables defined over a p-adic field with odd p. The conductors and newforms for representations of G are defined by using a certain family of open compact subgroups of G. In this paper, we determine the conductors of non-supercuspidal representations of G, and give explicit newforms in induced representations for non-supercuspidal generic representations.

p进局部域上酉群的简单超尖点表示的Rankin-Selberg伽马因子研究

On the Rankin-Selberg gamma factor of simple supercuspidal representations of the unitary group for $p$-adic local fields

本文研究p进域上拟分裂非分歧偶酉群的简单超尖点表示，计算其经驯分歧特征扭曲的秩1Rankin-Selberg伽马因子，给出非二进情形的内视提升推导，得到全新的二进情形结果，为推广Langlands参数研究奠定基础。

The Spherical Hecke algebra, partition functions, and motivic integration

This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition fu…

Endoscopic lifting of simple supercuspidal representations of unramified U(N) to GL(N)

Let U(N) be the quasi-split unitary group in N variables for a quadratic unramified extension of p-adic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.

On local newforms for unramified U(2,1)

Let G be the unramified unitary group in three variables defined over a p-adic field F of odd residual characteristic. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K_n} of G to define the local newforms for representations of G as the K_n-fixed vectors. We prove the existence of local newforms for generic representations and the multiplicity one property of the local newforms for admissible representations.

On L-factors attached to generic representations of unramified U(2,1)

Let G be the unramified unitary group in three variables defined over a p-adic field of odd residual characteristic. In this paper, we establish a theory of newforms for the Rankin-Selberg integral for G introduced by Gelbart and Piatetski-Shapiro. We describe L and epsilon-factors defined through zeta integrals in terms of newforms. We show that zeta integrals of newforms for generic representations attain L-factors. As a corollary, we get an explicit formula for epsilon-factors of generic representations.

On epsilon factors attached to supercuspidal representations of unramified U(2,1)

Let G be the unramified unitary group in three variables defined over a p-adic field F of odd resudual characteristic. Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G. In this paper, we formulate a conjecture on L- and epsilon-factors defined through zeta integrals in terms of local newforms for G, which is an analogue of the result by Casselman and Deligne for GL(2). We prove our conjecture for the generic supercuspidal representations of G.

非分歧酉群$\rm{U}(1,1)$的不可约超尖点表示的分支规则

Branching rules for irreducible supercuspidal representations of unramified $\mathrm{U}(1,1)$

本文确定了剩余特征$p≠2$的$p$-进域上非分歧拟分裂酉群$\rm{U}(1,1)(F)$的所有不可约超尖点表示限制到极大紧子群的分支规则，给出两个应用验证相关猜想，并结合前期结果完成对该群所有不可约光滑表示分支规则的描述。

On some zeta-integrals for unramified representations of GSp(4)

This article is a companion to several works of the author and others on the arithmetic of automorphic forms for GSp(4), and their associated L-functions and Galois representations. These works require, at various points, an input from smooth representation theory over p-adic local fields: the computation of values of the unique trilinear form on an unramified representation of GSp(4) x GL(2) x GL(2) invariant under the diagonal GL(2) x GL(2), for various different input data. In this note, we carry out these local computations, using a general formula for Shintani functions due to Gejima.

对于SL(2)×GL(2)的无分歧计算

The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$

本文通过使用Casselman-Shalika公式，提供了对于SL(2)×GL(2)的Shimura型Rankin-Selberg积分的无分歧计算的新证明。

Local newforms for generic representations of unramified ${\rm U}_{2n+1}$ and Rankin-Selberg integrals

Recently Atobe-Oi-Yasuda established the newform theory for irreducible tempered generic representations of unramified ${\rm U}_{2n+1}$ over non-archimedean local fields. In this paper we extend their result to every irreducible generic representations and compute the dimensions of the spaces of oldforms. We also compute the Rankin-Selberg integrals attached to newforms and oldforms under a natural assumption on the $γ$-factors defined by these integrals.

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 …

将包含Steinberg的超特殊子群的全部表示进行分类

Classify all representation which contains a Steinberg in its hyperspecial subgroup

本文研究了不可约光滑表示的分类，其固定空间包含Steinberg表示，证明此类表示必为Iwahori球形，且每个主系列贡献一个唯一子商包含Steinberg表示。

非分歧 U(1,1) 主级数表示的分支律

Branching rules for principal series representations of unramified U(1,1)

本文构造非分歧拟分裂酉群 U(1,1) 极大紧子群的一族不可约表示，并由此给出所有主级数表示限制到该紧子群时的无重数分支律分解，进而回应了文献中的若干公开猜想。

A note on Integral Satake isomorphisms

We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.

关于 $p$-adic Coxeter 轨道的一个分解

On a decomposition of $p$-adic Coxeter orbits

本文研究非阿基米德局部域上未约化群 ${\bf G}$ 的 $p$-adic Deligne-Lusztig 空间 $X_w(b)$ 的几何，证明了当 ${\bf G}$ 为经典群、$b$ 基本且 $w$ 为 Coxeter 元时，$X_w(b)$ 可分解为某个整 $p$-adic Deligne-Lusztig 空间的平移的不交并。

Geometric L-packets of Howe-unramified toral supercuspidal representations

We show that $L$-packets of toral supercuspidal representations arising from unramified maximal tori of $p$-adic groups are realized by Deligne--Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series represe…

A Cauchy--Harish-Chandra integral for a dual pair over a p-adic field, the definition and a conjecture

For a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank. The purpose of this article is to transfer this construction to the p-adic case. We provide the precise definition of the integral kernel operator and formulate a conjecture.

First steps towards $p$-adic Langlands functoriality

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 …

Analytic vectors in continuous p-adic representations

Given a compact p-adic Lie group G over a finite unramified extension L/Q_p let G_0 be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic G_0-representations that coincides with passage to analytic vectors in case L=Q_p. On the other hand, we study the functor "passage to analytic vectors" and its derived functors over general basefields. As an application we determine the higher analytic vectors in certain locally analytic induced representations.

关于$p$进域$K$上$\n\thrm{GL}_n(K)$模表示的猜想与结果

Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$

本文研究$p$进域上$\n\thrm{GL}_n(K)$的模表示，提出模$p$自守形式空间中光滑表示的结构猜想，证明了$n=2$非分歧情形下的部分结果与新有限长度结论。

Coxeter 型 $p$ 进 Deligne-Lusztig 概形的上同调

The cohomology of $p$-adic Deligne-Luszitg schemes of Coxeter type

本文确定了局部非阿基米德域上 Coxeter 型 $p$-Deligne-Lusztig 概形闭 Drinfeld 层的上同调，证明环面权空间集中于单一上同调度数且为成对非同构不可约表示，并探讨了 Moy-Prasad 商的极大性与 Kirillov 轨道方法的关系。

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…

p进Chevalley群的上同调

Cohomology of $p$-adic Chevalley groups

本文在Coxeter数标准假设下，利用分次李代数的新展示和正特征旗簇凝聚上同调，计算了p进Chevalley群及其Iwahori子群的上同调代数，并讨论了与Morava稳定化子群相关的GL_n(K)内形式。

Spherical functions on spherical varieties

Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicit…

A Satake isomorphism in characteristic p

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in generalisations of Serre's conjecture on the modularity of mod p …

G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range

Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf{F}$. Fix a representation $\overlineρ$ of the absolute Galois group of an unramified extension of $\mathbf{Q}_p$, valued in $G(\mathbf{F})$. We study the crystalline deformation ring for $\overlineρ$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine-Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result impro…

On the cohomology of integral $p$-adic unipotent radicals

Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.