On types for unramified p-adic unitary groups

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.

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.

Theta correspondence for almost unramified representations of unitary groups

In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number $-1$.

Local newforms for generic representations of unramified odd unitary groups and Fundamental Lemma

In this paper, we establish the theory of local newforms for irreducible tempered generic representations of unramified odd unitary groups over a non-archimedean local field. For the proof, we prove an analogue of the fundamental lemma for our compact groups.

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.

关于非分歧素数处具有极大抛物子群水平结构的酉 Shimura 簇的 Kudla-Rapoport 猜想

On the Kudla-Rapoport conjecture for unitary Shimura varieties with maximal parahoric level structure at unramified primes

本文通过 Deligne-Lusztig 簇的 Tate 猜想版本证明 Kudla-Rapoport 猜想，并在任意维数的若干新情形下无条件成立。

非分歧偶酉群的通用表示的局部新形式 I：偶导子情形

Local newforms for generic representations of unramified even unitary groups I: Even conductor case

本文针对非分歧偶酉群，在偶导子情形下构造紧开子群，并借助局部 Gan-Gross-Prasad 猜想、Rankin-Selberg 积分和 theta 对应建立不可约 tempered 通用表示的局部新形式理论。

Behavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups

We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non quasi-split inner forms.

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。

On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters

We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's o…

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.

Unitary representations of GL(n,K) distinguished by a Galois involution, for K a p-adic field

Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,ρ)$ is distinguished if and only if the cuspidal representation $ρ$ is $η^{k-1}$-distinguished, for $η$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished c…

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…

Unitary equivalences for reductive p-adic groups

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of automorphisms of the root datum). It is known that there is a large class of such examples (detailed in the paper). As a consequence, we establish relations between the unitary duals of different groups, in the spirit of endoscopy.

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.

维数 $d \leq 5$ 的紧幂零 p-adic 李群的酉对偶与矩阵系数

Unitary dual and matrix coefficients of compact nilpotent p-adic Lie groups with dimension $d \leq 5$

本文显式计算了维数 $d \leq 5$ 的紧幂零 p-adic 李群的酉对偶与矩阵系数，并作为应用给出了 Vladimirov 次 Laplace 算子的谱定理，证明其为全局亚椭圆算子的非平凡示例。

Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature $(1,n-1)$

In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…

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.

Unitary Dual of p-adic U(5)

We study the parabolically induced complex representations of the unitary group in 5 variables, $ U(5), $ defined over a p-adic field. Let $F$ be a p-adic field. Let $E : F$ be a field extension of degree two. Let $Gal(E : F ) = \{ 1, σ\}.$ We write $ σ(x) = \overline{x} \; \forall x \in E. $ Let $ E^* := E \setminus \{ 0 \} $ and let $ E^1 := \{x \in E \mid x \overline{x} = 1 \}$. $U(5) $ has three proper standard Levi subgroups, the minimal Levi subgroup $ M_0 \cong E^* \times E^* \times E^1 $ and the two maximal Levi subgroups $ M_1 \cong GL(2, E) \times E^1 $ and $ M_2 \cong E^* \times U(3…

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。

关于酉Friedberg--Jacquet周期的p-adic插值

On the p-adic interpolation of unitary Friedberg--Jacquet periods

本文建立了某些酉Shimura簇的高阶Coleman理论的函子性，并借此构造了插值酉Friedberg--Jacquet周期的p-adic解析函数。

On the socles of certain parabolically induced representations of $p$-adic classical groups

In this paper, we consider representations of $p$-adic classical groups parabolically induced from the products of shifted Speh representations and unitary representations of Arthur type of good parity. We describe how to compute the socles (the maximal semisimple subrepresentations) of these representations. As a consequence, we can determine whether these representations are reducible or not. In particular, our results produce many unitary representations, which are called complementary series.

将实单位群和p-adic辛群和正交群的Arthur包联系起来

Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

本文通过建立实单位群与p-adic辛群和正交群的Arthur包之间的显式对应关系，利用Zuckerman翻译 functor和Jacquet functor的关系，实现对实单位群Arthur包的计算，并关联Kazhdan-Lusztig多项式与微局部几何。

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.

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。

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 …

On the pro-$p$-Iwahori invariants of supersingular representations of unramified $U(2, 1)$

Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ over a non-archimedean local field $F$ of odd residue characteristic $p$. In this paper, for any supersingular representation of $G$ that contains the Steinberg weight, we prove its pro-$p$-Iwahori invariants, as a right module over the pro-$p$-Iwahori--Hecke algebra of $G$, is \emph{not} simple.

Unitary spherical spectrum for split classical groups

This paper gives a complete classification of the unitary irreducible spherical representations of split real and p-adic groups. The results were obtained around 2000, the changes to the new version are expository.

Local unitary periods and relative discrete series

Let $F$ be a $p$-adic field ($p\neq 2$), let $E$ be a quadratic Galois extension of $F$, and let $n \geq 2$. We construct representations in the discrete spectrum of the $p$-adic symmetric space $H \backslash G$, where $G = \mathbf{GL}_{2n}(E)$ and $H = \mathbf{U}_{E/F}(F)$ is a quasi-split unitary group over $F$.

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。

完美体单位ary Shimura 变种与 $p$-adic Eichler-Shimura 映射 I

Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

本文通过完美体构造和收敛Eichler-Shimura映射，研究单位ary群上的$p$-adic自动机形与特征种几何及$p$-adic $L$-函数的关系。

Description of unitary representations of the group of infinite $p$-adic integer matrices

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices over a residue ring modulo $p^k$. Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

On the group of infinite $p$-adic matrices with integer elements

Let $G$ be an infinite-dimensional real classical group containing the complete unitary group (or complete orthogonal group) as a subgroup. Then $G$ generates a category of double cosets (train) and any unitary representation of $G$ can be canonically extended to the train. We prove a technical lemma about the complete group $GL$ of infinite $p$-adic matrices with integer coefficients, this lemma implies that the phenomenon of automatic extension of unitary representations to trains is valid for infinite-dimensional $p$-adic groups.

特殊正交群和酉群的法尔格斯 - 肖尔茨对应与内窥镜分类

Fargues-Scholze correspondence and endoscopic classification for special orthogonal and unitary groups

研究在特定条件下特殊正交群和酉群的法尔格斯 - 肖尔茨对应，证明其与经典对应半简化一致，以此为基础构造对应、推导猜想并建立志村簇新挠消失结果。

$p$-进分裂 $\mathrm{SO}_{2n+1}$ 与 $\mathrm{Sp}_{2n}$ 的酉对偶：好奇偶情形（并略微超出）

Unitary dual of $p$-adic split $\mathrm{SO}_{2n+1}$ and $\mathrm{Sp}_{2n}$: The good parity case (and slightly beyond)

本文刻画 $p$-进分裂 $\mathrm{SO}_{2n+1}$ 与 $\mathrm{Sp}_{2n}$ 中好奇偶不可约表示的酉性，证明其等价于 Arthur 型，并给出酉性检验算法及离散自守谱可能局部分量的集合。

Generic representations in L-packets

We give the details of the construction of a map to restate a conjectural expression about adjoint group action on generic representations in L-packets. We give an application of the construction to give another proof of the classification of the Knapp-Stein R-group associated to a unitary unramified character of a torus. Finally we prove the conjecture for unramified L-packets.

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。

关于 p 进古典群的互补 Arthur 表示与酉对偶

On the complementary Arthur representations and unitary dual for p-adic classical groups

本文针对特征零的非阿基米德局部域上连通约化群的酉对偶结构猜想，给出辛群和分裂奇特殊正交群互补 Arthur 表示的刻画，得到\(\mathrm{GL}_N\)（\(N = 2,3\)）不可约自对偶尖点自守表示局部分量的限制。

Transfer of Plancherel Measures for Unitary Supercuspidal Representations between p-adic Inner Forms

Let $F$ be a $p$-adic field of characteristic 0, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $Π_{i=1}^{r} SL_{n_i} \subseteq M \subseteq Π_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under \textit{the local Jacquet-Langlands type correspondence} between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Mui{ć} and Savin (2000)…

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。

Locally analytic vectors of unitary principal series of GL_2(Qp)

The p-adic local Langlands correspondence for GL2(Qp) attaches to any 2-dimensional irreducible p-adic representation V of the absolute Galois groups of Qp an admissible unitary representation Pi(V) of GL2(Qp). The unitary principal series of GL2(Qp) are those Pi(V) corresponding to trianguline representations. In this article, for p>2, using the machinery of Colmez, we determine the space of locally analytic vectors for all non-exceptional unitary principal series of GL2(Qp) by proving a conjecture of Emerton.

秩2 p进酉群的模p表示的Gelfand-Kirillov维数

Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2

本文研究秩2 p进酉群的模p表示的Gelfand-Kirillov维数，在半单且足够一般的模L参数条件下，结合已有结果证明了相关容许光滑表示的Gelfand-Kirillov维数等于对应局部域在Q_p上的次数。

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 …

The Whittaker-Shintani functions for symplectic groups

In this note, we give a formula for the Whittaker-Shintani functions for the p-adic symplectic groups, which is a generalization of the Zonal spherical functions and Whittaker functions. We then use the formula to give an alternative proof of a conjecture given by T.Shintani on the unramified calculation of L-function for $Sp_{2n}\times GL_{1}$.

关于Arthur表示与单位元双的关系

On Arthur representations and the unitary dual

本文提出一个关于Arthur表示与单位元双结构的新猜想，并通过算法和验证支持其完备性。

The Kottwitz conjecture for unitary PEL-type Rapoport--Zink spaces

In this paper we study the cohomology of PEL-type Rapoport-Zink spaces associated to unramified unitary similitude groups over $\Q_p$ in an odd number of variables. We extend the results of Kaletha-Minguez-Shin-White to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapport-Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.

酉群上 Fourier-Jacobi 模型的 Whittaker-Shintani 函数

Whittaker-Shintani functions for Fourier-Jacobi models on unitary groups

本文陈述并证明了 p 进酉群和一般线性群上 Fourier-Jacobi 模型的 Whittaker-Shintani 函数公式，并以此证明了非分歧的 Ichino-Ikeda 猜想。

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…