Bernhard Krötz, Job J. Kuit, Henrik Schlichtkrull
We give an example of a semisimple symmetric space $G/H$ and an irreducible representation of $G$ which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.
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Let G/H be a unimodular real spherical space which is either absolutely spherical or wave-front. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real spherical space. Final version. To appear in Acta Math.
We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
Non-tempered Ext Branching Laws for the $p$-adic General Linear Group
本文研究 $p$ 进一般线性群中 Arthur 型表示的 Ext 分支律，并在离散序列表示及其 Aubert–Zelevinsky 对偶乘积的情形下建立局部非缓增 Gan–Gross–Prasad 猜想的 Ext 类比。
Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces
本文建立了离散子群增长率、不变微分算子联合谱的多面体界与矩阵系数衰减之间的精确联系，并利用Quint增长指示函数完全刻画了局部对称空间上$L^2$空间的温性，推广至所有Borel Anosov子群。
This paper has two aims. The first is to give a description of irreducible tempered representations of classical p-adic groups which follows naturally the classification of irreducible square integrable representations modulo cuspidal data obtained by C. Moeglin and the author. The second aim of the paper is to give description of an invariant (partially defined function) of irreducible square integrable representation of a classical p-adic group (defined by C. Moeglin using embeddings) in terms of subquotients of Jacquet modules. As an application, we describe behavior of partially defined fu…
The tempered spectrum of the similitude groups of non-degenerate symplectic, hermitian, or split orthogonal forms defined over $p$\snug-adic groups of characteristic zero is studied. The components of representations induced from discrete series of proper parabolic subgroups are classified in terms of $R$\snug-groups. Multiplicity one is proved. The tempered elliptic spectrum is identified, and the relation between elliptic characters appearing in a given induced representation is determined. Those irreducible tempered representations which are not elliptic and not fully induced from elliptic …
We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder \cite{Re}). The former is completely different than the Lusztig-Shoji algorithm \cite{Sh, L}, and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyz…
The Mackey bijection as a stratified equivalence
本文研究Mackey双射在实紧致群及其Cartan运动群 tempered 表示论中的分层等价性质，探讨其拓扑性质及在C*-代数分层上的行为。
Zariski dense non-tempered subgroups in higher rank of nearly optimal growth
本文利用含全测地超平面的闭双曲流形基本群，在 SO(n,2) 中构造出 Zariski 稠密、非缓增且增长近乎最优的离散 Anosov 子群。
In this article, we give a natural geometric interpretation of Kirillov's conjecture for tempered representations in the framework of orbit method. This interpretation can also be considered as a first generalization of Duflo's conjecture to non-discrete series representations.
Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be maximal compact. For a tempered representation $π$ of $G$, we realise the restriction $π|_K$ as the $K$-equivariant index of a Dirac operator on a homogeneous space of the form $G/H$, for a Cartan subgroup $H<G$. (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of $G$, so that we obtain an explicit version of Kirillov's orbit method for $π|_K$. In a companion paper, we use this realisation of $π|_K$ to give a geometric expression for the…
We study ${\rm Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm …
Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $π$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $π|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and fo…
We examine the reducibility of induced from discrete series representations of $SU_n$ over a $p$--adic field of characteristic zero. Some results are given for groups sharing derived group. We give a relationship between the $R$-groups for $U)n$ and $SU_n.$ Nonabelian $R$-groups are discussed. Elliptic representations are examined and in the generic case we give an explicit description f those $L$--packets containing non-discrete elliptic tempered representations. We exhibit $L$--packets containing both elliptic and non-elliptic components.
Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid
本文提出Satake群oid的三种不同视角，通过拓扑、李理论和几何方法，为Harish-Chandra原理提供新证明。
In this paper we prove a version of a trace Paley--Wiener theorem for tempered representations of a reductive $p$--adic group. This is applied to complete certain investigation of Shahidi on the proof that a Plancherel measure is invariant of a $L$--packet of discrete series.
Well-tempered symmetric spaces
本文研究实约化对称空间的Plancherel测度支集，引入良态对称空间类，用Cartan子群特征描述其支集与Fell拓扑，并应用于描述相关C*-代数及其K-理论。
Square integrability of regular representations on reductive homogeneous spaces
本文研究了可约齐性空间上$L^2(G/H)$作为$G$的单位亚表示的条件，并推导出不存在离散级数的结论。
Let $G$ be a reductive group and $θ$ an involution on $G$, both defined over a $p$-adic field. We provide a criterion for $G^θ$-integrability of matrix coefficients of representations of $G$ in terms of their exponents along $θ$-stable parabolic subgroups. The group case reduces to Casselman's square-integrability criterion. As a consequence we assert that certain families of symmetric spaces are strongly tempered in the sense of Sakellaridis and Venkatesh. For some other families our result implies that matrix coefficients of all irreducible, discrete series representations are $G^θ$-integrab…
Schwartz spaces on L-monoids: non-Archimedean
该研究在非阿基米德局部域上，结合相关假设完成了Braverman-Kazhdan-Ngô纲领，且该纲领对一般线性群可无条件成立。
On the geometric approach to the discrete series
本文通过几何方法建立了离散系列表示的参数化与特征公式的对应关系，并推导出Blattner猜想。
We combine the geometric realization of principal series representations of the author, with the Bott--Borel--Weil Theorem for direct limits of compact groups of Natarajan, Rodriguez-Carrington and the author, obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm--preserving representations on a Banach space or unitary representations on a Hilbert space. We specialize the results to diagonal embedding direct limit group…
On the rate of exponential decay of coefficients on homogeneous spaces
本文研究了齐性空间上系数指数衰减率，提出了一种基于表示论和群论的指数定义，并为$L^2(G/H)$的温和性提供了新的准则，扩展了现有结果。
For a connected reductive group $ G $ defined over a number field $ k $, we construct the Schwartz space $ \mathcal{S}(G(k)\backslash G(\mathbb{A})) $. This space is an adelic version of Casselman's Schwartz space $ \mathcal{S}(Γ\backslash G_\infty) $, where $ Γ$ is a discrete subgroup of $ G_\infty:=\prod_{v\in V_\infty}G(k_v) $. We study the space of tempered distributions $ \mathcal{S}(G(k)\backslash G(\mathbb A))' $ and investigate applications to automorphic forms on $ G(\mathbb A) $. In particular, we study the representation $ \left(r',\mathcal{S}(G(k)\backslash G(\mathbb{A}))'\right) $…
Limit Multiplicity For Unitary Groups and The Stable Trace Formula
该研究利用Arthur的迹公式稳定化及相关表示内窥分类，给出酉群$U(a,b)$部分非tempered表示的极限重数上界，并应用于酉群余紧算术子群的上同调增长研究。
Zariski density of discrete subgroups via critical exponents and unitary representations
该研究利用酉表示理论，证明临界指数足够接近对称空间体积增长熵的离散子群具有Zariski稠密性，并确定了三类双曲空间等距群下的临界指数最优值，推广了Borel稠密性定理。
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