Vincent Secherre
Let F be a non-Archimedean local field of characteristic 0, and let D be a finite dimensional central division algebra over F. We prove that any unitary irreducible representation of a Levi subgroup of GL(m,D), with m a positive integer, induces irreducibly to GL(m,D). This ends the classification of the unitary dual of GL(m,D) initiated by Tadic.
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Let $F$ be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadić's classification of the unitary dual of $\mathrm{GL}_{2n}(F)$, we classify irreducible unitary representations of $\mathrm{GL}_{2n}(F)$ that have nonzero linear periods, in terms of Speh representations that have nonzero periods. We also give a necessary and sufficient condition for the existence of a nonzero linear period for a Speh representation.
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 Malle-Navarro conjecture relates central block theoretic invariants in two inequalities. In this paper, we prove the conjecture for the 2-blocks and the unipotent 3-blocks of the general linear and unitary groups in non-defining characteristic. Moreover, we show that the conjecture holds for the unipotent 3-blocks of quotients of central subgroups of the special linear and unitary groups.
On some conjectures of the unitary dual of $U(p,q)$
本文通过引入基本情形概念研究 $U(p,q)$ 的酉对偶，并据此证明了 Salamanca-Riba–Vogan 1998年猜想及 Vogan 2023年的 FPP 猜想。
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups II: Comparison of the relative trace formulae
本文通过比较相对迹公式，证明了单位ary群上傅里叶-雅可比周期的广义Gan-Gross-Prasad猜想及其改进。
A note on the Howe Duality conjecture for symplectic-orthogonal and unitary pairs
本文通过加强经典群退化主级数 $I(s,χ)$ 在非负实参数情形的已有结果，给出了辛-正交对与酉对 Howe 对偶猜想的新证明。
In a recent preprint, Sakellaridis and Venkatesh considered the spectral decomposition of the space $L^2(X)$, where $X = H\G$ is a spherical variety and $G$ is a real or $p$-adic group, and stated a conjecture describing this decomposition in terms of a dual group $\check{G}_X$ associated to $X$. The main purpose of this paper is to verify the above conjecture in many cases when $X$, has low rank. In particular, we demonstrate this conjecture for many cases when $X$ has rank 1, and also some cases when $X$ has rank 2 or 3.
In this paper, we prove a conjecture of Wei Zhang on comparison of certain local spherical characters from which we draw some consequences for the Ichino-Ikeda conjecture for unitary groups.
The Burger-Sarnak method shows that the restriction of an automorphic representation of a reductive group to a reductive subgroup has automorphic support. Clozel has conjectured a qualitative refinement of this result, which were first verified and quantified in the $GL_n$ case by Venkatesh. We give a proof of this conjecture in the classical group cases.
The distribution of the unipotent modules (in non-defining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.
Lapid and Mao formulated a conjecture on an explicit formula of Whittaker Fourier coefficients of automorphic forms on quasi-split classical groups and metaplectic groups as an analogue of Ichino-Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this paper, we study even unitary group case. Indeed, we prove this local identity over $p$-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prov…
We state and prove an extension of the global Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture to the case of some Eisenstein series on unitary groups $U_n\times U_{n+1}$. Our theorems are based on a comparison of the Jacquet-Rallis trace formulas. A new point is the expression of some interesting spectral contributions in these formulas in terms of integrals of relative characters. As an application of our mains theorems, we prove the global Gan-Gross-Prasad and the Ichino-Ikeda conjecture for Bessel periods of unitary groups.
Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincide with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broué's abelian defect groups conjecture for …
We establish an explicit Plancherel decomposition for $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$ where $E/F$ is a quadratic extension of local fields of characteristic zero by making use of a local functional equation for Asai $γ$-factors. We also give two applications of this Plancherel formula: first to the global Ichino-Ikeda conjecture for unitary groups by completing a comparison between local relative characters that was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikeda conjecture on formal degrees in the case of unitary groups.
Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Z…
The main goal of this paper is to study the unitary Friedberg-Jacquet period through their connection with the unitary Shalika period. Locally we study the multiplicity of unitary Friedberg-Jacquet periods for discrete series. Globally we prove one direction of a conjecture of Xiao-Zhang, stating that the non-vanishing of a global unitary Friedberg-Jacquet period implies the non-vanishing of the central value of the (twisted) standard L-function.
In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(π)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields and thus are only new when $F=\mathbb{R}$.
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups I: Coarse expansions of the relative trace formulae
本文首次证明了单位群上Fourier-Jacobi周期的全局Gan-Gross-Prasad猜想，并建立了相对迹公式的大规模展开。
We study the Mathieu Conjecture for $SU(2)$ using the matrix elements of its unitary irreducible representations. We state a conjecture for the particular case $SU(2)$ implying the Mathieu Conjecture for $SU(2)$.
Let F be a finite field or a local field of any characteristic. If A is a finite dimensional associative nilpotent algebra over F, the set 1+A of all formal expressions of the form 1+x, where x ranges over the elements of A, is a locally compact group with the topology induced by the standard one on F and the multiplication given by (1+x)(1+y)=1+(x+y+xy). We prove a result conjectured by Eugene Gutkin in 1973: every unitary irreducible representation of 1+A can be obtained by unitary induction from a 1-dimensional unitary character of a subgroup of the form 1+B, where B is an F-subalgebra of A…
Let F be a number field, A_F its ring of adeles, and let π_n and π_{n+1} be irreducible, cuspidal, automorphic representations of SO_n(A_F) and SO_{n+1}(A_F), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to π_n and π_{n+1} is equivalent to the non-vanishing of L(1/2, π_n x π_{n+1}). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period int…
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems
本文证明了关于单位ary群上的Fourier-Jacobi周期的Gan-Gross-Prasad猜想，并通过谱展开和零阶情况的减少完成了主要定理的证明。
Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.
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。
In this paper we consider the unitary symmetric spaces of the form X=U(p,q)/U(1)U(p,q-1) and their discrete series representations. Inspired by the work of A.Venkatesh and Y.Sekellarides on L-groups of p-adic spherical spaces we formulate and prove natural relative branching laws for the restriction to smaller subgroups of the same type and corresponding unitary spaces.We think of this as steps to formulation and proving Gan Gross Prasad conjectures for unitary spaces. Using period integral and some results of T.Kobayashi we prove an analogue of thesis conjectures.
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_n\times U_{n+1}$ in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called $*$-generic, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods. The other, built upon Zeta integrals, is expressed in terms of fu…
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。
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.
We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some interesting integrals over SU(n). Our investigations were motivated by geometric complexity theory.
We show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favourable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.
The FPP Conjecture for Real Reductive Groups
该研究通过运用旗簇上$\u2702$-模的整体生成性质与霍奇滤过，证明了实约化群的FPP猜想，给出了实约化群酉对偶的强上界。
We construct characteristic-free bases and BGG resolutions of unitary simple modules of quiver Hecke algebras and Cherednik algebras. We hence solve and vastly generalise Berkesch-Griffeth-Sam's conjecture, calculate the Castelnuovo-Mumford regularity of ideals of of certain highly symmetric linear subspace arrangements, and compute Dirac homology of Cherednik algebras.
Let $G$ be $Sp(2n, \mathbb{R})$ or $SO^*(2n)$. We compute the Dirac index of a large class of unitary representations considered by Vogan in Section 8 of [Vog84], which include all weakly fair $A_{\mathfrak{q}}(λ)$ modules and (weakly) unipotent representations of $G$ as two extreme cases. We conjecture that these representations exhaust all unitary representations of $G$ with nonzero Dirac cohomology. In general, for certain irreducible unitary module of an equal rank group, we clarify the link between the possible cancellations in its Dirac index, and the parities of its spin-lowest $K$-type…
Spherical Characters in Families: the unitary Gan-Gross-Prasad case
本文研究了族中球面特征的变分，提出了其有理性与亚纯性质的猜想，并在酉 Gan-Gross-Prasad 情形下证明了这些猜想。
Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan-Gross-Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.
The unitary dual of $U(n,2)$
本文完整分类了n≥3时U(n,2)的酉对偶，并识别出其中的弱公平A_q(λ)-模与特殊幂零表示，从而验证了Trapa-Vogan猜想。
In this paper we first construct natural filtrations on the full theta lifts for any real reductive dual pairs. We will use these filtrations to calculate the associated cycles and therefore the associated varieties of Harish-Chandra modules of the indefinite orthogonal groups which are theta lifts of unitary lowest weight modules of the metaplectic double covers of the real symplectic groups. We will show that some of these representations are special unipotent and satisfy a K-type formula in a conjecture of Vogan.
We define variants of PEL type of the Shimura varieties that appear in the context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.
Irreducibly represented Lie groups and Nebbia's CCR conjecture on trees
本文研究局部紧群代数结构与酉表示的联系，刻画不可约表示李群，并为小迷向群的全不连通群建立公理框架，推进树上 Nebbia CCR 猜想及相关酉对偶拓扑描述。
We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence.
We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of $\mathrm{SL}(2,\mathbb{R})$.
We lift any (infinitesimal) unitary irreducible representation of $GL_n(\mathbb{R})$ to a family of representations that strongly contracts to a certain type of (infinitesimal) unitary irreducible representations of $\mathbb{R}^n\rtimes {M}_n$, with $M_n$ being the mirabolic subgroup of $GL_n(\mathbb{R})$. For the case of $n=2$ we obtain the full unitary dual of $\mathbb{R}^2\rtimes {M}_2$ as a strong contraction. We demonstrate the role of the Kirillov conjecture and Kirillov model for these contractions.
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。
This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori-Hecke algebras. This is commonly referred to as Kottwitz's Conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even di…
Compatibility of the Fargues--Scholze correspondence for unitary groups
本文证明了酉群的 Fargues--Scholze 对应与 Mok 的局部 Langlands 对应相容，并结合谱作用验证了范畴形式猜想，进而推出 Fargues 特征层猜想和最强 Kottwitz 猜想。
In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.
Local newforms and spherical characters for unitary groups
本文证明了类似Jacquet-Rallis基本引理的平滑转移，计算了Ichino-Ikeda猜想中的局部球面特征标，并推出酉群局部新形式的存在性。
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic 0 and let $G=U(n)$, $H=U(m)$ be unitary groups of hermitian spaces $V$ and $W$. Assume that $V$ contains $W$ and that the orthogonal complement of $W$ is a quasisplit hermitian space (i.e. whose unitary group is quasisplit over $F$). Let $π$ and $σ$ be smooth irreducible representations of $G(F)$ and $H(F)$ respectively. Then Gan, Gross and Prasad have defined a multiplicity $m(π,σ)$ which for $m=n-1$ is just the dimension of $Hom_{H(F)}(π,σ)$. For $π$ and $σ$ tempered, we state and prove an integral formula f…
We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.
We study Whittaker--Fourier coefficients of automorphic forms on a quasi-split unitary group. We reduce the analogue of the Ichino--Ikeda conjectures to a conjectural local statement using the descent method of Ginzburg--Rallis--Soudry.
This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $SL_n(εq)$ in the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the $\ell$-weights of $SL_n(εq)$ can be obtained from the $\ell$-weights of $GL_n(εq)$ when $\ell\nmid\mathrm{gcd}(n,q-ε)$, and verify the Alperin weight conjecture for $SL_n(εq)$ under the condition $\ell\nmid\mathrm{gcd}(n,q-ε)$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $…
We use $p$-adic families of automorphic forms for an unitary group in three variables, containing some non-tempered forms constructed by Rogawski, to prove some cases of the Bloch-Kato conjectures.
A proof of the Kudla-Rapoport conjecture for Krämer models
本文证明了分歧处酉 Rapoport-Zink 空间 Krämer 模型的 Kudla-Rapoport 猜想，确立了特殊闭链算术相交数与修正导出局部密度的精确等式，并据此放宽了酉 Shimura 簇算术 Siegel-Weil 公式的局部假设。
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。
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.
We prove a conjecture of Colmez concerning the reduction modulo $p$ of invariant lattices in irreducible admissible unitary $p$-adic Banach space representations of $GL_2(Q_p)$ with $p\ge 5$. This enables us to restate nicely the $p$-adic local Langlands correspondence for $GL_2(Q_p)$ and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.
In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental l…
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。
Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields
本文探讨了自守L函数临界值与单位群上算术归一化自守形式周期的乘法关系，构造了CM自守动机并验证了Deligne猜想，利用IINH公式和上同调自守形式的杯积分析，证明了自守周期可分解为协调上同调形式的乘积。
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。
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