U. K. Anandavardhanan, C. S. Rajan
We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square integrable representation of GL(n,E). As a consequence we characterise reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GL(n,E) in terms of GL(n,F)-distinguishedness.
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Local Transfer and Reducibility of Induced Representations of $p$-adic Groups of Classical Type
本文研究经典型 p-进群由一般超尖点表示诱导的表示的可约性点，利用局部函子性转移在酉轴内外给出统一结果，并借助已建立的全局一般转移证明局部转移的存在性。
We provide a family of representations of GL(2n) over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp(2n)- distinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, re- lies on the theory of automorphic forms. The results of this paper together with later works by the authors imply that the family of representations studied in this paper contains all irreducible, unitary representations of the general linear group that are distin- guished b…
In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $Sp_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$ for $F$ a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of $U_{2n}({\mathbb A}_k)$ with nonzero period in…
Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution
本文利用局部方法证明了酉群区分的超尖点表示等价于其 Galois 不变性，适用于复数与 $l$-模情形，并证明了相关同态空间维数至多为一。
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representa…
Distinguished Simple Supercuspidal Representations of $p$-adic $\text{GL}(n)$
本文研究了 $p$-adic $\text{GL}(n)$ 的区分简单超 cuspidal 表示，通过定义最大简单类型和扭曲伽马因子给出等价条件，并证明扭曲伽马因子在特定点的集合可确定表示是否被 $\text{GL}(n, \text{F})$ 区分。
Tempered Symplectic Distinction For Quaternionic $GL_n(D)$: A Conjecture of PRASAD
本文证明了 Prasad 关于四元数一般线性群上温和表示辛区分的修正猜想，给出了可区分温和表示的精确刻画。
By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In each case, we show that the integrals achieve local formal $L$-functions defined by Langlands parameters, when the test vector is associated to the newform. We give the relation between local periods involving essential Whittaker functions and special values of formal $L$-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also…
First explicit reciprocity law for unitary Friedberg--Jacquet periods
本文证明了当自动形式π由单位Friedberg-Jacquet周期区分时，相关动机的Bloch-Kato Selmer群消失，为单位群的互反律提供了首个显式结果。
If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(σ,d)$ be the unique pair associated to …
In this paper we gives the Langlands parameters of Langlands' packets of discrete series using the twisted endoscopy as explained by Arthur; this holds for orthogonal, symplectic, unitary and G-Spin groups and gives the most simple proof available. We have assume that the groups are quasi-split but this is just for simplicity. The proof explaines first what is the classification from the representation's theory point of view; this gives the Langlands' packets purely in terms of representation theory. And then using the theory of L-function of Shahidi and the doubling method of Rallis and Piate…
We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ being the base change of a stable cuspidal packet of the quasi-split unitary group associated to the quadratic extension $E/F$, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of $\mathrm{GL}_{n,X'}$ is nonvanishing if and only if the corresponding local system $E$ on $X'$ is conjugate self-dual with respect to an étale double cover $X'/X$ of curves,…
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation $π$ of $GL(n,E)$ is said to be distinguished with respect to $GL(n,F)$ if it admits a non-trivial linear form that is invariant under the action of $GL(n,F)$. It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the $F$-points of the mirabolic subgroup when $π$ is unitary and generic. In this paper, we prove that the essential vector of [JPSS81…
On the irreducibility of $p$-adic Banach principal series of $p$-adic reductive groups
研究p进约化群在p进巴拿赫空间上主系列的不可约性，通过多种方法，对酉诱导表示获最优准则，对GL_n(F)等获施耐德猜想变体，还将问题归结为光滑诱导表示和特定G。
The intersection of local Arthur packets for even orthogonal groups and applications
本文提出算法计算p-adic域上偶数正交群的局部Arthur包及其交集，并验证表示归属，进而证明多个相关猜想。
In this paper we study the question of determining when an irreducible admissible representation of ${\rm GL}_n(D)$ admits a symplectic model, that is when such a representation has a linear functional invariant under ${\rm Sp}_n(D)$, where $D$ is a quaternion division algebra over a non-Archimedian local field $k$ and ${\rm Sp}_{n}(D)$ is the unique non-split inner form of the symplectic group ${\rm Sp}_{2n}(k)$. We show that if a representation has a symplectic model it is necessarily unique. For ${\rm GL}_2(D)$ we completely classify those representations which have a symplectic model. Glob…
A trace Paley-Wiener theorem for $\mathrm{GL}_{n}(F)\backslash \mathrm{GL}_{n}(E)$, with $E/F$ a quadratic extension of non-Archimedean local fields
该研究将p进约化群的经典迹Paley-Wiener定理推广到非阿基米德局部域二次扩张下的相对情形，得到GL_n(F)\GL_n(E)上相对贝塞尔分布的标量Paley-Wiener定理，并结合局部朗兰兹对应与基变换刻画了相对特征的性质。
We introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of Langlands et.al. (in the quadratic case) and the relative trace formula of Jacquet and Lai. Certain matching statements relating this twisted relative trace formula to a relative trace formula are also proven (including the relevant undamental lemma in the "biquadratic case"). Using recent work of Jacquet, Lapid and their collaborators and the Rankin-Selberg integral representation of the Asai $L$-function (obtained by Flicker using the theory of Jacquet, Piatetskii-Shapiro, and Shalika)…
Subconvexity for $L$-functions on ${\rm U}(n) \times {\rm U}(n+1)$ in the depth aspect
本文研究深度层面下酉 Gan–Gross–Prasad 对的中心 L 函数次凸性；利用 Ichino–Ikeda 周期公式将 L 值转化为自守周期，并通过 Iwaniec–Sarnak 放大方法估计周期，证明了 GL_{n+1}×GL_n 基变换 L 函数的幂次次凸界。
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