Ramanujan duals and automorphic spectrum

Density of Self-Dual Automorphic Representations of GL_n(A_Q)

We study the number $N_{\mathrm{sd}}^K(λ)$ of self-dual cuspidal automorphic representations of $GL_N(\mathbb{A_Q})$ which are $K$-spherical with respect to a fixed compact subgroup $K$ and whose Laplacian eigenvalue is $\leq λ$. We prove Weak Weyl's Law for $N_{\mathrm{sd}}^K(λ)$ in the form that there are positive constants $c_1, c_2$ (depending on $K$) and $d$ such that $c_1λ^{d/2}\leq N_{\mathrm{sd}}^K(λ)\leq c_2λ^{d/2}$ for all sufficiently large $λ$. When $N=2n$ is even and $K$ is a maximal compact subgroup at all places, we prove Weyl's Law for the number of self-dual representations, i…

$Sp_{4n}(\mathbb A)$ 的 CAP 表示与正交群 $O_{2n}(\mathbb A)$ 非平凡自守特征相关的一个显式构造

An Explicit Construction of CAP Representations of $Sp_{4n}(\mathbb A)$ associated to Non-trivial Automorphic Characters of Orthogonal Groups $O_{2n}(\mathbb A)$

本文利用 theta 对应和 Howe 对偶性，通过构造正交群的非平凡自守特征并提升，显式构造了 $Sp_{4n}$ 的一族 CAP 表示，并刻画了其局部分量。

Local distinction, quadratic base change and automorphic induction for $\mathrm{GL_n}$

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual/conjugate-dual representations. When applied to semi-simple representations of the Weil-Deligne group $W'_F$ of a non Archimedean local field $F$, and further translated in terms of representations of $\mathrm{GL_n}(F)$ via the local Langlands correspondence when $F$ has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When $F$ has residual characteristic different…

$GL_N$ 上自对偶自守表示的根号数等分布

Root Number Equidistribution for Self-Dual Automorphic Representations on $GL_N$

本文研究了全实域上自守表示的根号数等分布，利用迹公式与内窥分类证明了辛及共轭自对偶表示在特定条件下的等分布，并推导了相关Galois表示的等分布结果。

Residual automorphic forms and spherical unitary representations of exceptional groups

Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by Moeglin and for the exceptional Chevalley group G_2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E_6, E_7, E_8, and F_4. In particular we prove Arthur's conjecture that the spherical constituent of an unramified principal series of a Chevalley group over any local field of characteristic zero is unitarizable if its Langlands parameter coincides with …

Eigenvarieties for classical groups and complex conjugations in Galois representations

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is "odd". We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are "many" points correspon…

仿射Jacobi-Trudi恒等式与$q,t$-Rogers-Ramanujan恒等式

Affine Jacobi-Trudi Identities and $q,t$-Rogers-Ramanujan Identities

猜想正交和辛Schur函数的对偶Jacobi-Trudi恒等式的仿射类比，用于推导Rogers-Ramanujan恒等式的t-模拟，包括经典、Andrews-Gordon及GOW恒等式，并证明任意高度矩形分区Schur函数的仿射对偶Jacobi-Trudi恒等式。

The Burger-Sarnak Method and Operations on the Unitary Duals of Classical 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.

On quadratic distinction of automorphic sheaves

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,…

自守剩余谱的一个简单构造

A simple construction of the automorphic residual spectrum

该研究针对数域上分裂半单线性代数群，通过正则化球Borel Eisenstein级数，遵循Kazhdan和Okounkov的哲学，结合等变积分公式证明其满足朗兰兹平方可积准则，进而给出Arthur幺正猜想的简单统一证明。

On automorphic sheaves on Bun_G

Let X be a smooth projective connected curve over an algebraically closed field k of positive characteristic. Let G be a reductive group over k, γbe a dominant coweight for G, and E be an \ell-adic \check{G}-local system on X, where \check{G} denotes the Langlands dual group. Let \Bun_G be the moduli stack of G-bundles on X. Under some conditions on the triple (G,γ,E) we propose a conjectural construction of a distinguished E-Hecke automorphic sheaf on \Bun_G. We are motivated by a construction of automorphic forms suggested by Ginzburg, Rallis and Soudry in [6,7]. We also generalize Laumon's …

对偶对$\boldsymbol{F}_{4}\times \boldsymbol{PGL}_{2}$的整体例外θ对应

The global exceptional theta correspondence for the dual pair $\mathbf{F}_{4}\times \mathbf{PGL}_{2}$

本文研究F₄×PGL₂的例外θ对应，建立了F₄上一类一级自守表示族的存在性，证明PGL₂上权k≥12的尖点本征形式的θ提升非零，还定义了PGL₂上的例外θ级数并证明其张成一级尖点形式空间。

The functorial source problem via dimension data

For an automorphic representation $π$ of Ramanujan type, there is a conjectural subgroup $\mathcal{H}_π$ of the Langlands L-group $^{L}G$ associated to $π$, called the {\it functional source} of $π$. The functorial source problem proposed by Langlands and refined by Arthur intends to determine $\mathcal{H}_π$ via analytic and arithmetic data of $π$. In this paper, we consider the functorial source problem of automorphic representations of a split group, a unitary group, or an orthogonal group which do not come from endoscopy and have minimal possible ramification. In this setting, $\mathcal{H}…

Double Descent in Classical Groups

Let ${\bf A}$ be the ring of adeles of a number field $F$. Given a self-dual irreducible, automorphic, cuspidal representation $τ$ of $\GL_n(\BA)$, with trivial central characters, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group $G$. We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan \cite{CFGK17}, which represent the standard $L$-functions for $G\times \GL_n$. Our results are valid also for …

关于函数域上的广义 Ramanujan 与 Arthur 猜想

On the generalized Ramanujan and Arthur conjectures over function fields

本文利用 V. Lafforgue 的 Galois 参数化和 Deligne 的 Frobenius 权重理论，证明了函数域上单群尖点自守表示在非分歧拟分裂位处的缓和性，结果与 Shahidi 和 Arthur 的广义猜想一致。

Spherical varieties, functoriality, and quantization

We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical $G$-variety $X$, one associates a dual group ${^LG_X}$ and an $L$-value (encoded in a representation of ${^LG_X}$), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ${^LG_X}\to {^LG_Y}$ of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approac…

Unitary Dual of GL_n at archimedean places and global Jacquet-Langlands correspondence

In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over $\bbR$, $\bbC$ or $\bbH$ of independent interest.

Some bounds on unitary duals of classical groups - non-archimeden case

In the first part of the paper we give some bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical p-adic groups. Roughly, it can show up only if the central character of the inducing irreducible cuspidal representation is dominated in an appropriate way by the square root of the modular character of minimal parabolic subgroup. For representations supported by fixed parabolic subgroup, a more precise bound is given. There are also bounds for specific Bernstein components. A number of these upper bounds are best possible. Th…

Archimedean zeta integrals on U(n,1)

For a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions. In this paper we explicitly calculate archimedean zeta integrals of this type for U(n,1), assuming that the corresponding archimedean component of the automorphic representation is a holomorphic discrete series.

Quantization of Fields and Automorphic Representations

In this paper we use the quantization of fields based on Geometric Langlands Correspondence \cite{diep1} to realize the automorphic representations of some concrete series of groups: for the affine Heisenberg (loop) groups it is reduced to the construction of the affine Kac-Moody representation by the Weyl relations in Fock spaces. For the solvable and nilpotent groups following the construction we show that it is the result of applying the constructions of irreducible unitary representation via the geometric quantization and the construction of positive energy representations and finally, for…

Eisenstein级数与自守表示

Eisenstein series and automorphic representations

本文为实单李群上的Eisenstein级数和自守形式提供导论，强调表示论，涵盖Fourier展开、Langlands常数项公式、Casselman-Shalika公式及Hecke算子与Langlands纲领的联系，并探讨其在弦理论中的物理意义。

Geometric cycles in compact Riemannian locally symmetric spaces of type IV and automorphic representations of complex simple Lie groups

Let G be a connected complex simple Lie group with maximal compact subgroup U. Let g be the Lie algebra of G, and X = G/U be the associated Riemannian globally symmetric space of type IV. We have constructed three types of arithmetic uniform lattices in G, say of type 1, type 2, and type 3 respectively. If g is not equal to b_n, n>0, then for each 0<i<4, there is an arithmetic uniform torsion-free lattice Γin G which is commensurable with a lattice of type i such that the corresponding locally symmetric space Γ\ X has some non-vanishing (in the cohomology level) geometric cycles, and the Poinc…

Level one algebraic cusp forms of classical groups of small ranks

We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO_7, SO_8, SO_9 (and G_2) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL_n with trivial infinitesimal c…

幂零锥上凝聚层的Langlands对偶实现

A Langlands dual realization of coherent sheaves on the nilpotent cone

本文证明了幂零锥形式邻域上的等变拟凝聚层与环路群上的Steinberg-Whittaker D-模之间的幺半群等价，并推广至仿射Hecke范畴的谱与自守实现，证实了Bezrukavnikov的猜想。

Umbral Moonshine

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain …