Sharp Fourier type and cotype with respect to compact semisimple Lie groups

The local character expansion near a tame, semisimple element

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected nor that the underlying field has characteristic zero.

关于半单李群上Hodge Laplacian的Schwartz估计

On the Schwartz estimate for Hodge Laplacians on semisimple Lie groups

本文将对称空间上的Schwartz估计推广到半单李群，证明了Hodge Laplacian与Dirac算子的相应估计，并给出李代数上同调的Kuga引理版本，以利用Fourier分析研究向量丛截面的热问题。

Vector-valued Hausdorff-Young inequality on compact groups

The main purpose of this paper is to study the validity of the Hausdorff-Young inequality for vector-valued functions defined on a non-commutative compact group. The natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some questions are posed.

球函数的傅里叶正性 I：分裂环面和球主级数

Fourier positivity for spherical functions I: split tori and spherical principal series

本文证明了局部域上半单线性代数群G的球函数在分裂环面A上的傅里叶正性，针对G的酉主级数参数，并给出SL_n(F)情形下对角分裂环面上傅里叶变换的显式递推公式及指数下界。

Fourier Coefficients of Automorphic Forms and Integrable Discrete Series

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, …

Bessel operators on Jordan pairs and small representations of semisimple Lie groups

We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier …

Spherical functions for small $K$-types

For a connected semisimple real Lie group $G$ of non-compact type, Wallach introduced a class of $K$-types called small. We classify all small $K$-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small $K$-type $(π,V)$ can be expressed as a product of hyperbolic cosines and a Heckman-Opdam hypergeometric function. As an application, the inversion formula for the spherical transform on $G\times_K V$ is obtained from Opdam's theory on hypergeometric Fourier transforms.

有限域上对称群的酉与模傅里叶变换

Unitary and Modular Fourier Transforms of the Symmetric Group over Finite Fields

本文在有限域上构造循环群与对称群的傅里叶变换，通过因式分解、Peirce分解及酉实现，推广DFT并在SageMath中实现。

Plancherel Theorem and the Left Ideals of the Group Algebra for the Jacobi Group

Let G be the three dimensional connected real semisimple Lie group and let KAN be the Iwasawa decomposition of G.Let J be the Jacobi group, which is the semidirect product of the two groups Heisenberg group with G. The Jacobi group plays an important role in Quantum Mechanics. The purpose of this paper is to define the Fourier transform in order to obtain the Plancherel theorem for the group J. To this end a classification of all left ideals of the group algebra for the Jacobi group

Paley-Wiener Theorems with respect to the spectral parameter

One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic analysis is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations. We give a short overview of the Fourier transform on such spaces and then ask if one can describe the image of the space of smooth compactly supported functions in terms of the spectral parameter, i.e., the parameterization of th…

Branching of some holomorphic representations of SO(2,N)

In this paper we consider the analytic continuation of the weighted Bergman spaces on the Lie ball $$\mathscr{D}=SO(2,n)/S(O(2) \times O(n))$$ and the corresponding holomorphic unitary (projective) representations of SO(2,n) on these spaces. These representations are known to be irreducible. Our aim is to decompose them under the subgroup SO(1,n) which acts as the isometry group of a totally real submanifold $\mathscr{X}$ of $\mathscr{D}$. We give a proof of a general decomposition theorem for certain unitary representations of semisimple Lie groups. In the particular case we are concerned wit…

Local noncommutative De Leeuw Theorems beyond reductive Lie groups

Let $Γ$ be a discrete subgroup of a unimodular locally compact group $G$. In Math. Ann. 388, 4251-4305 (2024), it was shown that the $L_p$ norm of a Fourier multiplier $m$ on $Γ$ can be bounded locally by its $L_p$-norm on $G$, modulo a constant $c(A)$ which depends on the support $A$ of $m$. In the context where $G$ is a connected Lie group with Lie algebra $\mathfrak{g}$, we develop tools to find explicit bounds on $c(A)$. We show that the problem reduces to: 1) The adjoint representation of the semisimple quotient $\mathfrak{s} = \mathfrak{g}/\mathfrak{r}$ of $\mathfrak{g}$ by the radical $…

紧 $\mathrm{Spin}_8$ 的例外 Siegel–Weil 定理

Exceptional Siegel-Weil theorems for compact $\mathrm{Spin}_8$

本文研究四元数 $E_8$ 中 $G_E\times S_E$ 对偶对，利用极小表示计算平凡表示的提升，证明其对应退化 Eisenstein 级数，并用于 $G_2$ 模形式算术性。