典型群的若干单模与正交几何和埃尔米特几何的$p$-秩

Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials

Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset \mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra $J\subset V$. We consider representations of $H$ that are gotten by the generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extens…

点到线聚合物与正交Whittaker函数

Point-to-line polymers and orthogonal Whittaker functions

本文通过正交Whittaker函数与罗宾逊-舍嫩斯坦-克努特对应关系，研究了点到线、点到半线及半空间限制下的定向聚合物模型，并推导了相关渗流问题的新公式。

Differential Recursion Relations for Laguerre Functions on Hermitian Matrices

In our previous papers \cite{doz1,doz2} we studied Laguerre functions and polynomials on symmetric cones $Ω=H/L$. The Laguerre functions $\ell^ν_{\mathbf{n}}$, $\mathbf{n}\in\mathbfΛ$, form an orthogonal basis in $L^{2}(Ω,dμ_ν)^{L}$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $(π_ν, \mathcal{H}_ν)$ of the automorphism group $G$ corresponding to a tube domain $T(Ω)$. In this article we consider the case where $Ω$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie al…

Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups

Let $G$ be an irreducible Hermitian Lie group and $D=G/K$ its bounded symmetric domain in $\mathbb C^d$ of rank $r$. Each $γ$ of the Harish-Chandra strongly orthogonal roots $\{γ_1, \cdots, γ_r\}$ defines a Heisenberg parabolic subgroup $P=MAN$ of $G$. We study the principal series representations $\Ind_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from $P$. We find the complementary series, reduction points, and unitary subrepresentations in this family of representations.