The inversion formula and holomorphic extension of the minimal representation of the conformal group

A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

This article is devoted to branching problems for holomorphic discrete series representations of a conformal group $G$ of a tube domain $T_Omega$ over a symmetric cone $Ω$. More precisely, we analyse restrictions of such representations to the conformal group $G'$ of a tube domain $T_{Ω'}$ holomorphically embedded in $T_Ω$. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. To do so, a stratification space for a symmetric cone is introduced. This structure put the light on a new functional model, called the \emph{s…

Inversion of Rankin-Cohen operators via Holographic Transform

The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models, whereas holographic operators increase it. Various expansions in classical analysis can be interpreted as particular occurrences of these transforms. From this perspective, we investigate two remarkable families of differential operators: the Rankin-Cohen operators and the holomorphic Juhl conformally covariant operators. Then we establish for the corresponding symmetr…

多面Painlevé I：不规则共形块、拓扑递归与全纯反常方法

Many-faced Painlevé I: irregular conformal blocks, topological recursion, and holomorphic anomaly approaches

该研究围绕Painlevé I tau函数的多种数学物理解释，发展统一数学框架，提出等价性猜想，完成Whittaker态构造与拓扑递归配分函数性质证明，推动相关理论的计算与解的存在性验证。

Resonance relations, holomorphic trace functions and hypergeometric solutions to qKZB and Macdonald-Ruijsenaars equations

The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equation…

Infinite dimensional geometry of $M_1=Diff_+(S^1)/PSL(2,R)$ and $q_R$--conformal symmetries

A geometric interpretation of approximate ($HS$-projective or $TC$-projective) representations of the Witt algebra $w^C$ by $q_R$-conformal symmetries in the Verma modules $V_h$ over the Lie algebra $sl(2,C)$ is established and some their characteristics are calculated. It is shown that the generators of representations coincide with the Nomizu operators of holomorphic $w^C$-invariant hermitean connections in the deformed holomorphic tangent bundles $T_h(M_1)$ over the infinite-dimensional Kähler manifold $M_1=\Diff_+(S^1)/PSL(2,R)$, whereas the deviations $A_{XY}$ of the approximate represent…

Positive Energy Representations, Holomorphic Discrete Series and Finite-Dimensional Irreps

Let G be a semi-simple non-compact Lie group with unitary lowest/highest weight representations. We consider explicitly the relation between three types of representations of G: positive energy (unitary lowest weight)representations, (holomorphic) discrete series representations and non-unitary finite-dimensional irreps. We consider mainly the conformal groups SO(n,2) treating in full detail the cases n=1,3,4.