Weyl calculus on the Fock space and Stratonovich-Weyl correspondence for Heisenberg motion groups

A survey of Heisenberg categorification via graphical calculus

In this expository paper we present an overview of various graphical categorifications of the Heisenberg algebra and its Fock space representation. We begin with a discussion of "weak" categorifications via modules for Hecke algebras and "geometrizations" in terms of the cohomology of the Hilbert scheme. We then turn our attention to more recent "strong" categorifications involving planar diagrammatics and derived categories of coherent sheaves on Hilbert schemes.

Calculus of Operators: Covariant Transform and Relative Convolutions

This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is systematically illustrated by a representative collection of examples. Keywords: Lie groups and algebras, convolution operator, relative convolution, representation theory, pseudo-differential operators, PDO, singular integral operator, SIO, the Heisenberg group, SL(2,R), induced representation, Fock-Segal-Bargmann (FSB) representation, Shrodinger representation, covar…

广义菱形群的复Weyl对应

Complex Weyl correspondence for a generalized diamond group

本文针对广义菱形群，在Fock空间上构造典型表示，研究并证明Berezin对应与复Weyl对应的协变性，给出显式符号公式，并联系余伴随轨道，确立其为Stratonovich-Weyl对应。

通过有限群和McKay对应的顶点表示

Vertex representations via finite groups and the McKay correspondence

本文基于有限群及虚特征标，利用圈积表示环构造Fock空间与顶点算子，还原特征标表，并在 $Γ$ 为 $SU_2$ 有限子群时给出 $ADE$ 型仿射与环面李代数基本表示的群论实现，提出McKay对应的新形式。

An equivalence between truncations of categorified quantum groups and Heisenberg categories

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described abo…