Hermitian Clifford Analysis and Its Connections with Representation Theory

TL;DR

This paper explores the Hermitian Clifford analysis and its relation to representation theory, questioning whether the holomorphic and anti-holomorphic Dirac operators are natural generalizations of the orthogonal Dirac operator in complex spaces.

Contribution

It critically examines the limitations of existing representation-theoretic constructions for Hermitian Dirac operators and suggests directions for developing Clifford analysis in complex and CR structures.

Findings

Representation theory challenges with unitary group representations.

Limitations of the generalized gradient construction for Hermitian Dirac operators.

Proposed development of Clifford analysis over complex vector spaces or CR structures.

Abstract

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on representation theory of Lie groups is the natural way to construct such a Dirac-type operator because applied to a Riemannian spin manifold it provides the Atiyah-Singer Dirac operator. This method, however, does not apply to these Hermitian Dirac operators because the representations of the unitary group used are not irreducible, causing problems in considering invariance under a group larger than U(n). This motivates either the development of Clifford analysis over a complex vector space with respect to a Hermitian inner product or the development of Dirac-type operators on Cauchy-Riemann structures.