Quasi Regular Functions in Quaternionic Analysis

TL;DR

This work develops the quaternionic analogue of regular function theory by introducing quasi regular and quasi anti regular functions, deriving their K-type decompositions, reproducing kernels, and invariant pairings. It shows that, although these spaces lack a unitary structure, they admit rich invariant pseudounitary forms and admit three-component Poincaré-restricted decompositions, along with explicit reproducing formulas and kernel expansions. A central theme is the construction and factorization of intertwining operators between spaces of quaternionic functions, via reproducing kernels and Clifford-algebraic (fermionic) formalisms, culminating in a detailed account of equivariant maps that interrelate ${\cal W}$, ${\cal U}$, and ${\cal W}'$. These results provide structural tools for conformal-invariant quaternionic algebras and open pathways to operator factorizations relevant to Clifford-analytic representations and potential physical interpretations. The Cayley transform and various subgroup restrictions further elucidate the representation-theoretic anatomy of these spaces, clarifying how the quasi-regular framework interplays with biharmonic and anti-regular sectors under symmetry reductions.

Abstract

We study a new class of functions that arise naturally in quaternionic analysis, we call them "quasi regular functions". Like the well-known quaternionic regular functions, these functions provide representations of the quaternionic conformal group. However, unlike the regular functions, the quasi regular ones do not admit an invariant unitary structure but rather a pseudounitary equivalent. The reproducing kernels of these functions have an especially simple form: (Z-W)^{-1}. We describe the K-type bases of quasi regular functions and derive the reproducing kernel expansions. We also show that the restrictions of the irreducible representations formed from the quasi regular functions to the Poincare group have three irreducible components. Our interest in the quasi regular functions arises from an application to the study of conformal-invariant algebras of quaternionic functions. We also introduce a factorization of certain intertwining operators between tensor products of spaces of quaternionic functions. This factorization is obtained using fermionic Fock spaces constructed from the quasi regular functions.

Theorems & Definitions (98)

• Proposition 1: Proposition 34 in ATMP
• Theorem 2
• Lemma 3: Lemma 24 in ATMP
• Proposition 4
• Lemma 5
• Definition 6
• Proposition 7
• Remark 8
• proof
• Theorem 9