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In this paper, we study left and right n -regular functions that originally were introduced in [I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics II, accepted in Adv. Theor. Math. Phys]. When n = 1 , these functions are the usual quaternionic left and right regular functions. We show that n -regular functions satisfy most of the properties of the usual regular functions, including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy–Fueter type reproducing formulas. Arguably, these Cauchy–Fueter type reproducing formulas for n -regular functions are quaternionic analogues of Cauchy's integral formula for the n th-order pole f (n − 1) (w) = (n − 1) ! 2 π i ∮ f (z) d z (z − w) n . We also find two expansions of the Cauchy–Fueter kernel for n -regular functions in terms of certain basis functions, we give an analogue of Laurent series expansion for n -regular functions, we construct an invariant pairing between left and right n -regular functions and we describe the irreducible representations associated to the spaces of left and right n -regular functions of the conformal group and its Lie algebra. [ABSTRACT FROM AUTHOR]

We show that conformal transformations on the generalized Minkowski space $$\mathbb {R}^{p,q}$$ map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when $$p$$ or $$q$$ is $$0$$ , and that this action has exactly three orbits if $$p, q \ne 0$$ . Then we extend these results to hyperboloids and affine planes of arbitrary dimension. These properties generalize the well-known properties of Möbius (or fractional linear) transformations on the complex plane $$\mathbb {C}$$ . [ABSTRACT FROM AUTHOR]

This is a continuation of a series of papers [2-4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper, we continue to study the quaternionic analogs of Cauchy's formula for the second-order pole. These quaternionic analogs are closely related to regularization of infinities of vacuum polarization diagrams in 4D quantum field theory. To add some flexibility, especially when dealing with Cauchy's formula for the second-order pole, we introduce a one-parameter deformation of quaternionic analysis. This deformation of quaternions preserves conformal invariance and has a geometric realization as anti de Sitter space sitting inside the 5D Euclidean space. We show that many results of quaternionic analysis--including the Cauchy-Fueter formula--admit a simple and canonical deformation. We conclude this paper with a deformation of the quaternionic analogs of Cauchy's formula for the second-order pole. [ABSTRACT FROM AUTHOR]

In the series of papers [2, 3], we approach quaternionic analysis from the point of view of representation theory of the conformal group and its real forms. This approach has proved very fruitful and pushed further the parallel with complex analysis and developed a rich theory. In [3] we study the counterparts of Cauchy-Fueter and Poisson formulas on the spaces of split quaternions and Minkowski space and show that they solve the problem of separation of the discrete and continuous series on and the imaginary Lobachevski space . In particular, we introduce an operator PlR, compute its effect on the discrete and continuous series components of the space of functions and obtain a surprising formula for the Plancherel measure of . The proof is based on a transition to the Minkowski space and some pretty lengthy computations. In this paper, we introduce an operator on and show that its effect on the discrete and continuous series components can be easily computed using the Schrödinger model for the minimal representation of O(p,q) (with p=q=3) and the results of Kobayashi–Mano from [7], particularly their computation of the integral expression for the operator . This provides an independent verification of the coefficients involved in the formula for PlR. This paper once again demonstrates a close connection between quaternionic analysis and representation theory of various O(p,q)’s. [ABSTRACT FROM PUBLISHER]