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The Second Order Pole over Split Quaternions

This is an addition to a series of papers [FL1, FL2, FL3, FL4], 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 develop split quaternionic analogues of certain results from [FL4]. Thus we introduce a space of functions ${\cal D}^h \oplus {\cal D}^a$ with a natural action of the Lie algebra $\mathfrak{gl}(2,\mathbb H_{\mathbb C}) \simeq \mathfrak{sl}(4,\mathbb C)$, decompose ${\cal D}^h \oplus {\cal D}^a$ into irreducible components and find the $\mathfrak{gl}(2,\mathbb H_{\mathbb C})$-equivariant projectors onto each of these irreducible components.