In the talk we introduce a quaternionic analogue of the Heisenberg group and explain the relation between this group and the quaternionic analogous of the Siegel upper half space.

We discuss regular functions, that are counterpart of complex holomorphic functions for quaternionic setting. The Hardy space is the space of regular functions in the Siegel upper half space with $L^2$ boundary values.

We construct the Cauchy-Szegö kernel for the Cauchy-Szegö projection integral operator from the space of $L^2$-integrable functions defined on the boundary of the quaternionic Siegel upper half space to the space of boundary values of the quaternionic regular functions of the Hardy space over the quaternionic Siegel upper half space.

We also present the fundamental solution for a hypoelliptic operator related to the boundary of the Siegel upper half space.

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