Harmonic Analysis on Exceptional Domain \(E_{6(-14)}/U(1)Spin(10)\)

Let

$$\begin{aligned} \mathcal {D}_{16}=\left\{ Z\in \mathcal {M}_{1,2}(\mathfrak {C}^{c}):\;\begin{array}{lll} 1-\left\langle Z,Z \right\rangle +\left\langle Z^{\sharp },Z^{\sharp }\right\rangle>0,\\ 2-\left\langle Z,Z \right\rangle \; >0\end{array}\right\} \end{aligned}$$

be an exceptional domain of non-tube type and let \(\mathcal {U}_{\nu }\) and \(\mathcal {W}_{\nu }\) the associated generalized Hua operators. In this paper, we determine the explicit formula of the action of the group \( E_{6(-14)}\) on \(\mathcal {D}_{16}\). We characterized the \(L^{p}\)-range, \(1\le p < \infty \) of the generalized Poisson transform on the Shilov boundary of the domain \(\mathcal {D}_{16}\).