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

Let \(\begin{aligned} \mathcal {D}_{16}=\left\{ Z\in \mathcal {M}_{1,2}(\mathfrak {C}^{c}):\;\begin{array}{lll} 1-\left\langle Z,Z \right\rangle +\left\langle Z^{\sharp },Z^{\sharp }\right\rangle>0,\\ 2-\left\langle Z,Z \right\rangle \; >0\end{array}\right\} \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{D}}_{16}=\left\{Z,\in ,{\mathcal{M}}_{1,2},({\mathfrak{C}}^c),:,,\begin{array}{c}1-〈Z,,,Z〉+〈Z^{♯},,,Z^{♯}〉>0,\hfill \\ 2-〈Z,,,Z〉>0\hfill \end{array}\right\}\end{array}$$ be an exceptional domain of non-tube type and let \(\mathcal {U}_{\nu }\) ${\mathcal{U}}_{\nu }$ and \(\mathcal {W}_{\nu }\) ${\mathcal{W}}_{\nu }$ the associated generalized Hua operators. In this paper, we determine the explicit formula of the action of the group \( E_{6(-14)}\) $E_{6(-14)}$ on \(\mathcal {D}_{16}\) ${\mathcal{D}}_{16}$ . We characterized the \(L^{p}\) $L^p$ -range, \(1\le p < \infty \) $1\le p<\infty$ of the generalized Poisson transform on the Shilov boundary of the domain \(\mathcal {D}_{16}\) ${\mathcal{D}}_{16}$ .