Spinorial representation of submanifolds in $S{L}_n(\mathbb{C})\mathrm{/}SU(n)$

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G\mathrm{/}H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G\mathrm{.}$ This in particular includes $S{L}_n(\mathbb{C})\mathrm{/}SU(n),$ and extends the previously known spinorial representation of a surface in ${\mathbb{H}}^3$ if $n=2.$ We also recover the Bryant representation of a surface with constant mean curvature 1 in ${\mathbb{H}}^3$ and its generalization for a surface with holomorphic right Gauss map in $S{L}_n(\mathbb{C})\mathrm{/}SU(n)\mathrm{.}$ As a new application, we obtain a fundamental theorem for the submanifold theory in that spaces.