Spinorial representation of submanifolds in $SL_n(\mathbb{C})/SU(n)$

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G/H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G.$ This in particular includes $SL_n(\mathbb{C})/SU(n),$ 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 $SL_n(\mathbb{C})/SU(n).$ As a new application, we obtain a fundamental theorem for the submanifold theory in that spaces.