Tensor invariants, Saturation problems and Dynkin automorphisms

Let $G$ be a connected almost simple algebraic group with a Dynkin automorphism $\sigma$. Let $G_\sigma$ be the connected almost simple algebraic group associated to $G$ and σ. We prove that the dimension of the tensor invariant space of $G_\sigma$ is equal to the trace of $\sigma$ on the corresponding tensor invariant space of $G$. We prove that if $G$ has the saturation property then so does $G_\sigma$. In particular, we show that the spin group $Spin(2n + 1)$ is of saturation property with factor $2$, which strengthens the results of Belkale-Kumar and Sam in the case of type $B_n$.
This is a joint work with Jiuzu Hong.