Tensor invariants, saturation problems, and Dynkin automorphisms

Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let Gσ be the connected almost simple algebraic group associated with G and σ. We prove that the dimension of the tensor invariant space of Gσ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does Gσ. As a consequence, we show that the spin group Spin(2n+1) has saturation factor 2, which strengthens the results of Belkale-Kumar [1] and Sam [28] in the case of type Bn.