Tensor invariants, Saturation problems, and Dynkin automorphisms

Abstract

Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let G_σ be the connected almost simple algebraic group associated to G and σ. We prove that the dimension of the tensor invariant space of G_σ 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) is of saturation property with factor 2, which strengthens the results of Belkale-Kumar and Sam in the case of type B_n.

27 pages. Rewrite introduction and add some applications

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