Tensor Product Decompositions and Open Orbits in Multiple Flag Varieties

For a connected semisimple algebraic group G, we consider some special infinite series of tensor products of simple G-modules whose G-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open G-orbits in multiple flag varieties and address the problem of classifying such series. Let G be a connected simply connected semisimple algebraic group. In this paper we establish a close interrelation between some special series of tensor products of simple G-modules whose G-fixed point spaces are at most one-dimensional and multiple flag varieties of G that contain open G-orbit. Motivated by this intimate connection with geometry, we then address the problem of classifying such series. Starting with the basic definition and examples in Sections 1 and 2, we introduce necessary notation in Section 3 and then formulate our main results in Section 4. Other results and proofs are contained in the remaining part of paper. Below all algebraic varieties are taken over an algebraically closed field k of characteristic zero. 1. Basic definition Fix a choice of Borel subgroup B of G and maximal torus T ⊂ B. Let P++ be the additive monoid of dominant characters of T with respect to B. Put P≫ := P++ \ {0}. For λ ∈ P++, denote by Eλ a simple G-module of highest weight λ and by λ ∗ the highest weight of dual G-module E ∗ λ . Let Pλ be the G-stabilizer of unique B-stable line in Eλ. If μ, λ1, . . . , λd ∈ P++, denote by c μ λ1,...,λd the multiplicity of Eμ inside Eλ1 ⊗ . . . ⊗ Eλd , i.e., the Littlewood–Richardson coefficient dim ( HomG(Eμ, Eλ1 ⊗ . . . ⊗ Eλd) ) . Denote respectively by ̟1, . . . ,̟r and α1, . . . , αr the systems of fundamental weights of P++ and simple roots of G with respect to T and B enumerated as in [Bo2]. Let respectively Date: August 25, 2006. 2000 Mathematics Subject Classification. 20G05, 14L30.