Let \(G\) be a simply connected semisimple algebraic group over an algebraically closed field of characteristic zero. This paper considers some special series of tensor products of simple \(G\)-modules whose \(G\)-fixed point spaces have dimension less than or equal to one, and relates these modules to \(G\)-orbits in multiple flag varieties.
Specifically, let \(B\) be a Borel subgroup and \(T\subset B\) a maximal torus. Then the simple modules \(E_\lambda\) of \(G\) are indexed by dominant weights \(\lambda\) of \(T\) with respect to \(B\). Call a \(d\)-tuple \((\lambda_1,\dots,\lambda_d)\) of nontrivial dominant weights ‘primitive’ if for every \((n_1,\dots,n_d)\in\mathbb{Z}_{\geq 0}^d\), the \(G\)-fixed point space in \(E_{n_1\lambda_1}\otimes\cdots\otimes E_{n_d\lambda_d}\) has dimension at most one.
Given a dominant weight \(\lambda\), let \(P_\lambda\) denote the parabolic subgroup which is the stabilizer in \(G\) of the unique \(B\)-stable line in \(E_\lambda\). One of the main results of the paper says the following: If the multiple flag variety \(G/P_{\lambda_1}\times\cdots\times G/P_{\lambda_d}\) contains an open \(G\)-orbit, then \((\lambda_1,\dots,\lambda_d)\) is a primitive \(d\)-tuple. The author also gives a partial converse to this statement and discusses classification problems for primitive \(d\)-tuples and such open orbits, obtaining upper bounds on \(d\) for their existence.