On the quasi-irreducibility and complete quasi-reducibility of some reductive prehomogeneous vector spaces (Q638753)

Fix an algebraically closed field \(k\) of characteristic zero. Let \(G\) be a linear \(k\)-group and \(\rho: G \to \mathrm{End}_k(V)\) be a finite-dimensional rational representation. We say a triplet \((G,\rho,V)\) is a prehomogeneous vector space, abbreviated as a PV, if \(V\) has a Zariski open orbit under \(G\). We define the dual triplet \((G, \rho^*, V^*)\) by taking the contragredient representation of \(\rho\); the dual triplet is not a PV in general. A rational function \(f \in k(V)^\times\) is called a relative invariant if it is a joint eigenfunction under the \(G\)-action. If the rational map \(\nabla \log f: V \to V^*\) is dominant, we call \(f\) a non-degenerate relative invariant. When non-degenerate relative invariant exist, we say \((G,\rho,V)\) is regular and in this case \((G, \rho^*, V^*)\) is a PV. In this paper, the author only considers the PV's with \(G\) reductive. Following H. Rubenthaler, a regular PV \((G,\rho,V)\) is called Q-irreducible (``Q'' for ``quasi'') if no proper subrepresentation of \(\rho\) gives rise to a regular PV. It is called completely Q-reducible if \(\rho\) is a direct sum of subrepresentations which give rise to Q-irreducible PV's. In this paper, Hamada introduced the notion of a regualr PV of general type, which are obtained from a regular PV by some castling transform (see 1.1, 1.2 in this paper). This gives a general recipe to construct Q-irreducible PV. He then takes up the classification of simple, 2-simple of type I and 3-simple of nontrivial type PV's by T. Kimura et al., and identifies the Q-irreducible (resp. completely Q-reducible) ones of non-general type. In the last section, Hamada treats the PV's which appear in M. Sato's classification. Proofs are given only for the difficult cases.