Decomposition of reductive regular prehomogeneous vector spaces

TL;DR

This paper introduces the concept of quasi-irreducible prehomogeneous vector spaces (PV) and demonstrates that any regular PV can be decomposed into these Q-irreducible components, providing a new structural understanding.

Contribution

It defines quasi-irreducible PVs, proves their intrinsic nature in completely Q-reducible PVs, and classifies Q-irreducible PVs of parabolic type.

Findings

Q-irreducible PVs are intrinsically defined.

Any regular PV decomposes into Q-irreducible PVs.

Classification of Q-irreducible PVs of parabolic type.

Abstract

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a connected reductive algebraic group over C. If $V= \oplus_{i=0}^{n}V_{i}$ is a decomposition of V into irreducible representations, then, in general, the PV's $(G,V_{i})$ are no longer regular. In this paper we introduce the notion of quasi-irreducible PV (abbreviated to Q-irreducible), and show first that for completely Q-reducible PV's, the Q-isotopic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of quasi-irreducible PV's. Finally we classify the quasi-irreducible PV's of parabolic type.