Quotients of some prehomogeneous vector spaces (Q1359016)

This paper gives a ``reduction'' of prehomogeneous vector spaces over an algebraically closed field \(\mathbb{C}\) and its application to the construction of relative invariants. Let \((G\times H,\rho,V)\) be a prehomogeneous vector space where \(G\times H\) is a product of two algebraic groups. If \(V//G:=\text{Spec}(\mathbb{C}[V]^G)\) is a vector space and if the action \(\overline\rho\) of \(H\) on \(V//G\) is prehomogeneous, then the properties of \((G\times H,\rho,V)\) are expected to be obtained from those of the prehomogeneous vector space \((H,\overline\rho,V//G)\). The author of this paper carried out the construction of the relative invariants of the prehomogeneous vector spaces \((\text{SL}_5\times\text{GL}_3,\Lambda_2\otimes\Lambda_1)\) and \((\text{Spin}_{10}\times\text{GL}_3\), half-spin rep. \(\otimes\Lambda_1)\).