On the convergence of the zeta function for certain prehomogeneous vector spaces
Abstract
Let (G,V) be an irreducible prehomogeneous vector space defined over a number field k, P in k[V] a relative invariant polynomial, and X a rational character of G such that P(gx)=X(g)P(x). Let Vkss=x ∈ Vk such that P(x) is not equal to 0. For x in Vkss, let Gx be the stabilizer of x, and Gx0 the connected component of 1 of Gx. We define L0 to be the set of x in Vkss such that Gx0 does not have a non-trivial rational character. We study the zeta function for (G,V).
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