Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms (Q1322365)

The purpose of this paper is to extend the range of zeta functions associated with prehomogeneous vector spaces. For an algebraic number field \(k\), we take a prehomogeneous vector space \((G_ k, V_ k)\) defined over \(k\). Roughly speaking, zeta functions associated with \((G_ k, V_ k)\) are obtained as a Dirichlet series whose coefficients are numbers of \(G_ k\)-orbits in \(V_ k\setminus S_ k\) (\(S_ k\) is a singular set of \((G_ k, V_ k)\)). However such definition excludes the cases related to the automorphic forms, for example Hecke's \(L\)-function. In this paper, the author tries to reconstruct the old general theory so that we can deal with the zeta functions in a wider class including them. The author generalizes the theory of prehomogeneous vector spaces with symmetric structure of \(K_ \varepsilon\)-type and the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character.