A Computation of the Gamma Matrix of a Family of p-Adic Zeta Integrals Original Research Article

Igusa has discussed the general form for the functional equation of the family of p-adic zeta integrals associated to a certain class of prehomogeneous vector spaces. The gamma matrix which occurs in this functional equation is known in only a small number of cases. We compute the gamma matrix for the zeta integrals associated to the prehomogeneous vector space of n × n symmetric matrices over a p-adic field k, with the action of GLn(k) given by x maps to gxtg. The computation is carried out by finding the γ-factors for a family of intermediate functional equations. This last has applications so computing the points of reducibility of certain families of degenerate principal series representations of classical groups, to the study of Eisenstein series on these same groups, and to non-vanishing of theta lifts.