Local zeta functions on Hermitian forms and its application to local densities (Q1271157)

Let \(k\) be a nonarchimedean local field of characteristic 0. Let * be an involution on \(k\) and denote by \(k_0\) the fixed field by *. Assume that \(k\) is unramified over \(k_0\). Let \(G_n= GL_n(k)\) and \(K_n= GL_n(O)\) where \(O\) is the ring of integers of \(k\). Let \(V_n\) be the set of Hermitian matrices. Denote by \(S(V_n)\) the space of Schwartz-Bruhat functions on \(V_n\). For \(\varphi\in S(V_n)\), \(s\in \mathbb{C}^n\) and a character \(\chi\) of \(k_0^x\), consider local zeta functions \(Z(\varphi, \chi,s)\) and \(Z^*(\varphi, \chi,s)\). Let \(S(K_n\setminus X_n)\) be the subspace of \(S(V_n)\) consisting of all \(K_n\)-invariant functions with support contained in \(X_n= V_n\cap G_n\). As for \(\varphi\in S(K_n\setminus X_n)\), functions \(Z(\varphi, \chi,s)\) are closely investigated by \textit{Y. Hironaka} [J. Math. Soc. Japan 51, 553-581 (1999)]. The zeta functions \(Z(\varphi, \chi,s)\) (resp. \(Z^* (\varphi,\chi, s))\) are those of the prehomogeneous vector spaces \((P_n, \rho, V_n)\) (resp. the dual of \((P_n, \rho, V_n))\), where \(P_n\) is the subgroup of \(G_n\) consisting of lower triangular matrices and \(\rho(g)\upsilon= g\upsilon g^*\) \((g\in P_n\), \(\upsilon\in V_n)\). When one decomposes these zeta functions into \(2^n\) sub-zeta-functions according to the \(P_n\)-orbit decomposition of \(V_n\) by a general theory of prehomogeneous vector spaces, it is known that there exists some functional equation between them. In this paper, an explicit description of such a functional equation is given. Further, as an application explicit expressions of local densities of integral representations of non degenerate unramified hermitian matrices with entries in the ring of \(p\)-integers is given.