Local zeta functions on Hermitian forms and its application to local densities
type: Item
instance of: scholarly article
title: Local zeta functions on Hermitian forms and its application to local densities
zbMATH Open document ID: 0932.11072
author: Yumiko Hironaka
published in: Journal of Number Theory
publication date: 2000-03-19T00:00:00Z
review text: 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.
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11S90
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 1221694
zbMATH Keywords: Hermitian forms
zbMATH Keywords: Schwartz-Bruhat functions
zbMATH Keywords: local zeta functions
zbMATH Keywords: prehomogeneous vector spaces
zbMATH Keywords: local densities
zbMATH Keywords: integral representations
zbMATH Keywords: hermitian matrices
MaRDI profile type: Publication
full work available at URL: jnth.1998.2237
OpenAlex ID: W2036249456
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DOI: 10.1006/JNTH.1998.2237
description: scientific article; zbMATH DE number 1221694