Fourier coefficients of certain Eisenstein series (Q5895261)

Let \(M_ n({\mathbb{Z}})\) be the set of n by n matrices with entries in the ring of rational integers \({\mathbb{Z}}\). For \(\gamma,\delta \in M_ n({\mathbb{Z}})\), we write \((\gamma,\delta)=1\) if (\(\gamma\delta\)) is a lower \(n\times 2n\) submatrix of some element of Sp(n;\({\mathbb{Z}})\). Put \(H_ n:=\{z\in M_ n({\mathbb{C}}) |\) \({}^ tz=z\), Im z\(>0\}\). Fixing such a pair \(\gamma\) and \(\delta\), a kind of Eisenstein series is defined: \[ E(z,s,k;(\gamma,\delta)):=\sum \det (cz+d)^{-k} abs(\det (cz+d))^{- 2s}\quad (z\in H_ n,\quad s\in {\mathbb{C}}), \] where (c,d) runs over a certain set whose elements are defined by a congruence condition with respect to \(\gamma\) and \(\delta\) and the modulus \(q\geq 3.\) The purpose of the present paper is to study Dirichlet series \[ \zeta (h;k,(\gamma,\delta);s)=\sum_{c}\sum_{d}\det (c)^{-k} abs(\det (c))^{-2s} e(q^{-1} tr hc^{-1}d), \] which appear as the Fourier coefficients of E(z,s,k;(\(\gamma\),\(\delta\))). \(\zeta\) (h;k,(\(\gamma\),\(\delta\));s) is expressed as \[ 2\phi (q)^{- 1}\sum_{\chi mod q,\chi (-1)=(-1)^ k}\prod_{p| q}b_ p((p^{k+2s}\chi (p))^{-1},h)\times \prod_{i}b_{p_ i}((p_ i^{k+2s}(\prod_{j\neq i}\chi_ j)(p_ i))^{- 1};h,\chi,(\gamma,\delta)), \] where \(q=\prod q_ i\), \(q_ i\) is a power of a prime \(p_ i\), \(\chi_ i\) is a Dirichlet character modulo \(q_ i\) such that \(\chi =\prod \chi_ i\) and \(\chi\) is a Dirichlet character modulo q. The factor \(b_{p_ i}\) is expressed by using an auxiliary function \(B_ p(x;h,\chi;(\gamma,\delta),Q)\). The main result of the paper is described as the precise shapes of \(B_ p(x;h,\chi;(\gamma,\delta),Q)\) according to the variations of the parameters involved. The details will appear elsewhere.