Arithmetic Stratifications and Partial Eisenstein Series

Abstract

Let P\G and Q\H be generalized flag varieties over a number field F.In this paper we study certain locally trivial fibre bundles Y_ηover P\G having Q\H as general fibre, and determine the arithmetic stratification of Y _ηwith respect to a line bundle. The arithmetic stratification is defined in terms of height zeta functions and the height zeta function of a stratum is of the form

$$\sum\limits_{{\gamma \in P(F)\backslash G(F)}} {{{e}^{{\langle s\lambda ,{{H}_{P}}(\gamma )\rangle }}}E_{Q}^{{Q{{w}^{{ - 1}}}{{Q}_{0}}}}(s\mu ,\eta ({{p}_{\gamma }})),}$$

where $E_{Q}^{{Q{{w}^{{ - 1}}}{{Q}_{0}}}}$ is a “partial Eisenstein series” associated to the Schubert cell Q∖Qw ⁻¹ Q ₀. The computation of the constant term of these gives estimates that allow one to determine the abcissa of convergence of the height zeta function of the stratum.