Schubert Eisenstein Series

We define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements of a particular Schubert cell, indexed by an element of the Weyl group. They generally not fully automorphic. We will develop some results and methods for GL3 that may be suggestive about the general case. The six Schubert Eisenstein series are shown to have meromorphic continuation and some functional equations. The Schubert Eisenstein series Es1s2 and Es2s1 corresponding to the Weyl group elements of order three are particularly interesting: at the point where the full Eisenstein series is maximally polar, they unexpectedly become (with minor correction terms added) fully automorphic and related to each other. AMS Subject Classification: 11F55. We define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements coming from a particular Schubert cell. More precisely, let G be a split semisimple algebraic group over a global field F , and let B be a Borel subgroup. The usual Eisenstein series are sums over B(F )\G(F ), that is, over the integer points in the flag variety X = B\G. Given a Weyl group element w, one may alternatively consider the sum restricted to a single Schubert cell Xw. This is the closure of the image in X of the double coset BwB. If w = w0, the long Weyl group element, then Xw = X so this contains the usual Eisenstein series as a special case. The notion of Schubert Eisenstein series seems a natural one, but little studied. The purpose of this paper is to look closely at the special case where G = GL(3) that suggest general lines of research for the general case. The Schubert Eisenstein series is not automorphic, so its place in the spectral theory is less obvious. An immediate question is whether the Schubert Eisenstein series, like the classical ones have analytic continuation. We will prove this when G = GL(3) and we hope that it is true in general. We will observe some other interesting phenomena on GL(3), to be described below. We will begin by supplying some motivation for this investigation. Recently it has been observed that Fourier-Whittaker coefficients of Eisenstein series are multiple Dirichlet series which may often be expressed as sums over Kashiwara crystals.