Modules with a Demazure flag

Abstract

A Demazure module can be described as the space of global sections of a suitable line bundle on a Schubert variety. A problem posed by the author in 1985 was to show that the tensor product of a one-dimensional Demazure module with an arbitrary one admits a Demazure flag, that is, a filtration whose quotients are Demazure modules. This was shown by P. Polo (who called such filtrations “excellent”) in a large number of cases including positive characteristic and by O. Mathieu for all semisimple algebraic groups first in zero characteristic and later in arbitrary characteristic. This paper settles this question in the context of a Kac-Moody algebra with symmetric simply-laced Cartan datum and in arbitrary characteristic. The method combines the corresponding “combinatorial excellent filtration” established independently by P. Littelmann et al. and the author with the globalization techniques of G. Lusztig and M. Kashiwara. In principle the method applies to an arbitrary symmetrizable Kac-Moody algebra; but for technical reasons it is necessary to use a positivity result of Lusztig which applies to only the simply-laced case.