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.