THE ALGEBRA OF QUANTUM BOSONS, THE SCHUBERT FILTRATION, AND LUSZTIG BASES

The ''Schubert filtration'', defined by means of the quantum Weyl group, is considered in Drinfel'd-Jimbo quantum algebras. A description is obtained for this filtration in terms of linear relations determined by the algebra of ''quantum bosons'', and also in terms of certain projection operators. The connection between this filtration and Lusztig bases and Kashiwara crystal bases is considered.