Semicanonical bases for Schur algebras

The Schur algebras are realized in the ring of constructible functions on the generalized Steinberg varieties attached to the n-step flag varieties. The semicanonical bases for Schur algebras are defined and shown to behave in the simplest possible way under the transfer maps. Such a property of the semicanonical basis is conjectured in Lusztig (2002) [L6, 5.5].