Semicanonical bases for Schur algebras. (Q986521)

Fix a pair of positive integers \((d,n)\). Let \(G=\text{GL}_d(\mathbb{C})\). The Steinberg variety \(\widetilde{\mathcal F}\) consists of pairs \((x,F)\), where \(x\) is a nilpotent endomorphism of \(\mathbb{C}^n\) and \(F\) is a partial flag of length \(n\) so that \(x\) maps \(F_i\) into \(F_{i-1}\). The generalized Steinberg variety \(Z\) or \(Z_d\) is now defined as consisting of triples \((x,F,F')\) with both \((x,F)\) and \((x,F')\) in \(\widetilde{\mathcal F}\). A \(\mathbb{C}\)-valued function \(f\) on \(Z\) is called constructible if it takes only finitely many values \(v\) and each \(f^{-1}(v)\) is a constructible subvariety. Let \(M_G(Z)\) consist of the constructible functions on \(Z\) that are constant on \(G\)-orbits. It is an algebra with convolution product. The author shows that there is an isomorphism of algebras from the Schur algebra \(S(n,d)\) to \(M_G(Z)\). One defines a natural basis on \(M_G(Z)\) and transports it to the Schur algebra. The resulting basis is the semicanonical basis. Using the geometry of the \(Z_d\) it is shown that the `transfer' map \(S(n,d+n)\to S(n,d)\) sends an element of the semicanonical basis of \(S(n,d+n)\) either to zero or to an element of the semicanonical basis of \(S(n,d)\). This confirms a conjecture of Lusztig.