A generalization of Steinberg's cross section. (Q2892816)

Let \(G\) be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element \(w\) of minimal length \(r\) a subvariety \(V\) of \(G\) isomorphic to an affine space of dimension \(r\) which meets the regular unipotent class \(Y\) in exactly one point. In this paper this is generalized to the case where \(w\) is replaced by any elliptic element in the Weyl group of minimal length \(d\) in its conjugacy class, \(V\) is replaced by a subvariety \(V'\) of \(G\) isomorphic to an affine space of dimension \(d\), and \(Y\) is replaced by a unipotent class \(Y'\) of codimension \(d\) in such a way that the intersection of \(V'\) and \(Y'\) is finite. The proofs use quantum groups and canonical bases. The base ring is often just a commutative ring \(A\) and an affine space of dimension \(d\) is treated through the set \(A^d\) of its \(A\)-valued points. Relevant maps between affine spaces are described by polynomials with integral coefficients. Thus one gets a version over \(\mathbf Z\). Some twisted cases are treated also.