Tilting modules for classical groups and Howe duality in positive characteristic
We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra. To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field k, we set in correspondence another series of classical groups (usually the same one). Denote by G1 (m) the group of rank m from the first series and by G2 (n) the group of rank n from the second series. For any pair (Ga(m),G2(n)) we construct the Gl(m) x G2(n)-module M(m,n). The construction of M(m, n) is independent of characteristic; for char k -~ 0, the actions of Gl(m) and G2(n) on M(m,n) form a reductive dual pair in the sense of Howe.