Tilting modules for classical groups and howe duality in positive characteristics

Abstract

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. We prove that M(m,n) is a tilting Gl(m)- and G2(n)-module and that Endvl(m) M(m, n) is generated by G2(n) and vice versa. The existence of such a module provides much information about the relations between the category /(:l(m, n) of rational G1 (rn)-modules with highest weights bounded in a certain sense by n and the category /C2(m,n) of rational G2(n)-modules with highest weights bounded in the same sense by rn. More specifically, we prove that there is a bijection of the set of dominant weights of Gl(m)-modules from/Cl(m, n) to the set of dominant weights of G2(m)-modules from K:2(m, n) such that Ext groups for induced G)(m)-modules from/Cl(m, n) are isomorphic to Ext groups for corre- sponding Weyl modules over G2(n). Moreover, the derived categories DblCI (m, n) and DblC2(m, n) appear to be equivalent. We also use our study of the modules M(m, n) to find generators and relations for the algebra of all G-invariants in A'(V "1 @(V*)'2), where G =GL,,, Sp2,~, Om and V is the natural G-module.