Tilting modules for classical groups and Howe duality in positive characteristic
To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field \(k\), the authors set in correspondence another series of classical groups. Let \(G_1(m)\) denote a group of rank \(m\) from the first series and \(G_2(n)\) a group of rank \(n\) from the second series. For any pair \((G_1(m),G_2(n))\) the authors construct a \(G_1(m)\times G_2(n)\)-module \(M(m,n)\). For \(\text{char }k=0\) the actions of \(G_1(m)\) and \(G_2(n)\) on \(M(m,n)\) form a reductive dual pair in the sense of Howe. It is shown that \(M(m,n)\) is a tilting \(G_1(m)\)- and \(G_2(n)\)-module and that \(\text{End}_{G_1(m)}M(m,n)\) is generated by \(G_2(n)\) and vice versa. The authors consider the category \({\mathcal K}_1(m,n)\) of rational \(G_1(m)\)-modules with highest weights bounded in a certain sense by \(n\) and the category \({\mathcal K}_2(m,n)\) of rational \(G_2(n)\)-modules with highest weights bounded in a certain sense by \(m\). It is shown that there is a bijection of the set of dominant weights of \(G_1(m)\)-modules from \({\mathcal K}_1(m,n)\) to the set of dominant weights of \(G_2(n)\)-modules from \({\mathcal K}_2(m,n)\) such that Ext groups for induced \(G_1(m)\)-modules from \({\mathcal K}_1(m,n)\) are isomorphic to Ext groups for corresponding Weyl modules over \(G_2(n)\). The derived categories \(D^b{\mathcal K}_1(m,n)\) and \(D^b{\mathcal K}_2(m,n)\) appear to be equivalent. These results are applied to finding generators and relations for the algebra of all \(G\)-invariants in \(\wedge^\bullet(V^{n_1}\oplus(V^*)^{n_2})\), where \(G=GL_m\), \(Sp_{2m}\), \(O_m\), and \(V\) is the natural \(G\)-module.
scientific article; zbMATH DE number 934930
scientific article