Injective and tilting resolutions and a Kazhdan-Lusztig theory for the general linear and symplectic group

Let k be an algebraically closed field of characteristic p>0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. We also give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig type polynomials. This builds further on work of Cox-De Visscher and Brundan-Stroppel.