A refined Weyl character formula for comodules on GL (2, A)

Let A be any commutative unital ring and let GL (2, A) be the general linear group scheme on A of rank 2. We study the representation theory of GL (2, A) and the symmetric powers Symᵈ (V), where (V, ) is the standard right comodule on GL (2, A). We prove a refined Weyl character formula for Symᵈ (V). There is for any integer d 1 a (canonical) refined weight space decomposition Symᵈ (V) ᵢ Symᵈ (V) ⁱ where each direct summand Symᵈ (V) ⁱ is a comodule on N GL (2, A). Here N is the schematic normalizer of the diagonal torus T GL (2, A). We prove a character formula for the direct summands of Symᵈ (V) for any integer d 1. This refined Weyl character formula implies the classical Weyl character formula. As a Corollary we get a refined Weyl character formula for the pull back Symᵈ (V K) as a comodule on GL (2, K) where K is any field. We also calculate explicit examples involving the symmetric powers, symmetric tensors and their duals. The refined weight space decomposition exists in general for group schemes such as GL (n, A) and SL (n, A). The methods introduced in the paper may have applications to the study of finite rank torsion free comodules on SL (n, Z) and GL (n, Z). There is no "highest weight theory" or "complete reducibility property" such comodules, and we want to give a definition of the notion "good filtration" for such modules. Such a study may have applications to the study of groups G such as SL (n, k) and GL (n, k) and quotients G/H where k is an arbitrary field and H G is a closed subgroup.