Category ${\mathscr{C}}_{k}$ of multi-loop algebra representations versus modular representations: Questions of G. Lusztig

Let G be a connected simply-connected simple algebraic group over an algebraically closed field of characteristic p > 0 with a fixed maximal torus T and a fixed Borel subgroup B containing T . Let X+ be the set of dominant characters of T . Consider the category C = C(G) of finite dimensional rational representations of G. The simple objects of C, up to isomorphism, are indexed by X+; let Lλ be the simple object indexed by λ ∈ X+. Let E 0 λ be the Weyl module indexed by λ ∈ X+. The Weyl modules form another basis of the Grothendieck group G(C) of C. Hence, for any λ ∈ X+, we can write