Let $\mathfrak{g}$ be a semi-simple Lie algebra.

So in characteristic $0$, the Grothendieck group of a block of category $\mathcal{O}$ is given by the classes of the Verma modules. Unlike the simples (or projectives), Verma modules are very easy to understand explicitly, and one can easily perform computations with them (eg. involving translation functors).

In characteristic $p$, the analogous object is the category of modules with a fixed central character: one must specify both a Harish-Chandra character, and a character of the Frobenius (or p-) center. Now each simple is a quotient of a baby Verma module; **however** the classes of the baby Verma's no longer give a basis in the Grothendieck group -- in fact, they all have the same class!

I'm looking for a basis for the Grothendieck group of this category, where the corresponding objects are easy to understand (unlike the simples or projectives). In fact, I'd be happy with a collection of objects whose images span the Grothendieck group. I'm most interested in the case where the Frobenius character is 0, but the Harish-Chandra character is singular.

I've heard that the Weyl modules (coming from the char $p$ version of Borel-Weil theory) might be useful here; though I'm not entirely sure if they can be defined in this generality. This link might be relevant: Weyl modules and reduction modulo $p$.