Donkin's tilting module conjecture

Let $G$ be a reductive algebraic group over a field $k$ of characteristic $p$, let $G_rT$ be the corresponding Frobenius kernel together with a maximal torus, and let $X_r$ denote the set of $p^r$-restricted dominant weights. Let $\rho$ be the sum of the fundamental weights, let $w_0$ be the longest element of the Weyl group, and for $u$ let $T( u)$ denote the indecomposable tilting module of highest weight $u$. For u\renewcommand{\arraystretch}{1.2}\begin{array}{c}\end{array} in $X_r$, set

$$\hat{\nu}=2(p^r-1)\rho+w_0\nu,$$

and let $\widehat{Q}_r(\nu)$ be the $G_rT$-projective cover of the corresponding simple module $\widehat{L}_r(\nu)$. Donkin's tilting module conjecture. For all $\lambda\in X_r$,

$$T(\hat{\lambda})|_{G_rT}\cong\widehat{Q}_r(\lambda).$$

Equivalently, in the symmetric formulation,

$$T((p^r-1)\rho+\lambda)|_{G_rT}\cong\widehat{Q}_r((p^r-1)\rho+w_0\lambda)$$

for all $\lambda\in X_r$. This conjecture identifies the restriction of the relevant indecomposable tilting modules with the projective covers for the Frobenius kernel and torus; its resolution concerns the structure of tilting modules in modular representation theory.