Explicit calculations in an infinitesimal singular block of SL n

Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$ . For each $n \geq 1$ , there exists a singular block in the category of $G_1$ -modules, which contains precisely $n+1$ irreducible modules. We are interested in the ‘lift’ of this block to the category of $G_1T$ -modules. Imposing only mild assumptions on $p$ , we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.