On some extensions of p-restricted completely splittable GL(n)-modules

Abstract

In this paper, we calculate the space $Ext^1_{GL(n)}(L_n(\lambda),L_n(\mu))$, where GL(n) is the general linear group of degree $n$ over an algebraically closed field of positive characteristic, $L_n(\lambda)$ and $L_n(\mu)$ are rational irreducible GL(n)-modules with highest weights $\lambda$ and $\mu$ respectively, the restriction of $L_n(\lambda)$ to any Levi subgroup of GL(n) is semisimple, $\lambda$ is a $p$-restricted weight and $\mu$ does not strictly dominate $\lambda$.