Let $K$ be a field of characteristic zero. Consider the polynomial ring $S=K[X_{i,j}]_{1\le i\le m,1\le j\le n}$ on the entries of a generic $m\times n$ matrix $X=(X_{i,j})$. Let $I_p$ be the ideal in $S$ generated by $p\times p$ minors of $X$. I explain how to calculate completely the local cohomology modules $H^i_{I_p}(S)$. I will also explain why the problem is interesting. It turns put the result allows to classify the maximal Cohen-Macaulay modules of covariants for the action of $SL(n)$ on the set of $m$ $n$-vectors. It also allows to describe the equivariant simple $D$-modules, where $D$ is the Weyl algebra of differential operators on the space of $m\times n$ matrices. This is a joint work with Claudiu Raicu and Emily Witt. The relevant references are arXiv 1305.1719 and arXiv 1309.0617.