Representations of algebraic groups of type $C_n$ with small weight multiplicities

We find lower estimates for the maximal weight multiplicities in irreducible representations of algebraic groups of type $C_n$ in characteristic $p\leq7$. If $n\geq8$ and $p\ne2$, then for an irreducible representation such multiplicity is either at least $n-4-[n]_4$, where $[n]_4$ is the residue of $n$ modulo 4, or all weight multiplicities are equal to 1. For $p=2$ the situation is more complicated and for every $n$ and $l$ there exists a class of representations with the maximal weight multiplicity equal to $2^l$. For symplectic groups in characteristic $p>7$ and spinor groups similar results were obtained earlier. Bibl. – 15 titles.