Representations of dimensions $(p^n\pm 1)/2$ of the symplectic group of degree $2n$ over a field of characteristic $p$

Abstract

The irreducible representations ${\varphi }_n^1$ and ${\varphi }_n^2$ of the symplectic group $G_n=S{p}_{2n}(P)$ over an algebraically closednfield $P$ of characteristic $p>2$ with highest weights ${\omega }_{n-1}+\frac{p-3}{2}{\omega }_n$ and $\frac{p-1}{2}{\omega }_n$, respectively, are investigated. It is proved that the dimension of ${\varphi }_n^i$ ($i=1,2$) is equal to $(p^n+(-1{)}^i)/2$, all weight multiplicities of these representations are equal to $1$, their restrictions to the group $G_k$ naturally embedded into $G_n$ are completely reducible with irreducible constituents ${\varphi }_k^1$ and ${\varphi }_k^2$, and their restrictions to $S{p}_{2n}(p)$ can be obtained as the result of the reduction modulo $p$ of certain complex irreducible representations of the group $S{p}_{2n}(p)$.