Elementary covering numbers in odd-dimensional unitary groups

Abstract

Let $(K,\Delta)$ be a Hermitian form field and $n\geq 3$. We prove that if $\sigma\in U_{2n+1}(K,\Delta)$ is a unitary matrix of level $(K,\Delta)$, then any short root transvection $T_{ij}(x)$ is a product of $4$ elementary unitary conjugates of $\sigma$ and $\sigma^{-1}$. Moreover, the bound $4$ is sharp. We also show that any extra short root transvection $T_i(x,y)$ is a product of $12$ elementary unitary conjugates of $\sigma$ and $\sigma^{-1}$. If the level of $\sigma$ is $(0,K\times 0)$, then any $(0,K\times 0)$-elementary extra short root transvection $T_i(x,0)$ is a product of $2$ elementary unitary conjugates of $\sigma$ and $\sigma^{-1}$.