The maximal discrete extension of the Hermitian modular group

Let $\Gamma_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$, which coincides with normalizer $\Gamma_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$. The description involves $n$-torsion subgroup ideal class This is defined a particular $\widehat{\mathbb{K}}_n$ and can describe ramified primes it. case $n=2$ give explicit description, generalized Atkin-Lehner involutions. Moreover find natural characterization $SO(2,4)$.