The maximal discrete extension of the Hermitian modular group

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