A fundamental domain of Ford type for some subgroups of the orthogonal group

Abstract

We initiate a study of the spectral theory of the locally symmetric space $X=\Gamma\backslash G/K$, where $G=SO(3,Complex)$, $\Gamma=SO(3,Z[i])$, $K=SO{3}$. We write down explicit equations defining a fundamental domain for the action of $\Gamma$ on $G/K$. The fundamental domain is well-adapted for studying the theory of $\Gamma$-invariant functions on $G/K$. We write down equations defining a fundamental domain for the subgroup $\Gamma_Z=\SO(2,1)_Z$ of $\Gamma$ acting on the symmetric space $G_R/K_R$, where $G_R$ is the split real form $\SO(2,1)$ of $G$ and $K_R$ is its maximal compact subgroup $\SO(2)$. We formulate a simple geometric relation between the fundamental domains of $\Gamma$ and $\Gamma_Z$ so described. We then use the previous results compute the covolumes of of the lattices $\Gamma$ and $\Gamma_Z$ in $G$ and $G_R$.