Some aspects of harmonic analysis on locally symmetric spaces related to real-form embeddings

Let $$G =\mathrm{{ SO}}_{3}(\mathbb{C})$$ , Γ = SO3(ℤ[i]), K = SO(3), and let X be the locally symmetric space Γ ∖ G ∕ K. In this paper, we present a relationship between the heat kernel on SL3(ℂ) and SO3(ℂ). We write down explicit equations defining a fundamental domain for the action of Γ on G ∕ K. The fundamental domain is well adapted for studying the theory of Γ-invariant functions on G ∕ K. We write down equations defining a fundamental domain for the subgroup $${\Gamma }_{\mathbb{Z}} =\mathrm{ SO}{(2,1)}_{\mathbb{Z}}$$ of Γ acting on the symmetric space $${G}_{\mathbb{R}}/{K}_{\mathbb{R}}$$ , where $${G}_{\mathbb{R}}$$ is the split real form SO(2, 1) of G and $${K}_{\mathbb{R}}$$ is its maximal compact subgroup SO(2). We formulate a simple geometric relation between the fundamental domains of Γ and $${\Gamma }_{\mathbb{Z}}$$ so described. Both the formula for the heat kernel and the fundamental domains are designed to aid in a detailed study of the spectral theory of X and the embedded subspace $${X}_{\mathbb{R}} = {\Gamma }_{\mathbb{Z}}\setminus {G}_{\mathbb{R}}/{K}_{\mathbb{R}}$$ .