Abstract
This dissertation is concerned with the lower bound for the first eigenvalue λ₁(T) of Laplace operator Δ on a locally symmetric space T G / K, where G = SO_2n+1(C) with n ≥ 5, T is a lattice in G and K is a maximal compact subgroup of G. It is well known that Casimir operator C acts by a scalar -λ_π on each irreducible admissible representation π of G by Diximir's Schur Lemma. We will show that there exists a spectral gap between the greatest lower bound of λ₁(T) for those lattices T arising from some imaginary quadratic field and the infimum λ₁(G) of λ_π, when π passes through all those nontrivial irreducible spherical unitary representation of G.