First eigenvalue of Laplace operator on locally symmetric space

Abstract

This paper is concerned with the lower bound for the first positive eigenvalue $\lambda _{1}(\Gamma )$ of the Laplace operator $\bigtriangleup$ on the space of square integrable functions on some locally symmetric space $\Gamma \backslash G / K$, where $G=SO_{2n+1}(\mathbb {C})$, $K$ is a maximal compact subgroup of $G$ and $\Gamma$ is a lattice in $G$ which arises from some imaginary quadratic field. It is well-known that the Casimir operator $\fancyscript{C}$ acts by a scalar $-\lambda _{\pi }$ on every irreducible admissible representation $\pi$ of $G$ by Diximir’s Schur Lemma. In this paper, we prove that when $n\ge 5$, there exists a spectral gap between $\lambda _{1} (\Gamma )$ and the infimum $\lambda _{1}(G)$ of $\lambda _{\pi }$ when $\pi$ passes through all non-trivial irreducible spherical unitary representations of $G$.