Analysis and Probability Seminar

We consider the joint spectrum of the algebra of invariant differential operators on a general higher rank locally symmetric space. We prove that the supremum of the real part of the joint spectrum with respect to a natural polyhedral norm is exactly described by the growth indicator function of the discrete subgroup. This generalizes a classical result by Elstrod, Patterson and Corlette which states that in the rank one case the bottom of the Laplace spectrum is determined by the critical exponent of the discrete subgroup.