Resonances and residue operators for symmetric spaces of rank one

AbstractLet X=G/K be a rank-one Riemannian symmetric space of the noncompact type and let −Δ be the Laplace–Beltrami operator on X. We show that the resolvent operator R(z) of Δ can be meromorphically continued across the spectrum and explicitly determine the poles, i.e. the resonances. Further we describe the residue operators in terms of finite-dimensional spherical representations of G. The result answers a question posed by M. Zworski in [M. Zworski, What are the residues of the resolvent of the Laplacian on non-compact symmetric spaces? Seminar held at the IRTG Summer School 2006, Schloss Reisensburg, 2006. Available at http://math.berkeley.edu/~zworski/reisensburg.pdf]. The rank of the residue operators is derived from a restricted root version of the Weyl dimension formula for spherical highest weight representations which we prove for arbitrary symmetric spaces of the noncompact type