Resonances for the Laplacian on Riemannian symmetric spaces: The case of 𝑆𝐿(3,ℝ)/𝕊𝕆(3)

Abstract

Abstract. We show that the resolvent of the Laplacian on SL(3, R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are infinite dimensional irreducible SL(3, R)-representations. We determine their Langlands parameters and wave front sets. Also, we show that precisely one of these representations is unitarizable. Alternatively, they are given by the differential equations which determine the image of the Poisson transform associated with the resonance.

Mathematical study of scattering resonances

“…For a subtle application of that see a recent paper by Datchev [52] where existence of arbitrarily wide resonance free strips for negative curvature perturbations of z → z + 1 \H 2 is established. For a recent analysis of a higher rank symmetric spaces and references to earlier works see Mazzeo-Vasy [177,178] and Hilgert-Pasquale-Przebinda [126].…”

On the space of 𝐾-finite solutions to intertwining differential operators

“…Let Π = {α, β} be the set of simple roots for the positive system ∆ + . We fix α and β in such a way that the root spaces g α and g β are given by g α = CE 12 and g β = CE 23 . We write ρ for half the sum of the positive roots.…”

“…A direct computation shows that (XY + Y X) ⊗ 1 − ρ satisfies the condition (C2), that is, it is annihilated by X α and X β in M ( ρ). (Here one may take X α and X β to be X α = E 12 and X β = E 23 , as g α = CE 12 and g β = CE 23 ; see Section 4.1.) Now the lemma follows.…”

“…5.13], as these two cases are factored through SL(3, R). The spherical representation in (a) is also recently realized in [12] as the range of a residue operator of SL(3, R).…”

Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces

“…In the last thirty years, the problem of the analytic or meromorphic extension of the resolvent of the Laplacian across its continuous spectrum has been studied for several classes of noncompact complete Riemannian manifolds. Examples include asymptotically hyperbolic manifolds, such as the real hyperbolic space and its convex co-compact quotients ( [16,7,5]), manifolds with asymptotically cylindrical ends ( [6,20]), and Riemannian symmetric or locally symmetric spaces of the noncompact type ( [6,7,1,21,17,18,23,19,12,13]).…”