Resonances of the Laplace operator on homogeneous vector bundles on symmetric spaces of real rank-one

Residue representations -- the rank one case

“…In [Rob22], the residue representations have been completely determined in terms of their Langlands parameters under the assumption that the resonances arise from the poles the trivial Plancherel density. In this section, we present an algorithm which allows us to compute the Langlands parameters of each residue representation, the Gelfand-Kirillov dimension of the space of these representations and their wave front set under no restriction on τ .…”

“…(2) Use [Mia79] or [Rob22] to compute the poles of the Plancherel density associated with σ. We denote them by ±λ σ k ∈ 1 2 , k ∈ + , where λ σ k < 0.…”

“…In this case, K = Spin(2n) and M = Spin(2n − 1). We are not considering G = Spin(2n + 1, 1), because there are no resonance in this case (see [Rob22]) .…”

“…For the Laplacian acting on line bundles over complex hyperbolic spaces, the resonances has been completely determined by Will in [Wil03]. The complete list of the resonances for the Laplacian acting on sections on E τ , when G/K if of rank one and τ is arbitrary was determined in [Rob22].…”

“…Moreover, in [Wil03], this has been also done for the case of SL(2, Ê), in the (line) bundle case. For the general rank one case, the residue representations were determined in [Rob22] under the assumption that the resonances arise from the poles the trivial Plancherel density, which implies that τ has to occur in the spherical principal series representations.…”