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

Hilgert, J.; Pasquale, A.; Przebinda, T.

doi:10.1090/ert/506

Resonances for the Laplacian on Riemannian symmetric spaces: The case of $\mathrm {SL}(3,\mathbb {R})/\mathrm {SO}(3)$
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by J. Hilgert, A. Pasquale and T. Przebinda

Represent. Theory 21 (2017), 416-457

DOI: https://doi.org/10.1090/ert/506

Published electronically: October 11, 2017

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Abstract:

We show that the resolvent of the Laplacian on $\mathrm {SL}(3,\mathbb {R})/\mathrm {SO}(3)$ can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb C$. 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 $\mathrm {SL}(3,\mathbb {R})$-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.

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