Journal of Lie Theory

no. 3
Isometric Actions of Quaternionic Symplectic Groups
Journal of Lie Theory, Volume 29 (2019) no. 3, pp. 755-786
Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by studying a natural non-associative algebra $\mathfrak{m}$ coming from the affine structure of $Sp(1) \backslash H$. We compute the automorphism group of $\mathfrak{m}$ and as a consecuence of this, we are able to compute the isometry group of $Sp(1) \backslash H$ at least up to connected components. Using these results, we obtain a uniqueness result on the structure of $Sp(1) \backslash H$ together with an isometric left $Sp(k,l)$-action and classify its finite volume quotients up to finite coverings. Finally, we classify arbitrary isometric actions of $Sp(k,l)$ into connected, complete, analytic, pseudo-Riemannian manifolds of dimension bounded by $\textrm{dim}(Sp(1) \backslash H)$ that admit a dense orbit.
DOI: 10.5802/jolt.1077
Classification: 22F30, 17B40, 53C24
Keywords: Pseudo-Riemannian manifolds, rigidity results, non-compact quaternionic symplectic groups 
@article{JOLT_2019_29_3_a6,
     author = {M. Sedano-Mendoza},
     title = {Isometric {Actions} of {Quaternionic} {Symplectic} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {755--786},
     year = {2019},
     volume = {29},
     number = {3},
     doi = {10.5802/jolt.1077},
     zbl = {1428.22017},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1077/}
}
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M. Sedano-Mendoza. Isometric Actions of Quaternionic Symplectic Groups. Journal of Lie Theory, Volume 29 (2019) no. 3, pp. 755-786. doi: 10.5802/jolt.1077

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