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We know some nice space-time have a lot of symmetries. It is said that

  • Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$

  • de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and

  • anti-de Sitter spacetime has $$SO(d-1,2)/SO(d-1,1).$$

e.g. see https://physics.stackexchange.com/a/75604/42982

One then is interested in unitary irreducible representations of the space-time symmetry group.

Question: Is this correct that the above is the symmetry of Minkowski, de Sitter spacetime, and anti-de Sitter spacetime? It this the same as the isometry of these spacetimes? How to show this is the complete symmetry?

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