Representation theoretic rigidity in \(PSL(2,\mathbb{R})\)
Let \(G\) be a connected simple Lie group with trivial center, let \(\Gamma\) be an abstract group, and let \(\iota_ 1\) and \(\iota_ 2\) be inclusions of \(\Gamma\) in \(G\). Assume that each of the images \(\iota_ j(\Gamma)\) is a lattice subgroup, meaning that \(\iota_ j(\Gamma)\) is discrete and that the \(G\)-invariant measure on \(G/\iota_ j(\Gamma)\) has total finite mass. One says that \(\iota_ 1\) and \(\iota_ 2\) are equivalent if there is some automorphism \(\rho\) of \(G\) such that \(\iota_ 2=\rho\circ \iota_ 1\). If \(G\) is not isomorphic to \(PSL_ 2(\mathbb{R})\) then the Mostow rigidity theorem says that \(\iota_ 1\) and \(\iota_ 2\) are equivalent. This result fails for \(PSL_ 2(\mathbb{R})\). The authors prove for \(PSL_ 2(\mathbb{R})\) the following Theorem. Suppose that \(\pi_ 1\) and \(\pi_ 2\) are irreducible representations of \(PSL_ 2(\mathbb{R})\), not in the discrete series. Then \(\pi_ 1\circ \iota_ 1\) and \(\pi_ 2\circ\iota_ 2\) are equivalent representations of \(\Gamma\) if and only if \(\iota_ 1\) and \(\iota_ 2\) are equivalent inclusions and \(\pi_ 1\) and \(\pi_ 2\) are equivalent representations of \(PSL_ 2(\mathbb{R})\). The essential step in the proof of this theorem is an analytic criterion for the equivalence of \(\iota_ 1\) and \(\iota_ 2\).
scientific article; zbMATH DE number 205190
scientific article