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Unitary Shimura correspondences for split real groups. (English) Zbl 1114.22009
Let \(G\) be the split real form of a simple and simply connected algebraic group. Then \(G\) has a unique nontrivial two-fold cover \(\widetilde G\), which is a nonlinear group. The paper under review establishes a natural correspondence between the pseudospherical complementary series CS\(({\tilde \delta},{\widetilde G})\) of \(\widetilde G\) and the pseudospherical complementary series CS\((\delta^\ell,G^\ell)\) of \(G^\ell\). Here, \(G^\ell\) is a linear group locally isomorphic to the dual group of \(G\). It is conjectured that this correspondence induces a bijection of CS\(({\tilde \delta},{\widetilde G})\) and CS\((\delta^\ell,G^\ell)\). In case \(G\) is simply laced or of type \(G_2\), one has \(G^\ell=G\) and this correspondence is simply \(J_{\widetilde G}({\tilde \delta},\nu)\leftrightarrow J_G(2\nu)\) (the spherical representation of \(G\)). This bijection is known a posterior for \(SL(n)\) and \(G_2\) by classification of the unitary duals. The main theorems proved in the paper say that this correspondence induces an injection from CS\(({\tilde \delta},{\widetilde G})\) to CS\((\delta^\ell,G^\ell)\), and it is a bijection when \(G= \)Sp\((2n,{\mathbf R})\). The results are important for studying unitary representations of semisimple Lie groups and have applications for studying automorphic forms.

MSC:
22E46 Semisimple Lie groups and their representations
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