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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:
 2.2e+47 Semisimple Lie groups and their representations
Keywords:
complementary series
Full Text:
References:
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