Cohomological induction in various categories and the maximal globalization conjecture

1999 ◽  
Vol 96 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Hon-Wai Wong
Keyword(s):  
Cohomological Induction

The Genuine Omega-regular Unitary Dual of the Metaplectic Group

2012 ◽  
Vol 64 (3) ◽  
pp. 669-704
Author(s):  
Alessandra Pantano ◽  
Annegret Paul ◽  
Susana A. Salamanca-Riba
Keyword(s):  
Tensor Product ◽  
Unitary Representations ◽  
Dimensional Representation ◽  
Oscillator Representation ◽  
One Dimensional ◽  
Metaplectic Group ◽  
Unitary Dual ◽  
Cohomological Induction

Abstract We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

scholarly journals Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

2020 ◽  
Vol 32 (4) ◽  
pp. 941-964 ◽  
Author(s):  
Jian Ding ◽  
Chao-Ping Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Parabolic Subgroups ◽  
Dirac Cohomology ◽  
Finiteness Result ◽  
Irreducible Unitary Representations ◽  
Cohomological Induction ◽  
Good Range

AbstractLet G be a connected complex simple Lie group, and let {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of {\widehat{G}^{\mathrm{d}}} come from {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of {\widehat{F}_{4}^{\mathrm{d}}}.

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