nLab
Zuckerman induction

Contents

Idea

Zuckerman induction is a special case of co-induced representations:

For 𝔤\mathfrak{g} a semisimple Lie algebra and KK a suitable algebraic? group, write (𝔤,K)Mod(\mathfrak{g}, K)Mod for the category of Harish-Chandra modules over (𝔤,K)(\mathfrak{g}, K). Then for i:TKi \colon T \hookrightarrow K an algebraic subgroup there is the corresponding forgetful functor i *:(𝔤,K)Mod(𝔤,T)Mod i^* \colon (\mathfrak{g}, K)Mod \to (\mathfrak{g}, T)Mod which restricts the representation along the inclusion.

This functor has a right adjoint coinduced representation functor

i *:(𝔤,T)Mod(𝔤,K)Mod i_* \colon (\mathfrak{g}, T)Mod \to (\mathfrak{g}, K)Mod

and this is called the Zuckerman functor. (MP, 1).

The discussion of the derived functor of this is sometimes called cohomological induction.

References

  • Dragan Miličić, Pavle Pandžić, Equivariant derived categories, Zuckerman functors and localization, from Geometry and Representation Theory of real and p-adic Lie Groups , J. Tirao, D. Vogan, J.A. Wolf, editors, Progress in Mathematics 158, Birkhäuser, Boston, 1997, 209-242, pdf

Created on November 25, 2012 at 04:37:19. See the history of this page for a list of all contributions to it.