nLab Zuckerman induction

Idea

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

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

This functor has a right adjoint coinduced representation functor

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

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