# nLab horocycle correspondence

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

In geometric representation theory the horocycle correspondence for $G$ a complex reductive group and $B \subset G$ a Borel subgroup is the correspondence of group quotients given by

$G//_{ad} G \stackrel{p}{\longleftarrow} G_{ad}//B \stackrel{\delta}{\longrightarrow} B \backslash G / B \,.$

This generalizes the Grothendieck-Springer correspondence. (Ben-Zvi & Nadler 09, example 1.2)

The pull-push integral transform of D-modules from right to left through this correspondence is the Harish-Chandra transform

$p_\ast \delta^! \;\colon\; \mathcal{D}(B\backslash G / B) \longrightarrow \mathcal{D}(G/_{ad}G)$

from the Hecke category on the left.

A (unipotent) character sheaf? (Lusztig 85) for $G$ is a simple object-direct summand of a D-module that is in the image of a simple object under this transform. (Ginzburg 89)

## References

Original articles include

• George Lusztig, Character sheaves I. Adv. Math 56 (1985) no. 3, 193-237.

• Victor Ginzburg, Admissible modules on a symmetric space. Orbites unipotentes et représentations, III. Astérisque No. 173-174 (1989), 9–10, 199–255.

Interpretation in the context of extended TQFT is in

Last revised on August 2, 2017 at 13:51:28. See the history of this page for a list of all contributions to it.