nLab horocycle correspondence

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)

Related concepts

• twistor correspondence, Penrose transform

• Schubert calculus – Correspndences

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

• David Ben-Zvi, David Nadler, The Character Theory of a Complex Group (arXiv:0904.1247)