nLab horocycle correspondence




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

G// adGpG ad//BδB\G/B. 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 *δ !:𝒟(B\G/B)𝒟(G/ adG) 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 GG is a simple object-direct summand of a D-module that is in the image of a simple object under this transform. (Ginzburg 89)


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

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