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2022 | OriginalPaper | Chapter

The Semi-infinite Intersection Cohomology Sheaf-II: The Ran Space Version

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Abstract

This chapter is a sequel to [Ga1]. We define the semi-infinite category on the Ran version of the affine Grassmannian and study a particular object in it, denoted \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\), which we call the semi-infinite intersection cohomology sheaf.
Unlike the situation of [Ga1], this \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\) is defined as the middle of extension of the constant (more precisely, dualizing) sheaf on the basic stratum, in a certain t-structure. We give several explicit descriptions and characterizations of \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\): we describe its !- and *- stalks; we present it explicitly as a colimit; we relate it to the IC sheaf of Drinfeld’s relative compactification \(\overline {\operatorname {Bun}}_N\); we describe \(\operatorname {IC}^{\frac {\infty }{2}}_{\operatorname {Ran}}\) via the Drinfeld–Plucker formalism.

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Appendix
Available only for authorised users
Footnotes
1
Technically, not constant but rather dualizing.
 
2
Note that even though the index category (i.e., \((\operatorname {Sch}^{\operatorname {aff}}_{\operatorname {ft}})_{/{\mathcal {Y}}}\)) is ordinary, the above limit is formed in the -category \(\operatorname {DGCat}\). This is how -categories appear in this paper.
 
3
The notion of universal local contractibility is recalled in Sect. A.1.8.
 
4
In the formula below \(-|{ }_{(X^\lambda \times X^I)^{\subset }}\) denotes the !-pullback along the projection (Xλ × XI)→ XI.
 
5
We are grateful to Lin Chen for pointing out a mistake in the statement of Proposition 2.5.3 in the previous version of the paper. The corrected argument is due to him.
 
6
Braden’s theorem extends from schemes to ind-schemes by an easy colimit argument.
 
7
The corresponding assertion would be false for the corresponding embedding \(\operatorname {SI}^{\leq 0}_{\operatorname {Ran}}\subset \operatorname {Shv}(\overline {S}{ }^0_{\operatorname {Ran}})\); this is a geometric counterpart of the fact that the local field is not compact, while the quotient of adeles by principal adeles is compact.
 
8
See Sect. A.2.4, where this notion is recalled.
 
9
Note also that the fully faithfulness of \((\operatorname {pr}^\lambda _{\operatorname {Ran}})^!\) has been already stated in Lemma 2.3.3; however, the argument given below will give an alternative proof of this fact.
 
10
The formalism described in this subsection (as well as the term) was suggested by S. Raskin.
 
Literature
[Bar]
go back to reference J. Barlev, Moduli spaces of generic data, arXiv:1204.3469. J. Barlev, Moduli spaces of generic data, arXiv:1204.3469.
[BFGM]
go back to reference A. Braverman, M. Finkelberg, D. Gaitsgory and I. Mirkovic, Intersection cohomology of Drinfeld compactifications, Selecta Math. (N.S.) 8 (2002), 381–418. A. Braverman, M. Finkelberg, D. Gaitsgory and I. Mirkovic, Intersection cohomology of Drinfeld compactifications, Selecta Math. (N.S.) 8 (2002), 381–418.
[BG2]
go back to reference A. Braverman and D. Gaitsgory, Deformations of local systems and Eisenstein series, GAFA 17 (2008), 1788–1850.MathSciNetMATH A. Braverman and D. Gaitsgory, Deformations of local systems and Eisenstein series, GAFA 17 (2008), 1788–1850.MathSciNetMATH
[DrGa]
[DS]
go back to reference V. Drinfeld and C. Simpson, B-Structures on G-bundles and Local Triviality, Mathematical Research Letters 2 (1995), 823–829.MathSciNetCrossRef V. Drinfeld and C. Simpson, B-Structures on G-bundles and Local Triviality, Mathematical Research Letters 2 (1995), 823–829.MathSciNetCrossRef
[FGV]
go back to reference E. Frenkel, D. Gaitsgory and K. Vilonen, Whittaker patterns in the geometry of moduli space of bundles on curves, Annals of Math. 153 (2001), no. 3, 699–748.MathSciNetCrossRef E. Frenkel, D. Gaitsgory and K. Vilonen, Whittaker patterns in the geometry of moduli space of bundles on curves, Annals of Math. 153 (2001), no. 3, 699–748.MathSciNetCrossRef
[Ga1]
go back to reference D. Gaitsgory, The semi-infinite intersection cohomology sheaf, arXiv:1703.04199, to appear in AIM. D. Gaitsgory, The semi-infinite intersection cohomology sheaf, arXiv:1703.04199, to appear in AIM.
[Ga2]
go back to reference D. Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve, arXiv:1505.02331. D. Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve, arXiv:1505.02331.
[Ga4]
go back to reference D. Gaitsgory, The local and global versions of the Whittaker category, arXiv: 1811.02468. D. Gaitsgory, The local and global versions of the Whittaker category, arXiv: 1811.02468.
[GR]
go back to reference D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry, volume I, AMS (2017). D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry, volume I, AMS (2017).
[Lu1]
go back to reference J. Lurie Higher Topos Theory, Princeton University Press (2009). J. Lurie Higher Topos Theory, Princeton University Press (2009).
Metadata
Title
The Semi-infinite Intersection Cohomology Sheaf-II: The Ran Space Version
Author
Dennis Gaitsgory
Copyright Year
2022
DOI
https://doi.org/10.1007/978-3-030-82007-7_6

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