The big projective module as a nearby cycles sheaf

Campbell, Justin

doi:10.1007/s00029-016-0257-7

Cited by 3 publications

(6 citation statements)

References 6 publications

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“…This generalizes some known relations between Whittaker and tilting/projective modules in the classic BGG category O [81][82][83]. For abelian theories, the conjecture is proven in [84].…”

Section: Jhep10(2016)108supporting

confidence: 65%

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Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

109

229

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Abstract:We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs-and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality -an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

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“…This generalizes some known relations between Whittaker and tilting/projective modules in the classic BGG category O [81][82][83]. For abelian theories, the conjecture is proven in [84].…”

Section: Jhep10(2016)108supporting

confidence: 65%

“…In particular, all projective and tilting modules in O C arise this way. In geometric representation theory, it is known that a non-degenerate Whittaker module over a semisimple Lie algebra can be averaged or degenerated to give the "big" projective module in the BGG category O [81][82][83]. Our analysis of Neumann b.c.…”

Section: Jhep10(2016)108mentioning

confidence: 99%

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

109

229

View full text Add to dashboard Cite

show abstract

“…The canonical map is given in these terms by . It follows from [Cam17, Example 4.3] that is the cohomologically normalized pullback of the unique indecomposable tilting sheaf on which extends . Moreover, the monodromy filtration satisfies , , and , where .…”

Section: First Proof Of Theorem 141mentioning

confidence: 99%

“…In [Cam17], the author introduced a similar construction in the situation of a finite-dimensional flag variety. Namely, we showed that the nearby cycles of a one-parameter family of nondegenerate Whittaker sheaves on a flag variety is the big projective sheaf, which is isomorphic to the tilting extension of the constant perverse sheaf on the big cell.…”

Section: Introductionmentioning

confidence: 99%

Nearby cycles of Whittaker sheaves on Drinfeld’s compactification

Campbell¹

2018

Compositio Math.

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In this article we study the perverse sheaf on Drinfeld's compactification obtained by applying the geometric Jacquet functor (alias nearby cycles) to a nondegenerate Whittaker sheaf. Namely, we describe its restrictions along the defect stratification in terms of the Langlands dual Lie algebra, in particular showing that this nearby cycles sheaf is tilting. We also describe the subquotients of the monodromy filtration using the Picard-Lefschetz oscillators introduced by S. Schieder, including a proof that the subquotients are semisimple without reference to any theory of weights. This argument instead relies on the action of the Langlands dual Lie algebra on compactified Eisenstein series (or what is essentially the same, cohomology of quasimaps into the flag variety) constructed by Feigin, Finkelberg, Kuznetsov, and Mirković.

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“…There have been several constructions of tilting objects as sheaves or D-modules, including certain averaging or limiting process (c.f. [7], [14], [3], [5]). In this paper, we construct the tilting object corresponding to the open Schubert cell, often referred as the big tilting, as a holomorphic Lagrangian brane in the Fukaya category F (T * B).…”

Section: Introductionmentioning

confidence: 99%