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Noncommutative resolutions of affine Schubert varieties in type A and canonical bases

  • Ilya Dumanski
  • Vasily Krylov
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Abstract

Given a resolution Gr̃underlineλ → Gr̅λ of an affine Schubert variety for GLn, we define its noncommutative version -- a sheaf of algebras on Gr̅λ, derived equivalent to Gr̃underlineλ as well as its Steinberg versions in both zero and positive characteristics. This, in particular, allows us to define the perversely-exotic t-structure on the derived category of equivariant coherent sheaves on Gr̃underlineλ, analogously to Bezrukavnikov--Mirković in the case of Springer resolution. We study the basis of classes of irreducible objects in the equivariant K-theory, and explicitly identify it with the (parabolic) Kazhdan--Lusztig canonical basis in a certain cell quotient. This allows us to relate it to the canonical basis for the quantum affine group. In the course of the proof, we establish some properties of coherent-constructible equivalences.

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Cite

BibTeXRIS
BibTeX
@misc{dumanski2026noncommutative,
  title = {{Noncommutative resolutions of affine Schubert varieties in type A and canonical bases}},
  author = {Ilya Dumanski and Vasily Krylov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.15981},
  eprint = {2606.15981},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15981},
}
RIS
TY  - UNPB
TI  - Noncommutative resolutions of affine Schubert varieties in type A and canonical bases
AU  - Ilya Dumanski
AU  - Vasily Krylov
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2606.15981
UR  - https://arxiv.org/abs/2606.15981
AB  - Given a resolution Gr̃^underlineλ → Gr̅^λ of an affine Schubert variety for GL_n, we define its noncommutative version -- a sheaf of algebras on Gr̅^λ, derived equivalent to Gr̃^underlineλ as well as its Steinberg versions in both zero and positive characteristics. This, in particular, allows us to define the perversely-exotic t-structure on the derived category of equivariant coherent sheaves on Gr̃^underlineλ, analogously to Bezrukavnikov--Mirković in the case of Springer resolution. We study the basis of classes of irreducible objects in the equivariant K-theory, and explicitly identify it with the (parabolic) Kazhdan--Lusztig canonical basis in a certain cell quotient. This allows us to relate it to the canonical basis for the quantum affine group. In the course of the proof, we establish some properties of coherent-constructible equivalences.
ER  -