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Maffei's action and symplectic Springer action for quiver varieties

Yaochen Wu

Abstract

We examine the relationship between the actions of two Weyl groups on the cohomology of a smooth quiver variety: the Maffei's action of the Weyl group associated to the quiver, and the symplectic Springer action of the Namikawa-Weyl group of the affine quiver variety. We show there is a natural map from the former group to the latter, which is an embedding in favorable situations, and this map intertwines their actions on the cohomology. This answers a question raised in arXiv:1904.10497.

Maffei's action and symplectic Springer action for quiver varieties

Abstract

We examine the relationship between the actions of two Weyl groups on the cohomology of a smooth quiver variety: the Maffei's action of the Weyl group associated to the quiver, and the symplectic Springer action of the Namikawa-Weyl group of the affine quiver variety. We show there is a natural map from the former group to the latter, which is an embedding in favorable situations, and this map intertwines their actions on the cohomology. This answers a question raised in arXiv:1904.10497.
Paper Structure (18 sections, 27 theorems, 60 equations)

This paper contains 18 sections, 27 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Quiver varieties
  3. Quiver Weyl group
  4. Poisson deformations and Namikawa-Weyl groups
  5. Goal and structure of the paper
  6. The groups $W$ and $W^{\mathrm{N}}$
  7. Generators of $W(v,w)$
  8. Symplectic leaves of quiver varieties
  9. The canonical decomposition
  10. Examples
  11. The symplectic Springer action
  12. Maffei's action and tautological line bundles
  13. Maffei's isomorphism
  14. Tautological line bundles
  15. Maffei's action on the cohomology
  16. ...and 3 more sections

Key Result

Theorem 1.5

namikawa2011poisson There is a commutative diagram where $B_Y = H^2(Y,\mathbb{C})$, and $m_X$, $m_Y$ are universal graded Poisson deformations of $X$ and $Y$ respectively, with $m_X^{-1}(0) = X, m_Y^{-1}(0) = Y$.

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: beauville2000symplectic
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6: losev2022deformations
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10: namikawa2010poisson
  • ...and 56 more