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Symplectomorphism group of $T^*(G_\mathbb{C}/B)$ and the braid group I: a homotopy equivalence for $G_\mathbb{C}=SL_3(\mathbb{C})$

Xin Jin

Abstract

For a semisimple Lie group $G_\mathbb{C}$ over $\mathbb{C}$, we study the homotopy type of the symplectomorphism group of the cotangent bundle of the flag variety and its relation to the braid group. We prove a homotopy equivalence between the two groups in the case of $G_\mathbb{C}=SL_3(\mathbb{C})$, under the $SU(3)$-equivariancy condition on symplectomorphisms.

Symplectomorphism group of $T^*(G_\mathbb{C}/B)$ and the braid group I: a homotopy equivalence for $G_\mathbb{C}=SL_3(\mathbb{C})$

Abstract

For a semisimple Lie group over , we study the homotopy type of the symplectomorphism group of the cotangent bundle of the flag variety and its relation to the braid group. We prove a homotopy equivalence between the two groups in the case of , under the -equivariancy condition on symplectomorphisms.

Paper Structure

This paper contains 22 sections, 26 theorems, 80 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation and set-up
  3. Main Theorem
  4. Acknowledgement
  5. Preliminaries and Set-ups
  6. Set-up for the symplectomorphism group
  7. Motivation for the definition
  8. Definition of $\mathrm{Sympl}_{\mathcal{Z}}^{G}(T^*\mathcal{B})$
  9. Moment maps
  10. Moment maps for the $G_\mathbb{C}$-action and $G$-action on $T^*\mathcal{B}$
  11. Construction of the surjective homomorphism $\beta_G:\mathrm{Sympl}_\mathcal{Z}^{G}(T^*\mathcal{B})\rightarrow B_{\mathbf{W}}, G=SU(n)$
  12. A study of certain loci in $\mu^{-1}(\mathrm{diag}(1,-1,0,...,0))$
  13. Construction of $\beta_G: \mathrm{Sympl}^G_\mathcal{Z}(T^*\mathcal{B})\rightarrow B_{\mathbf{W}}$
  14. Fiberwise Dehn twists and the surjectivity of $\beta_G$
  15. $\beta_G$ is a homotopy equivalence for $G=SU(3)$
  16. ...and 7 more sections

Key Result

Theorem 1.4

(1) There is a natural surjective group homomorphism (2) $\beta_G$ is a homotopy equivalence for $G=SU(2), SU(3)$.

Figures (5)

  • Figure 1: The fibers of $\mu$ over a Weyl chamber $W$ for $G=SU(2)$ and the Dehn twist. The Dehn twist moves the lower straight red line to the upper red curve.
  • Figure 2: The reduced spaces over $\mathbb{R}_+\cdot p, p=\mathrm{diag}(1,0,-1)\in i\mathfrak{t}$, and an illustration of one symplectomorphism for $G=SU(3)$. The reduced spaces have been rescaled to be of the same size. The union of the two arcs in the leftmost reduced space is the projection of the subregular Springer fibers. The symplectomophism restricts to the identity near the zero section of $T^*\mathcal{B}$, so fixes every point in the reduced spaces near the vertex of $W$. The arcs in the two reduced spaces on the right illustrate how the symplectomorphism moves the Springer fibers.
  • Figure 3: The transformation of the triangle $\overline{Q_1Q_2Q_3}$ under $\tau_{\alpha_1}$.
  • Figure 4: Local picture of $\Sigma_0$.
  • Figure 5: A vector field $X_\sigma$ on $\mathcal{U}_\sigma$ shrinking it towards $\sigma$.

Theorems & Definitions (57)

  • Conjecture 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 47 more