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Open Access Publications from the University of California

Symplectic approaches in geometric representation theory

  • Author(s): Jin, Xin
  • Advisor(s): Nadler, David
  • et al.
Abstract

We study various topics lying in the crossroads of symplectic topology and geometric representation theory, with an emphasis on understanding central objects in geometric representation theory via approaches using Lagrangian branes and symplectomorphism groups.

The first part of the dissertation focuses on a natural link between perverse sheaves and holomorphic Lagrangian branes.

For a compact complex manifold $X$, let $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotangent bundle of $X$. We prove that under the quasi-equivalence between $D^b_c(X)$ and $DFuk(T^*X)$ established by Nadler and Zaslow, holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.

The second part is motivated from general features of the braid group actions on derived category of constructible sheaves. 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.

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