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

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 $SU(3)$-equivariance condition on symplectomorphisms.

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Received 28 December 2015

Accepted 11 July 2018

Published 26 July 2019