Journal of Lie Theory 35 (2025), No. 1, 083--100
Copyright Heldermann Verlag 2025
The Affine Closure of T*(SLn/U)
Boming Jia
Yau Mathematical Sciences Center, Tsinghua University, Jingzhai, Beijing, P. R. China
jiabm@tsinghua.edu.cn
[Abstract-pdf]
We show that the affine closure $\overline{T^*(\mathrm{SL}_n/U)}$ has symplectic singularities, in the sense of Beauville. In the special case $n=3$, we show that the affine closure $\overline{T^*(\mathrm{SL}_3/U)}$ is isomorphic to the closure $\overline{\mathcal{O}}_\textrm{min}$ of the minimal nilpotent orbit $\mathcal{O}_{\textrm{min}}$ in $\mathfrak{so}_8$. Moreover, the quasi-classical Gelfand-Graev action of the Weyl group $W$ on $\overline{T^*(\mathrm{SL}_3/U)}$ can be identified with the restriction to $\overline{\mathcal{O}}_\textrm{min}$ of E.\,Cartan's triality action on $\mathfrak{so}_8$.
Keywords: Symplectic singularities, triality action.
MSC: 20G05, 17B10, 14M15.
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