The Geometry of the affine closure of $T^*(\mathrm{SL}_n/U)$

Publication date: 16 December 2021

Abstract: We show that the affine closure

o v e r l i n e T^{*} (m a t h r m {S L}_{n} / U)

has symplectic singularities, in the sense of Beauville. In the special case

n = 3

, we show that the affine closure

o v e r l i n e T^{*} (m a t h r m {S L}_{3} / U)

is isomorphic to the closure

o v e r l i n e {m a t h c a l O}_{e} x t r m m i n

of the minimal nilpotent orbit

m a t h c a l O_{e x t r m m i n}

in

m a t h f r a k {s o}_{8}

. Moreover, the quasi-classical Gelfand-Graev action of the Weyl group

W

on

o v e r l i n e T^{*} (m a t h r m {S L}_{3} / U)

can be identified with the restriction to

o v e r l i n e {m a t h c a l O}_{e} x t r m m i n

of the triality

S_{3}

-action on

m a t h f r a k {s o}_{8}

.

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