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Involutions on the Affine Grassmannian and Moduli Spaces of Principal Bundles
by Anthony Henderson

Vol. 13 No. 1 (2018) P.43~P.97
 DOI: https://doi.org/10.21915/BIMAS.2018103 10.21915/BIMAS.2018103

ABSTRACT

Let $G$ be a simply connected semisimple group over $\mathbb{C}$. We show that a certain involution of an open subset of the affine Grassmannian of $G$, defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of $\mathrm{SL}(2)$ on the framed moduli space of $\mathbb{G}_m$-equivariant principal $G$-bundles on $\mathbb{P}^2$. As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms $N\to G$ where $N$ is the normalizer of $\mathbb{G}_m$ in $\mathrm{SL}(2)$. When $G=\mathrm{SL}(r)$, the strata are Nakajima quiver varieties $\mathfrak{M}_0^{\mathrm{reg}}(\mathbf{v},\mathbf{w})$ of type D.

KEYWORDS
Affine Grassmannian, nilpotent cone, moduli space, quiver variety.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: Primary 14J60; Secondary 14M15, 17B08.

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