Towards the tropicalization of reductive groups

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero carrying the trivial valuation. In this article we discuss two candidates for what could be the tropicalization of $G$. Our first suggestion is the extended affine building associated to $G$. This perspective makes makes use of Berkovich's embedding of the extended affine building into the Berkovich analytic space $G^{\textrm{an}}$ and expands on work of Mumford by associating a toroidal bordification of $G$ to the choice of stacky fan in the building. We show that the natural retraction onto the building is compatible with the tropicalization map associated to a toroidal bordification. Our second suggestion is a Weyl chamber of $G$, a special instance of spherical tropicalization, where we think of $G$ as a spherical $G\times G$-variety with respect to left-right-multiplication. We show that the spherical tropicalization map may be identified with the toroidal tropicalization map associated to a wonderful compactification of $G$. This map also has a moduli-theoretic interpretation expanding on the compactifications of $G$ as moduli spaces of framed $\mathbb{G}_m$-equivariant principal bundles on chains of projective lines introduced by Martens and Thaddeus.