Spherical Tropicalization

We extend tropicalization of subvarieties of algebraic tori over a trivially valued algebraically closed field to subvarieties of spherical homogeneous spaces. We show the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, we show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. We provide examples of tropicalization of subvarieties of GL(n), SL(n), and PGL(n).