Spherical amoebae and a spherical logarithm map

Let G be a connected reductive algebraic group over C with a maximal compact subgroup K. Let $G/H$ be a (quasi-affine) spherical homogeneous space. In the first part of the paper, following Akhiezer's definition of spherical functions, we introduce a K-invariant map sLogΓ, t: G/H → Rˢ which depends on a choice of a finite set Γ of dominant weights and s = |Γ|. We call sLogΓ, t a spherical logarithm map. We show that when Γ generates the highest weight monoid of $G/H$, the image of the spherical logarithm map parametrizes K-orbits in $G/H$. This idea of using the spherical functions to understand the geometry of the space K G/H of K-orbits in $G/H$ can be viewed as a generalization of the classical Cartan decomposition. In the second part of the paper, we define the spherical amoeba (depending on Γ and t) of a subvariety Y of $G/H$ as sLogΓ, t(Y), and we ask for conditions under which the image of a subvariety Y ⊂ G/H under sLogΓ, t converges, as t → 0, in the sense of Kuratowski to its spherical tropicalization as defined by Tevelev and Vogiannou. We prove a partial result toward answering this question, which shows in particular that the valuation cone is always contained in the Kuratowski limit of the spherical amoebae of $G/H$. We also show that the limit of the spherical amoebae of $G/H$ is equal to its valuation cone in a number of interesting examples, including when $G/H$ is horospherical, and in the case when $G/H$ is the space of hyperbolic triangles.