Spherical amoebae and a spherical logarithm map

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$ 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 $sLog_{\Gamma, t}: G/H \to \mathbb{R}^s$ which depends on a choice of a finite set $\Gamma$ of dominant weights and $s = |\Gamma|$. We call $sLog_{\Gamma, t}$ a spherical logarithm map. We show that when $\Gamma$ 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 $K \backslash G/H$ 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 $\Gamma$ and $t$) of a subvariety $Y$ of $G/H$ as $sLog_{\Gamma, t}(Y)$, and we ask for conditions under which the image of a subvariety $Y \subset G/H$ under $sLog_{\Gamma, t}$ converges, as $t \to 0$, 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.