Spherical amoebae and a spherical logarithm map

Let $Y\subset(\mathbb{C}^{*})^{n}$ be an algebraic subvariety of the $n$-dimensional complex algebraic torus. The (classical) amoebae $\mathcal{A}_{t}(Y)$ of $Y$, for a varying real parameter $t$, have been much studied in the past few decades in several contexts, including in tropical and non-Archimedean geometry. The amoeba $\mathcal{A}_{t}(Y)$ is defined to be the image of $Y$ under the logarithm map $\operatorname{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ defined for fixed $t>0$ by

$$\operatorname{Log}_{t}(x_{1},\ldots,x_{n}):=(\log_{t}|x_{1}|,\ldots,\log_{t}|x_{n}|).$$

Note that the map $\operatorname{Log}_{t}$ is invariant under the action of the compact torus $(S^{1})^{n}\subset(\mathbb{C}^{*})^{n}$. In fact, with respect to the isomorphism $(x_{1},\ldots,x_{n})\mapsto((\theta_{1},\ldots,\theta_{n}),(r_{1},\ldots,r_{n}))$ where $x_{i}=r_{i}e^{i\theta_{i}}$ between $(\mathbb{C}^{*})^{n}$ and $(S^{1})^{n}\times\mathbb{R}_{>0}^{n}$, the logarithm map $\operatorname{Log}_{t}$ can be seen to be the projection onto the second component(s), followed by the usual logarithm function $r_{i}\mapsto\log_{t}r_{i}$.

For an algebraic subvariety $Y$ in $(\mathbb{C}^{*})^{n}$, it is well-known that the amoeba $\mathcal{A}_{t}(Y)$ stretches to infinity in several directions, also known as its tentacles. These tentacles indicate the directions in which $Y$ approaches infinity, or in other words, they parametrize the orbits at infinity in its tropical compactification. The precise statement is as follows (cf. [Jonsson]): as $t\to 0$, $\mathcal{A}_{t}(Y)$ converges to the tropical variety $\operatorname{trop}(Y)$ as subsets in $\mathbb{R}^{n}$, where convergence is defined here in the sense of Kuratowksi (see Definition 3.3). The theory of amoebae and tropicalization connects classical complex geometry with tropical geometry, which studies piecewise-linear objects in Euclidean space, and the study of these interactions is an active research area.

In this paper, we take initial steps toward generalizing the above to the non-abelian setting. More precisely, we choose to view the complex torus $(\mathbb{C}^{*})^{n}$ as a special case of a spherical homogeneous space in which the group is abelian. We then see whether, or how, the abelian story (as briefly outlined above) might be generalized for non-abelian spherical homogeneous spaces. Before further details, we first briefly recall the setting. Let $G$ be a complex connected reductive algebraic group. A homogeneous space $G/H$ of $G$ is called spherical if a Borel subgroup $B$ of $G$ has a dense orbit, with respect to the usual action by left multiplication. (See Section 1 for details.) Thus, the complex torus $(\mathbb{C}^{*})^{n}$ is the special case in which $G=(\mathbb{C}^{*})^{n}$ is the torus itself, $H=\{1\}$ is the trivial subgroup, and the Borel subgroup $B$ is again the torus $(\mathbb{C}^{*})^{n}$ itself.

One of our fundamental motivations for this paper is the desire to better understand the $K$-orbit space of a spherical homogeneous space $G/H$ as above, where $K$ is a maximal compact subgroup of $G$. Indeed, in the abelian case, the logarithm map $\operatorname{Log}_{t}$ can be viewed as the quotient map of $(\mathbb{C}^{*})^{n}$ by its maximal compact subgroup $(S^{1})^{n}$, and we can view $\mathbb{R}^{n}_{>0}$ as the parameter space of the $(S^{1})^{n}$-orbits in $(\mathbb{C}^{*})^{n}$. Similarly we may ask whether there exists a non-abelian, or spherical, analogue of the classical logarithm map on $G/H$ which allows us to naturally parametrize the $K$-orbit space $K\backslash G/H$ of $G/H$. This problem can be viewed as a generalization of the classical Cartan and polar decompositions in Lie theory. However, as far as we know, there is no general and concrete description of the $K$-orbit space $K\backslash G/H$ known. (For some important results in this direction see [Knopetal].) A main motivation for the present paper is the belief that there should be a natural isomorphism between $K\backslash G/H$ and the real closure of the valuation cone $\mathcal{V}_{G/H}$ (see Section 1.1 for the definition) of an arbitrary spherical variety $G/H$ (over $\mathbb{C}$). In the abelian case, the valuation cone is precisely $\mathbb{R}^{n}$, so the logarithm $\operatorname{Log}_{t}$ provides this natural isomorphism. We think of this paper as taking some steps forward in the search of such a natural isomorphism (see also Conjecture 6). (We also note that related ideas have appeared in [Feu].) However, it should be noted that the results we obtain are of a different nature than a direct generalization of the torus case. In contrast to the abelian situation in which $\operatorname{Log}_{t}$ is an isomorphism $K\backslash G/H\cong\overline{\mathcal{V}}_{G/H}$, what we see in our results is that, in the limit as $t\to 0$, the image of our spherical logarithm map $\operatorname{sLog}_{t}$ (to be defined precisely below) converges in an appropriate sense to the valuation cone. In particular, for fixed $t$, our $\operatorname{sLog}_{t}$ does not provide the desired canonical isomorphism.

In addition to the above, we were motivated by the notion of a spherical tropicalization map associated to a spherical homogeneous space $G/H$. Spherical tropical geometry was introduced and developed by Tevelev and Vogiannou [Vogiannou] and by the fourth author and Manon [Kaveh-Manon]. We believe that there should be a theory of spherical amoebae for $G/H$ which correspond to the classical amoebae in the abelian case, such that the “spherical amoebae converge to the spherical tropicalization” in an appropriate sense (see also Remark 3.2).

With these motivations in mind, we now describe more precisely the contributions of this article. Let $G/H$ be a spherical homogeneous space as above. For technical reasons we additionally assume that $G/H$ is quasi-affine. (This is not terribly restrictive, as we explain in Remark 1.6.) Let $\Lambda^{+}_{G/H}$ denote the semigroup of $B$-highest-weights of irreducible $G$-representations appearing in the coordinate ring $\mathbb{C}[G/H]$, considered as a $G$-module in the natural way. Following some rather little-known work of Akhiezer [Akhiezer-rk], we define a $K$-invariant spherical function $\phi_{\lambda}$ on $G/H$ associated to a $B$-weight $\lambda$ for $\lambda\in\Lambda^{+}_{G/H}$. We normalize these spherical functions by setting $\phi_{\lambda}(eH)=1$. Our first observations concerning Akhiezer’s spherical functions is the following; see Propositions 2.6 and 2.11 for precise statements.

Motivated by the above, let $\Gamma=\{\lambda_{1},\ldots,\lambda_{s}\}\subset\Lambda^{+}_{G/H}$. We define $\Phi_{\Gamma}:G/H\to\mathbb{R}_{\geq 0}^{s}$ by:

$$\Phi_{\Gamma}(x)=(\phi_{\lambda_{1}}(x),\ldots,\phi_{\lambda_{s}}(x))$$

(details are in Section 2). For a fixed $t>0$, we then define the spherical logarithm map (depending on $\Gamma$ and with base $t$) to be the map $\operatorname{sLog}_{\Gamma,t}:=\log_{t}(\Phi_{\Gamma}):G/H\to\mathbb{R}^{s}$ (see Definition 2.12). ¹¹1Note that the spherical tropicalization map defined by the fourth author and Manon in [Kaveh-Manon] naturally takes values in a $\mathbb{Q}$-vector space. However, since our spherical logarithm map uses a logarithm, we need to work with $\mathbb{R}$ coefficients. If we assume that the above subset $\Gamma$ generates $\Lambda^{+}_{G/H}$ as a semigroup, then from Theorem 1 it follows that we obtain a parametrization of $K\backslash G/H$, i.e., the points in the image $\operatorname{sLog}_{\Gamma,t}(G/H)$ parameterize $K$-orbits in $G/H$. This is a positive step in the broader program of understanding $K\backslash G/H$. We will have more to say about the relation between the image of $\operatorname{sLog}_{\Gamma,t}$ and the valuation cone $\mathcal{V}_{G/H}$ below.

The spherical functions used to define $\operatorname{sLog}_{\Gamma,t}$ have somewhat subtle properties. The next result, below, gives an inequality involving a product of two spherical functions. The essential point here is that spherical functions are not necessarily multiplicative: given two $B$-weights $\lambda$ and $\mu$, in general we can have $\phi_{\lambda}\phi_{\mu}\neq\phi_{\lambda+\mu}$ (see Example 2.10). However, it is worth emphasizing that when $G/H$ is horospherical the multiplicativity indeed holds (Lemma 2.5). When $G/H$ is not horospherical, our next result gives some information about the relation between the product $\phi_{\lambda}\phi_{\mu}$ and the other spherical functions. To state it, we need the following notation. For a $B$-weight $\lambda$, let $V_{\lambda}$ denote the irreducible $G$-representation with highest weight $\lambda$. Given a $G$-module $W$ we say $V_{\lambda}$ appears in $W$ if the decomposition of $W$ into irreducible $G$-representations contains a copy of $V_{\lambda}$. Here we view $\mathbb{C}[G/H]$ as a $G$-module. Given $V_{\lambda}$ and $V_{\mu}$ both appearing in $\mathbb{C}[G/H]$, we let $\langle V_{\lambda}\cdot V_{\mu}\rangle$ denote the $\mathbb{C}$-span of the set of products $f\cdot g\in\mathbb{C}[G/H]$ where $f\in V_{\lambda}$ and $g\in V_{\mu}$. We have the following; for a precise statement see Proposition 2.9.

As we have observed, our definition of $\operatorname{sLog}_{\Gamma,t}$ depends on the choice of $\Gamma$. In the special case when $G/H$ is horospherical and thus the spherical functions are multiplicative, we can define the spherical logarithm map $\operatorname{sLog}_{t}$ canonically (i.e. independently of the choice of $\Gamma\subset\Lambda^{+}_{G/H}$) by

$$\operatorname{sLog}_{\Gamma,t}(x):=\log_{t}(\Phi(x))\,\,\textup{ for }x\in G/H$$

where $\Phi(x)$ is defined by $\langle\Phi(x),\lambda\rangle=\phi_{\lambda}(x)$, $\forall\lambda\in\Lambda^{+}_{G/H}$. In this case, the image of the map $\operatorname{sLog}_{t}$ lands naturally in the $r$-dimensional vector space $\mathcal{Q}_{G/H}\otimes\mathbb{R}:=\operatorname{Hom}(\Lambda^{+}_{G/H},\mathbb{R})$, where $r$ is the rank of the spherical variety $G/H$. This is also the vector space in which the valuation cone $\mathcal{V}_{G/H}$ naturally lies (see Section 1, as well as Remark 2.13). As we mentioned above, this paper was partly motivated by the search for a natural isomorphism between $K\backslash G/H$ and the valuation cone. Since $\operatorname{sLog}_{\Gamma,t}(G/H)$ parametrizes $K\backslash G/H$, it is natural to ask for a relationship between it and $\mathcal{V}_{G/H}$. A choice of generating set $\Gamma\subset\Lambda^{+}_{G/H}$ gives a linear embedding of the valuation cone $\mathcal{V}_{G/H}$ into $\mathbb{R}^{s}$, the codomain of $\operatorname{sLog}_{\Gamma,t}$ in the general case, given by:

$$v\mapsto(\langle v,\lambda_{1}\rangle,\ldots,\langle v,\lambda_{s}\rangle),\quad v\in\mathcal{V}_{G/H}.$$

When comparing $\mathcal{V}_{G/H}$ with $\operatorname{sLog}_{\Gamma,t}(G/H)$ in what follows, we identify $\mathcal{V}_{G/H}$ with its image in $\mathbb{R}^{s}$ under the above embedding.

We now turn to the related question of spherical amoebae. Let $Y\subset G/H$ be a subvariety of $G/H$. We define the spherical amoeba of $Y$ to be the image $\operatorname{sLog}_{\Gamma,t}(Y)$ of $Y$ under the spherical logarithm. One of our motivations for this manuscript was to ask: do the spherical amoebae approach (in a suitable sense) the spherical tropicalization of $Y$ as $t\to 0$? Of course, as in the classical (abelian) case, one needs to be precise about what the word “approach” means. Following [Jonsson], here we take it to mean the convergence of subsets in the sense of Kuratowski (Definition 3.3). Moreover, in the non-abelian case, the image of the spherical homogeneous space $G/H$ under the spherical tropicalization map is not necessarily the entire codomain; instead, the image is the valuation cone $\mathcal{V}_{G/H}$. This is in contrast to the classical case $G=(\mathbb{C}^{*})^{n}$, in which the image is the entire vector space $\mathbb{R}^{n}$. We ask the following (cf. Question 3.4):

(1) Under what conditions do the images $\operatorname{sLog}_{\Gamma,t}(G/H)$ approach the valuation cone $\mathcal{V}_{G/H}$ as $t\to 0$ in the sense of Kuratowski convergence of subsets? (2) For $Y\subseteq G/H$ a subvariety, under what conditions do the spherical amoebae $\operatorname{sLog}_{\Gamma,t}(Y)$ approach $\operatorname{strop}(Y)$ as $t\to 0$ in the sense of Kuratowski convergence of subsets?

In Remark 3.6 we sketch a proof strategy to give a positive answer to Part (1) of the above question; moreover, as we discuss further below, in Section 4 we show that in many examples, the Kuratowski limit of $\operatorname{sLog}_{\Gamma,t}(G/H)$ is indeed the valuation cone $\mathcal{V}_{G/H}$. We expect that a proof of Part (2) to be more difficult, and (as in the torus case [Jonsson]), an answer might require tools from non-Archimedean geometry. Nevertheless we conjecture the following.

We also note that progress on (1) above would partially answer the broad motivating question of finding a relation between $K\backslash G/H$ and $\mathcal{V}_{G/H}$. In this paper, we prove some preliminary results which address the question (1) above. Firstly, we answer the “curve case”, as follows. The precise statement is Theorem 3.1.

Theorem 4 yields the following corollary (see Corollary 3.5).

Statement (1) in Corollary 5 shows that $\mathcal{V}_{G/H}$ is contained in the Kuratowski limit of the images $\operatorname{sLog}_{\Gamma,t}(G/H)$. When is the limit exactly $\mathcal{V}_{G/H}$? In Section 4, we give some examples for which, with appropriate choices of $\Gamma$, the images $\operatorname{sLog}_{\Gamma,t}(G/H)$ limits to $\mathcal{V}_{G/H}$ as $t\to 0$: (i) the basic (quasi) affine space $\operatorname{SL}_{n}(\mathbb{C})/U$ (cf. Example 1.1(4)), (ii) the “group case” with $\operatorname{SL}_{n}(\mathbb{C})\times\operatorname{SL}_{n}(\mathbb{C})$ acting on $\operatorname{SL}_{n}(\mathbb{C})$ (cf. Example 1.1(5)), (iii) $\operatorname{SL}_{n}(\mathbb{C})/\operatorname{SL}_{n-1}(\mathbb{C})$, and (iv) the space of hyperbolic triangles. These examples give some evidence that the answer to the question (1) above may be positive, and hence the $K$-orbits $K\backslash G/H$ is related, in an appropriate limit, to $\mathcal{V}_{G/H}$. In fact, the first author previously conjectured a tight relation between $K\backslash G/H$ and $\mathcal{V}_{G/H}$. Since the conjecture is not recorded elsewhere, we put it here.

As a special case we also state the following conjecture.

For historical context, we should note here that it has been known for some time that the theory of moment maps from symplectic geometry can be used to parametrize $K$-orbits for multiplicity-free spaces. The theory of multiplicity-free Hamiltonian spaces are related to this question of $K$-orbits on spherical $G/H$ for the following reason: if $X$ is smooth spherical variety embedded in a projective space which is equipped with a linear $K$-action and a $K$-invariant Hermitian structure (and hence has a $K$-invariant Kähler structure), then $X$ is a multiplicity-free Hamiltonian $K$-space. Thus, a $K$-equivariant embedding of $G/H$ into a projective space gives $G/H$ such a structure. Moreover, for a multiplicity-free Hamiltonian $K$-space $M$, it is known that the Kirwan map²²2 The Kirwan map is the composition of the $K$-moment map $\Phi:M\to\mathfrak{k}^{*}$ with the quotient map $\mathfrak{k}^{*}/K\cong\mathfrak{t}^{*}_{+}$ where $\mathfrak{t}$ denotes the Lie algebra of the (compact) maximal torus $T$ of $K$. provides a parametrization of the $K$-orbits in $M$ (see [Brion-moment, Proposition 5.1]). Thus, this Kirwan map provides another potential approach to Conjectures 6 and 7.

We now outline the contents of the paper. In Section 1 we briefly recall necessary background, including the definition of the spherical tropicalization map as in [Vogiannou, Kaveh-Manon]. Then in Section 2 we define the spherical functions, following Akhiezer, and prove several properties about them. We also define the notion of a spherical logarithm map. In Section 3 we give precise statements of the Kuratowski convergence of subsets, and formulate our questions concerning the convergence of spherical amoebae. We also prove the “curve case” recounted above, as well as the horospherical case. Finally, in Section 4 we compute spherical functions in several important examples, and in some examples, we also give some natural parametrizations of the $K$-orbit space $K\backslash G/H$.

In this section, we set some notation and briefly review some facts regarding spherical varieties.