Morphisms of the spherical data of spherical homogeneous spaces

Question

Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a spherical homogeneous space, then $X$ defines a spherical datum (Luna datum) $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\varsigma)$, see below.

The spherical datum is a functor. Indeed, assume we have two spherical homogeneous spaces $X_1$ and $X_2$ of $G$. A $G$-morphism $\varphi\colon X_1\to X_2$ induces a morphism $\mathcal L(\varphi)\colon \mathcal L(X_1)\to \mathcal L(X_2)$ (in the obvious sense, see below).

Losev's Theorem 1 from his paper Uniqueness property for spherical homogeneous spaces says that if there exists an isomorphism $\lambda\colon\mathcal L(X_1)\to \mathcal L(X_2)$, then there exists an isomorphism $\varphi\colon X_1\to X_2$.

Question. Assume we have two spherical homogeneous spaces $X_1$ and $X_2$ of $G$, and a morphism of spherical data $\lambda\colon \mathcal L(X_1)\to \mathcal L(X_2)$. Does there exist a $G$-morphism $\varphi\colon X_1\to X_2$ inducing $\lambda$, that is, such that $\lambda=\mathcal L(\varphi)$?

A positive answer to this question would strengthen Losev's theorem even in the case when $\lambda$ is an isomorphism.

We specify our version of the spherical datum of $X$. Let $B\subset G$ be a Borel subgroup, and let $T\subset B\subset G$ be a maximal torus. Let $S=S(G,T,B)$ denote the corresponding system of simple roots. Let ${\mathcal{P}}(S)$ denote the set of subsets of $S$.

Let ${\mathcal{X}}(B)$ denote the character group of $B$, and let $M\subset {\mathcal{X}}(B)$ denote the weight lattice of $X$. Set $N={\rm Hom}(M,{\mathbb{Z}})$, $N_{\mathbb{Q}}=N\otimes_{\mathbb Z}{\mathbb{Q}}$. Let $\mathcal{V}\subseteq N_{\mathbb{Q}}$ denote the valuation cone of $X$.

Let ${\mathcal{D}}$ denote the set of colors of $X$ (the set of $B$-invariant prime divisors of $X$). We have two maps: $$\rho\colon {\mathcal{D}}\to N,\qquad \varsigma\colon {\mathcal{D}}\to {\mathcal{P}}(S). $$ Here, for $D\in{\mathcal{D}}$, $$ \varsigma(D)=\{\alpha\in S\ | \ P_\alpha\cdot D\neq D\}, $$ where $P_\alpha\supset B$ denotes the parabolic subgroup corresponding to $\alpha\in S$. We set $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\varsigma)$.

If $\mathcal L(X_i)=(N_i,\mathcal V_i,\mathcal D_i, \rho_i,\varsigma_i)$ for $i=1,2$, then a $G$-morphism $\varphi\colon X_1\to X_2$ induces a morphism of spherical data $$ \mathcal L(\varphi)=(\ \lambda_N\colon N_1\to N_2,\quad \lambda_{\mathcal D}\colon\mathcal D_1\to\mathcal D_2\ ) $$ such that $(\lambda_N\otimes_{\mathbb Z} \mathbb Q)(\mathcal V_1)\subseteq \mathcal V_2$ and the corresponding diagrams for $\varsigma_i$ and for $\rho_i$ commute: for any $D_1\in\mathcal D_1$ we have $$\lambda_N(\rho_1(D_1))=\rho_2(\lambda_{\mathcal D}(D_1)),\qquad \varsigma_1(D_1)=\varsigma_2(\lambda_{\mathcal D}(D_1)). $$ This permits us to define an abstract morphism of spherical data $\lambda\colon \mathcal L(X_1)\to\mathcal L(X_2)$ and to ask our question above.

Answer 1

The situation is more complicated. First of all $\lambda_{\mathcal D}$ does not exist, since there may be colors in $X_1$ which map dominantly to $X_2$. The best way to describe morphisms $X_1\to X_2$ is to use some kind of Stein factorization $X_1\to X'\to X_2$ where $X_1\to X'$ has connected fibers and $X'\to X_2$ is a finite morphism. If $X_i=G/H_i$ then $X'=G/H'$ with $H'=H_1H_2^0$. The finite morphisms are just quotients by finite automorphism groups and are therefore handled by Losev's theorem on the automorphism group of spherical varieties.

The morphisms with connected fibers were described in my Hyderabad paper. They are classified by so-called colored subspaces. More precisely, let $\mathcal D^0\subseteq \mathcal D_1$ be the set of colors mapping dominantly to $X_2$. Let $N^0\subseteq N_1$ be the kernel of $\lambda_N$. Then $\mathcal D^0$ is an additional datum to $\lambda$ and $\lambda_{\mathcal D}$ is only defined on $\mathcal D':=\mathcal D_1\setminus\mathcal D^0$. The requirement is that $(N^0,\mathcal D^0)$ is a colored subspace, meaning that $N^0$ is generated as a convex cone by $N^0\cap\mathcal V_1$ and $\rho_1(\mathcal D^0)$.

So assembling both cases one gets: A spherical morphism is a pair $(\lambda_N,\lambda_{\mathcal D})$ where $\lambda_N$ is a homomorphism $N_1\to N_2$ and $\lambda_{\mathcal D}$ is a map defined on a subset $\mathcal D'$ of $\mathcal D_1$ to $\mathcal D_2$. The axioms are roughly

• $\rho_2\circ\lambda_{\mathcal D}(D)=\lambda_N\circ\rho_1(D)$ and $\varsigma_2\circ\lambda_{\mathcal D}(D)=\varsigma_1(D)$ for all $D\in{\mathcal D'}$
• $\lambda_N^{\mathbb Q}(\mathcal V_1)=\mathcal V_2$
• $(\ker\lambda_N^{\mathbb Q},\mathcal D_1\setminus\mathcal D')$ is a colored subspace.
• The spherical datum $(N_2,\mathcal V_2,\mathcal D_2)$ is obtained from the spherical datum $(\lambda_N(N_1),\mathcal V_2,\mathcal D')$ according to Losev's rules, i.e., certain spherical roots are doubled and certain colors are identified in an admissible way.

Edit: I just realized that a morphism between spherical data does not uniquely determine the morphism between spherical varieties. For example a twist by a central element of $G$ is not seen on the level of spherical data.