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.

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.