Horospherical type of a spherical variety

Question

In the following, I will fix $k$ a characteristic zero algebraically closed field, and $G$ a connected reductive group over $k$, $B$ a Borel subgroup of $G$, $T\subseteq B$ a maximal torus, $X$ a $G$-spherical variety, $A_{X}$ the maximal Cartan torus of $X$ as appeared in the local structure theorem.

In Knop's paper Weylgruppe und Momentabbildung, the notion of a horospherical type for a spherical variety is defined, see the definition after corollary 2.4. Let's denote $P=P(X)$ the associated parabolic of $X$, and $P^{-}$ the opposite parabolic subgroup, then the horospherical subgroup $S$ is characterized by $S\cap B=B_{0}$ for $B_{0}$ the generic stabilizer of the Borel subgroup, $N_{G}(S)=P^{-}$, and the horospherical type $\mathfrak{S}_{X}$ of $X$ is defined to be the conjugacy class of $S$.

Since I have the background from representation theory, I am wondering whether the horospherical subgroup $S$ is related to $X_{\phi}=G/T^{\prime}U^{-}$? here $T^{\prime}$ is a subtorus of $T$ such that $T/T^{\prime}\cong A_{X}$, $X_{\phi}$ is called the "most degenerate" boundary degeneration of $X$, here by boundary degenertation, I mean the boundary degenerations as defined in the Sakellaridis–Venkatesh paper Periods and harmonic analysis on spherical varieties.

Answer 1

As @LSpice pointed out, the group $S$ is not characterized by $N_G(S)=P^-$. Instead it is the normal subgroup of $P^-$ such that $P^-/S=T/T'$. Note also that $T'$ is in general not a subtorus of $T$ but just a closed subgroup.

This being said, you are right that $S=T'U^-$. This follows from Satz 2.7(2) of the paper you mentioned: An arbitrary homogeneous $G$-variety has a $1$-parameter deformation to a $G$-variety of the form $X_0=V\times G/S$. If $X$ is spherical then $\dim G/S=\dim X=\dim X_0$ (see, e.g., the first equality of Satz 7.1) and therefore $X_0=G/S$. Thus $X_0$ is a horospherical deformation of $X$ which is also the definition of the most degenerate boundary degeneration in the Sakellaridis–Venkatesh paper.