I am now studying on the horospherical variety. For example, I am observing $\mathrm{SL}_2(\mathbb{C})/U$ where $U$ is a unipotent subgroup $$ \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}. $$ We take $B$ to be the Borel subgroup consisting of the upper triangular matrix, and its unipotent radical is $U$. I found easily that the quotient $\mathrm{SL}_2(\mathbb{C})/U$ is isomophic to $\mathbb{C}^2 \setminus \{0\}$, mapping the matrix $$\begin{pmatrix}a & b \\ c & d\end{pmatrix}$$ to the point $(a,c)$. The color of $\mathrm{SL}_2(\mathbb{C})/U$, which is a $B$-stable but not $\mathrm{SL}_2(\mathbb{C})$-stable divisor is unique, $$ \{(x,0):x\in \mathbb{C}^*\} \in \mathbb{C}^2 \setminus \{0\}. $$ Now I am interested in the embedding of $\mathrm{SL}_2(\mathbb{C})/U$ into $\mathrm{Bl}_0(\mathbb{C}^2)$. The exceptional divisor $E$ is one example of the $\mathrm{SL}_2(\mathbb{C}^2)$-divisor. The paper I am reading says that $\mathrm{Bl}_0(\mathbb{C}^2)$ has no colors, but I am not understanding why the Zariski closure of the line $\{(x,0)\}$ in $\mathrm{Bl}_0(\mathbb{C}^2)$ is not the color. The zariski closure must contain the line $\{(x,0):x\in \mathbb{C}^*\}\in \mathbb{C}^2\setminus \{0\}$ and the point $(1:0)$ inside the exceptional divisor $E$. Considering the action of $B$, it looks the Zariski closure is stable under the $B$-action. The $\mathrm{SL}_2(\mathbb{C})$ action on the exceptional divisor must be exactly the action of it over $\mathrm{SL}_2(\mathbb{C})/B$, so $(1:0)$, which is represented by the matrix $$ \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}, $$ is fixed under the $B$-action. Hence, the Zariski closure is stable under $B$-action, at least under my description.
However, all the papers I am reading is saying that the color is empty. I don't find what I am missing.
I found the answer myself after searching on the internet for a while. I was misunderstanding the notion of colors. This is the reference I found: http://ronan.terpereau.perso.math.cnrs.fr/enseignement/overview_classification_spherical_and_compelxity-one_varieties.pdf
According to this reference, since $\mathrm{SL}_2(\mathbb{C})/U$ is densely embedded inside $\mathbb{A}^2_\mathbb{C}$, one can define a rational map $\mathbb{A}^2_\mathbb{C} \to \mathrm{SL}_2(\mathbb{C})/B=\mathbb{P}^1_\mathbb{C}$. Obviously this is not globally defined, and the indeterminancy is contained in the closure of the color $D_\alpha=\{(x,0)\}$. When this happens, we say that $\mathbb{A}^2_\mathbb{C}$ is colored by the color $D_\alpha$. The blow-up of $\mathbb{A}^2_\mathbb{C}$ along this rational map is called the decoloration. $\mathrm{Bl}_0(\mathbb{A}^2_\mathbb{C})$ has no color since $\mathrm{Bl}_0(\mathbb{A}^2_\mathbb{C}) \to \mathrm{SL}_2(\mathbb{C})/B$ is now a globally defined morphism.