I am going to describe a "degeneracy functor" $$\delta_\mathfrak{q} \ : \ \text{Par}(\text{SL}_n,B_n) \ \to \ \text{Par}(\text{SL}_{n-1},B_{n-1})$$ from parabolics of a big group to a smaller group (both containing given Borels), and my question is: what is the morally correct definition of this functor which extends at least to arbitrary reductive groups $G$?
The functor is attached to something like $$\mathfrak{q} \ = \ \left( \begin{array}{cccccc} \color{white}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} \\ & \color{yellow}{\blacksquare} & \color{yellow}{\blacksquare} & \color{yellow}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} \\ & \color{yellow}{\blacksquare} & \color{yellow}{\blacksquare} & \color{yellow}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} \\ &\color{yellow}{\blacksquare} & \color{yellow}{\blacksquare} & \color{yellow}{\blacksquare} & \color{white}{\blacksquare} & \color{white}{\blacksquare} \\ & & & & \color{white}{\blacksquare} & \color{white}{\blacksquare} \\ & & & & & \color{white}{\blacksquare} \end{array} \right),$$ where I suspect $\mathfrak{q}B$ should be viewed as a standard parabolic in $\text{SL}_n$.
The functor $\delta_\mathfrak{q}$ takes a parabolic $B_n\subseteq \color{blue}{P}\subseteq \text{SL}_n$ and "$\mathfrak{q}$-blockifies" it, e.g.
$$\left( \begin{array}{cccccc} \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ \color{blue}{\blacksquare}& \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ &\color{yellow}{\blacksquare}& \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & \color{yellow}{\blacksquare}& \color{yellow}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & & & \color{blue}{\blacksquare} \end{array} \right) \ \mapsto \ \left( \begin{array}{ccccc} \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ \color{blue}{\blacksquare}& \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & \color{yellow}{\blacksquare}& \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & & \color{blue}{\blacksquare} \end{array} \right)$$
$$\left( \begin{array}{cccccc} \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ \color{blue}{\blacksquare}& \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ &\color{green}{\blacksquare}& \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & \color{yellow}{\blacksquare}& \color{yellow}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & & & \color{blue}{\blacksquare} \end{array} \right) \ \mapsto \ \left( \begin{array}{ccccc} \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ \color{blue}{\blacksquare}& \color{green}{\blacksquare} & \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & \color{green}{\blacksquare}& \color{green}{\blacksquare} & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & \color{blue}{\blacksquare} & \color{blue}{\blacksquare} \\ & & & & \color{blue}{\blacksquare} \end{array} \right)$$ In the above, the parabolic $P$ on the left is drawn in $\color{blue}{\text{blue}}/\color{green}{\text{green}}$, and its image $\delta_\mathfrak{q}(P)$ on the right is also drawn in $\color{blue}{\text{blue}}/\color{green}{\text{green}}$. We draw $\mathfrak{q}$ on the left in $\color{yellow}{\text{yellow}}/\color{green}{\text{green}}$.
More precisely, the functor takes $P$, "adds in" the rest of $\mathfrak{q}$ if $P$ contains a root of $\mathfrak{q}$, then "contracts the $\text{GL}_3$ in $\mathfrak{q}$ to a copy of $\text{GL}_2$", i.e. $\delta_\mathfrak{q}(P_I)= P_{d(I)}$ where $d:[n]\to [n-1]$ is the degeneracy map induced by contracting the two simple roots in $\mathfrak{q}$, and $P_I$ is the standard parabolic attached to subset $I$ of roots.
To reiterate my question: is this an example of something that works for arbitrary (B,N)-pairs/Tits systems/reductive groups/Coxeter groups/[any class of such groups/root data bigger than just ADE]?
