I have a question about root set corresponding to $P_θ ∩ M_Ω$ where $θ$ is a subset of simple roots, $Ω=θ∪{α}$ where $α$ is a simple root and not in $θ$, $P_θ$ is a parabolic subgroup corresponding to $\theta$ and $M_Ω$ is a Levi factor of the parabolic subgroup $P_Ω$.
What is $P_θ ∩ M_Ω$?
Thanks!
I will prove something more general by using the Borel-Tits relative structure theory for connected reductive groups over fields. The key to everything is the dynamic description of parabolic subgroups of connected reductive groups over a general ground field; if this technique is not known to you then everything may seem mysterious, but once it is in your toolkit then matters become very tractable by "digging roots and lifting weights", as follows.
Let $G$ be a connected reductive group over an arbitrary field $k$, let $S$ be a split maximal $k$-torus, and let $\Phi^+$ be a positive system of roots in the relative root system ${}_k\Phi := \Phi(G,S)$. Let $\Delta$ be the basis of simple roots in $\Phi^+$, so the "parabolic" subsets of ${}_k\Phi$ containing $\Phi^+$ (in the sense of Bourbaki) are precisely the subsets $\Phi^+ \cup [I]$ where $I$ is a subset of $\Delta$ and $[I]$ is the set of roots whose expansion in $\Delta$ has vanishing coefficients outside $I$ (so for $I=\emptyset$ we recover $\Phi^+$).
For each subset $I \subset \Delta$ there is associated a unique parabolic $k$-subgroup $P_I$ of $G$ containing $S$ such that $\Phi(P_I,S) = \Phi^+ \cup [I]$ (so $P_{\emptyset}=P$ and $P_{\Delta}=G$). These $P_I$'s represent (without repetition) the $G(k)$-conjugacy classes of parabolic $k$-subgroups of $G$, and $P_I \subset P_{I'}$ if and only if $I \subset I'$. More specifically, if we fix a maximal $k$-torus $T$ in $P$ (so $T$ contains $S$, as minimality implies $P = Z_G(S) \ltimes U$ for $U = \mathscr{R}_u(P)$) then for each $I$ there is a unique Levi $k$-subgroup $L_I$ in $P_I$ containing $T$ (i.e., $P_I = L_I \ltimes U_I$ for $U_I = \mathscr{R}_u(P_I)$).
Suppose $J$ is a subset of $\Delta$ strictly containing $I$. What is $P_I \cap L_J$? (The question posed is the special case when $J - I$ has size 1 and $k$ is a finite extension of $\mathbf{Q}_p$, but this is no easier than the general case.) Observe that $L_J$ contains $S$ as a split maximal $k$-torus, with $\Phi(L_J,S)$ having $J$ as a basis for the positive system of roots $\Phi(L_I,S) \cap \Phi^+$. In particular, the parabolic $k$-subgroups of $L_J$ are labeled up to $L_J(k)$-conjugacy by subsets $J'$ of $J$ (represented by the image under $P_J \twoheadrightarrow L_J$ of the $k$-groups $P_{J'}$ for $J' \subset J$; all such $P_{J'}$ contain $U_J$ inside $U_{J'}$).
The claim is that $P_I \cap L_J$ is the parabolic $k$-subgroup of $L_J$ thereby "corresponding" to $J-I$. By the dynamic description of parabolic subgroups, $P_I = P_G(\lambda)$ where $\lambda$ is a cocharacter of $S$ orthogonal to all elements of $I$ but with positive pairing against all elements of $\Delta - I$. Thus, by considering the meaning of the dynamic method on geometric points, $P_I \cap L_J = P_{L_J}(\lambda)$. So this schematic intersection is a parabolic $k$-subgroup of the connected reductive $L_J$ containing $S$ (in particular, it is smooth and connected), so it is uniquely determined by the nontrivial $S$-weights which occur on its Lie algebra, and these are the ones in $\Phi(L_J,S)$ orthogonal to $\lambda$.
But $\Phi(L_J,S)$ is the set of roots "supported" in $J$, and those appearing on the Lie algebra of $P_{L_J}(\lambda)$ are the ones whose pairing against $\lambda$ is non-negative. This says just that the coefficients on members of $I$ are non-negative but imposes no conditions on coefficients of elements of $J \cap (\Delta - I) = J - I$, so in other words we have exactly the set of relative roots in the union of $[J] \cap \Phi^+$ and $[J-I]$. But that's exactly the parabolic $k$-subgroup of $L_J$ associated to $J-I$ among those containing the minimal parabolic $k$-subgroup of $L_J$ which contains $S$ and has set of $S$-roots $[J] \cap \Phi^+$.
