I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.


More generally, let $F$ be a field of characteristic 0, $L/F$ a quadratic extension, and $H$ be an $L/F$-Hermitian form on $L^n$. Write $G={\rm SU}(L^n,H)$, which is a semisimple $F$-group. I am looking for a reference where the relative system of simple roots and the conjugacy classes of $F$-parabolics in $G$ are explicitly computed. Here "relative" means with respect to a maximal split $F$-torus.


The case of a special *orthogonal* group is treated in Borel's 1966 paper "Linear algebraic groups" in the Boulder proceedings "Algebraic Groups and Discontinuous Subgroups", Proc. Sympos Pure Math. vol. 9. Borel writes that for an orthogonal group, the $F$-paraboics are the stabilizers of the $F$-rational isotropic flags.



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> **Question.** Is it true for $G={\rm SU}(L^n,H)$ that the $F$-parabolics in $G$ are the stabilizers of the $F$-rational isotropic flags in $(L^n, H)$ ?
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Parabolics and simple roots for a special unitary group: reference request
I have edited in some remarks from the comments ([1](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983612_385903) [2](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983785_385903) [3](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983788_385903)). To start, I am used to thinking of the stabiliser of a self-dual flag, but of course every self-dual flag has an isotropic flag as its “bottom half”, and every isotropic flag can be completed to a self-dual flag by tossing the duals “on top”. ([Yu - An introduction to explicit Bruhat–Tits theory](https://web.archive.org/web/20070824032813/http://www.math.purdue.edu/%7Ejyu/notes/banff2001.pdf) is where I first encountered the description of Bruhat–Tits buildings of symplectic groups in this ‘self-dual’ language.)


$\DeclareMathOperator\tr{tr}$Write $L^n = V^\* \oplus \overline V \oplus W$ for some $L$-vector space $V$, where $(W, H)$ is anisotropic and $H(v^\* \oplus \overline v) = \tr\_{L/F} \langle v^\*, v\rangle$. A choice of $L$-basis $(v\_i)\_{i \in I}$ for $V$ gives rise to a split torus $S$ in $G$, consisting of the transformations that preserve the $F$-line through each basis vector. Then $S$ is a maximal split torus in $G$, since its centraliser in $G$ is the product of the anisotropic group $\operatorname{SU}(W, H)$ with the torus consisting of the transformations that preserve the $L$-line through each basis vector.


Write $(v\_{i^\*})\_{i \in I}$ for the basis of $V^\*$ dual to $(v\_i)\_{i \in I}$. Put $J = I \cup I^\*$.


For each $i \in J$, the map $a\_i : S \to \operatorname{GL}\_1$ that sends $s \in S$ to $s v\_i/v\_i$, in the hopefully obvious notation, is a relative root if $W \ne 0$; and its root space consists of all those skew-adjoint endomorphisms of $L^n$ that annihilate all $v\_{i'}$ with $i' \ne i, i^\*$, and that carry $W$ into $L v\_i$.


For each pair $i, j \in J$ such that $i \ne j$, the map $a\_{i j} = a\_i - a\_j$ is a relative root; and its root space is the set of all skew-adjoint endomorphisms of $L^n$ that annihilate all $v\_{j'}$ with $j' \ne j, j^\*$, and that carry $L v\_j$ into $L v\_i$. (Note that $a\_{i i^\*} = 2a\_i$, so, to get the full root *algebra* for $a\_i$—if it is a relative root—we need to take the sum of the $a\_i$ and $a\_{i i^\*}$ root *spaces*.)


Since these spaces, together with $\operatorname C\_{\mathfrak{su}(L^n, H)}(S)$, span $\mathfrak{su}(L^n, H)$, we have found all relative roots. The relative root system is of type $\mathsf C\_{\dim(V)}$ if $W = 0$ and $\mathsf{BC}\_{\dim(V)}$ if $W \ne 0$. If we identify $I$ with $\{1, \dotsc, m\}$, then one choice of simple roots is the union of $\{a\_{i(i + 1)} \mathrel: 1 \le i < m\}$ with $\{a\_{m m^\*}\}$ if $W = 0$, or with $\{a\_m\} if $W \ne 0$.


I will now appeal to the parameterisation of (rational) parabolics by (rational) cocharacters, according to which we attach to a (rational) cocharacter $\lambda$ of $G$ the parabolic $P\_G(\lambda) = \{g \in G \mathrel: \text{$\lim\_{t \to 0} \lambda(t)g\lambda(t)^{-1}$ exists}\}$. (This parameterisation has the advantage that it also singles out a Levi component $M\_G(\lambda) = \operatorname{Cent}\_G(\lambda)$ of $P\_G(\lambda)$.)


Fix a rational cocharacter $\lambda$ of $G$. After replacing it by a $G$-conjugate, we may assume that it takes values in $S$; and, after replacing it by a further Weyl conjugate, we may assume that $\langle a\_{i j^\*}, \lambda\rangle \ge 0$ for all $i, j \in I$. Then $P\_G(\lambda)$ is the stabiliser of the flag
$$
\Bigl\{\bigoplus\_{\substack{j \in I \\ \langle a\_{i j}, \lambda\rangle \ge 0}} L v\_j \mathrel: i \in I\Bigr\}
$$
(where I've put $a\_{ii} = 0$ for each $i \in J$).


Since every isotropic flag can be conjugated into $V$, we have shown that the parabolics are precisely the stabilisers of isotropic flags.


