Split Reductive Root Systems, Bruhat Cells, and Parabolics — Examples

The root datum and root coordinates are those of [F1]: $\mathrm{\Phi }=\{\pm 2\chi \}$ with coroot ${\chi }^{\vee }$ and $⟨\alpha ,{\alpha }^{\vee }⟩=2$, so the root datum is the simply connected rank-one datum; the root groups are $U_{\pm \alpha }=U^{\pm }$, and the Lie algebra decomposes as ${\mathfrak{s}\mathfrak{l}}_2=\mathfrak{t}\oplus {\mathfrak{g}}_{\alpha }\oplus {\mathfrak{g}}_{-\alpha }$ with one-dimensional root spaces. The Weyl group is $W=N_G(T_2)\mathrm{/}{T}_2=\{1,s_{\alpha }\}\cong \mathbb{Z}\mathrm{/}2$ because ${\mathrm{S}\mathrm{L}}_2$ has exactly two Borel subgroups containing $T_2$, namely $B$ and $U^{-}{T}_2=B^{-}$.

Write $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with $ad-bc=1$. The cell $B$ is defined by $c=0$ and has dimension $2$. If $c\mathrm{\ne }0$, then $g=u_{\alpha }(a\mathrm{/}c)n_{\alpha }\mathrm{diag}(-c,-c^{-1})u_{\alpha }(d\mathrm{/}c)$ by multiplication. Thus $B{n}_{\alpha }B=U^{+}{n}_{\alpha }B$ is exactly $c\mathrm{\ne }0$, parametrized uniquely by $U^{+}\times T_2\times U^{+}$ and of dimension $3$. This proves the two-cell decomposition over $k$ and on $k$-points. The Gaussian cell $U^{-}{T}_2{U}^{+}$ is instead $a\mathrm{\ne }0$, as follows from $g=u_{-\alpha }(c\mathrm{/}a)\mathrm{diag}(a,a^{-1})u_{\alpha }(b\mathrm{/}a)$; its multiplication map is also an open immersion, but its image differs from $B{n}_{\alpha }B$.

Finally $G\mathrm{/}B\cong {\mathbf{P}}^1$ by [F3], since the quotient of the connected nonsolvable group ${\mathrm{S}\mathrm{L}}_2$ by the Borel subgroup is a homogeneous curve; the cell decomposition of $G\mathrm{/}B$ is $\{B\}\bigsqcup Y(s_{\alpha })$ with $Y(s_{\alpha })=U{w}_{0}B\mathrm{/}B\cong {\mathbf{A}}^{n(w_0)}={\mathbf{A}}^1$, so the two Bruhat cells of the flag variety correspond to the two $T_2$-fixed points of ${\mathbf{P}}^1$; the action of $G(k)$ factors through ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(k)=\mathrm{Aut}({\mathbf{P}}^1)(k)$.

The roots and root groups of ${\mathrm{G}\mathrm{L}}_n$ are as in [F1]: the weight of the coordinate function ${\chi }_i$ on $D_n$ is ${\chi }_i(\mathrm{diag}(x_1,\dots ,x_n))=x_i$; the adjoint action on $E_{ij}$ gives the weight ${\chi }_i-{\chi }_j$, and the root group $U_{ij}$ consists of the elementary matrices $I+a{E}_{ij}$, isomorphic to ${\mathbf{G}}_a$; the positive roots relative to $B$ are those with $i<j$, and the simple roots ${\alpha }_{i,i+1}$ form the base of the root system $A_{n-1}$.

Let $I\subseteq \{1,\dots ,n-1\}$ and let the complement of $I$ in the ordered set of simple roots be the partial sums listed, so that the integers $a_1,\dots ,a_s\ge 1$ sum to $n$. Choose a cocharacter with diagonal exponents constant within each block and strictly decreasing between consecutive blocks. Entrywise conjugation scales $x_{ij}$ by the exponent difference, so its limit subgroup is precisely the group with zero blocks below the diagonal and its limit kernel has identity diagonal blocks. By [F2] its zero simple-root pairings recover $I$. Thus $P_I$ is the group of invertible matrices preserving the flag of subspaces $0\subseteq k^{a_1}\subseteq k^{a_1+a_2}\subseteq \cdots \subseteq k^n$, i.e. the block upper triangular matrices with diagonal blocks of sizes $a_1,\dots ,a_s$; its unipotent radical $R_u(P_I)$ has identity diagonal blocks, zeros below and arbitrary blocks above and its Levi factor is the block diagonal ${\prod }_i{\mathrm{G}\mathrm{L}}_{a_i}$; the multiplication map $R_u(P_I)⋊L_I\to P_I$ is a group-scheme isomorphism by [F2], and $P_{\mathrm{\varnothing }}=B$ and $P_{\mathrm{\Delta }}=G$ because the corresponding flags are the complete flag and the trivial flag. The all-algebra matrix equations verify these stabilizers scheme-theoretically. For $n=1$ the simple-root set is empty, both endpoints are $G=B={\mathbf{G}}_m$, and its unipotent radical is trivial.

The maximal proper parabolics correspond to $I=\mathrm{\Delta }\setminus \{i\}$, i.e. to the stabilizers of a single subspace of dimension $i$: taking $a_1=i$, $a_2=n-i$ gives the block upper triangular group with two diagonal blocks, which is the stabilizer of $k^i\subseteq k^n$, and the general $P_I$ is the intersection of these stabilizers over the cuts $i\in \mathrm{\Delta }\setminus I$, not over $I$. For $n\ge 3$, the minimal proper parabolics strictly above $B$ are $P_{\{i\}}$, since singletons are the minimal nonempty proper subsets of the base. For $n\le 2$ there is no proper parabolic strictly between $B$ and $G$; in particular, for $n=2$ the singleton gives $G$, rather than a proper parabolic. Finally, the simple-root reflections act on the characters ${\chi }_i$ by adjacent transpositions. Since they generate the Weyl group, its faithful lattice action is precisely $S_n$. The Bruhat cells of $G\mathrm{/}B$ are indexed by $S_n$, and the length $n(w)$ equals the dimension of the cell by the length formula of [F3]. For the roots ${\chi }_i-{\chi }_j$ with $i<j$, the count $\mathrm{\mid }{\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{-}\mathrm{\mid }$ is the inversion number of $w^{-1}$. This equals the inversion number of $w$: $(i,j)\mapsto (w(j),w(i))$ bijects their inversion sets. Thus the count is $\mathrm{\#}\{(i,j):i<j,w(i)>w(j)\}$ over every field.

The Lie algebras agree and the abstract root systems agree: by [F2], $\mathrm{Lie}(G_1)\cong \mathrm{Lie}(G_2)\cong {\mathfrak{s}\mathfrak{l}}_2$, and both root systems are $\{\pm \alpha \}$, abstractly the root system $A_1$; the groups are not isomorphic because their centres differ, ${\mu }_2\mathrm{\ne }1$ by [F3].

The root data are nevertheless not isomorphic. Write $X_1=X(T_1)=\mathbb{Z}{\chi }_1$ and $X_2=X(T_2)=\mathbb{Z}{\chi }_2$, so a $\mathbb{Z}$-linear isomorphism $f:X_1\to X_2$ carries the generator ${\chi }_1$ to $\pm {\chi }_2$. An isomorphism of root data must carry ${\mathrm{\Phi }}_1$ onto ${\mathrm{\Phi }}_2$, in particular it must send the root ${\alpha }_1=2{\chi }_1$ to an element of ${\mathrm{\Phi }}_2=\{\pm {\chi }_2\}$. But $f(2{\chi }_1)=\pm 2{\chi }_2\mathrm{\notin }\{\pm {\chi }_2\}$ because ${\chi }_2$ generates a free $\mathbb{Z}$-lattice. Equivalently, ${\alpha }_1$ is divisible by $2$ in $X_1$ while ${\alpha }_2={\chi }_2$ is not divisible by $2$ in $X_2$: divisibility of a root in the character lattice is an invariant of root data. The same obstruction appears dually: $\mathbb{Z}{\alpha }_1^{\vee }=X_{\ast }(T_1)$ is the full cocharacter lattice while $\mathbb{Z}{\alpha }_2^{\vee }=2\mathbb{Z}{\chi }_2^{\vee }$ has index $2$ in $X_{\ast }(T_2)$. Hence there is no isomorphism of quadruples $(X,\mathrm{\Phi },X^{\vee },{\mathrm{\Phi }}^{\vee })$.

Therefore the Lie algebra ${\mathfrak{s}\mathfrak{l}}_2$ and the abstract root system $A_1$, which are the same for the two groups in characteristic $\mathrm{\ne }2$, do not determine the root datum or the group: the additional data are the position of the coroot lattice $\mathbb{Z}{\alpha }^{\vee }$ inside $X_{\ast }(T)$ and the divisibility of the root in $X(T)$, and these separate the simply connected group ${\mathrm{S}\mathrm{L}}_2$ from the adjoint group ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$.