Lemma 1.3 · Local structure and smooth Borel stabilizers for \(G/G^\theta\) in characteristic \(\neq2\)

This replaces [BLV86, Théorème 3.4] at both places where the paper uses it (§2.1, p. 8: \([L(X),L(X)]\subseteq H\); §3.4, p. 17: \((B_y)_u=L(X)\cap U\)). It also supplies the smoothness that the differential-form calculus of §3.3 and Lemmas 3.4.2, 5.1.1 needs (checklist 5).

Proof.

1. (a) \(H=G^{\langle\theta\rangle}\) is the fixed-point scheme of a group of order 2, which is invertible on \(k\), acting on a smooth affine scheme. It is therefore smooth (fixed points of a linearly reductive group; [CGP, Prop. A.8.10(2)], cited, not opened). \(G\to G/H\) is an \(H\)-torsor, hence smooth.

2. Roots of \(L\). Since \(T=(T^\theta)^\circ S\) and \(\alpha-\theta\alpha\) is trivial on \(T^\theta\), we have \(\alpha|_S=0\iff\theta\alpha=\alpha\). So \(\Phi_L=\{\alpha:\theta\alpha=\alpha\}\). For \(\alpha\in\Phi_L\), \(\theta\) preserves \(U_\alpha\) and acts on it by a sign \(\varepsilon_\alpha\).

3. (b) Suppose \(\varepsilon_\alpha=-1\) for some \(\alpha\in\Phi_L\). On the rank-one group \(G_\alpha=\langle U_\alpha,U_{-\alpha}\rangle\cong SL_2\) or \(PGL_2\), \(\theta\) is then conjugation by \(\alpha^\vee(\sqrt{-1})\), and \(G_\alpha\) contains a one-dimensional \(\theta\)-split torus \(S'\) (the Cayley transform of \(\alpha^\vee(\mathbb G_m)\); here \(2\in k^\times\)). \(S'\) centralizes \(S\) and \(S'\not\subseteq S\), so \(SS'\) is a larger \(\theta\)-split torus, a contradiction. Hence \(\varepsilon_\alpha=1\) for all \(\alpha\in\Phi_L\), and \(\theta\) fixes \([L,L]=\langle U_\alpha:\alpha\in\Phi_L\rangle\).

4. (c) Let \(\beta<0\). If \(\beta\notin\Phi_L\), then \(\theta\beta>0\), and for \(Y\in\mathfrak g_\beta\) we have \(Y=(Y+\theta Y)-\theta Y\in\mathfrak h+\mathfrak b\). If \(\beta\in\Phi_L\), then \(\mathfrak g_\beta\subseteq\mathfrak h\) by step 3. Together with \(\mathfrak t\subseteq\mathfrak b\) this gives \(\mathfrak g=\mathfrak b+\mathfrak h\). So the differential of \(B\to X\) at \(x_0\) is onto: \(Bx_0\) is open, \(B\to Bx_0\) is smooth, and \(B\cap H\) is smooth with Lie algebra \(\mathfrak b\cap\mathfrak h\).

If \(g\in B\cap H\), then \(g=\theta(g)\in B\cap\theta(B)=T\cdot\prod_{\beta>0,\theta\beta>0}U_\beta=T(U\cap L)\), since \(\beta>0,\ \theta\beta>0\) forces \(\beta\in\Phi_L\). With step 3 this gives \(B\cap H=T^\theta\ltimes(U\cap L)\).

5. (d) \(P\subseteq P(X)\): \(Px_0=U_PT[L,L]x_0=U_PTx_0\subseteq Bx_0\). Conversely, let \(\alpha\notin\Phi_L\) be simple with \(P_\alpha\subseteq P(X)\). Every unit on the \(P(X)\)-homogeneous \(X̊\) is a character of \(P(X)\) (Rosenlicht), hence orthogonal to \(\alpha^\vee\). But \(\mathcal X(X)_{\mathbb Q}=\{\chi:\theta\chi=-\chi\}\), and \(\alpha^\vee\) has non-zero projection to \(X_*(S)_{\mathbb Q}\), so some \(\chi\in\mathcal X(X)\) has \(\langle\alpha^\vee,\chi\rangle\ne0\). This is a contradiction, so \(P(X)=P\). \(L\) is a Levi of \(P\) with \([L,L]\subseteq H\), and \(L/(L\cap H)=T/T^\theta\).

On \(T^\theta\): \(\delta_{B_{x_0}}=|e^{2\rho_L}|=\delta(X)\). Since \(\theta\Phi(U_P)=-\Phi(U_P)\), \(2\rho_P\) is \(\theta\)-anti-invariant and hence trivial on \(T^\theta\). So \(\delta_B=|e^{2\rho_L+2\rho_P}|\) agrees with \(\delta(X)\) on \(T^\theta\), and \(\delta_{P(X)}=|e^{2\rho_P}|\) is trivial on \(T^\theta\).

6. (e) Cited input. Springer [Some results on algebraic groups with involutions, 1985], char \(\neq2\), not opened locally: every \(B\)-orbit contains \(y=gH\) such that \(\theta_y:=\operatorname{Int}(g)\theta\operatorname{Int}(g)^{-1}\) stabilizes a maximal torus \(T'\subseteq B\).

Then \(B_y=B\cap G^{\theta_y}=(B\cap\theta_yB)^{\theta_y}\) and \(B\cap\theta_yB=T'\ltimes U''\) with \(U''=\prod_{\beta\in\Phi^+\cap\theta_y\Phi^+}U_\beta\). Since \(H^1(\mathbb Z/2,V)=0\) for every \(k\)-vector space \(V\), filtering \(U''\) by \(\theta_y\)-stable subgroups with vector-group quotients shows that \((T'\ltimes U'')^{\theta_y}=T'^{\theta_y}\ltimes U''^{\theta_y}\), with \(U_1:=U''^{\theta_y}\) smooth and connected. \(T_1:=T'^{\theta_y}\) has component group of exponent 2. A weight of \((T'^{\theta_y})^\circ\) on \(\mathfrak g_\beta\) (\(\beta,\theta_y\beta>0\)) is \(\beta|\), which vanishes only if \(\theta_y\beta=-\beta\); that is impossible. The cocharacter \(\lambda=\check\rho+\theta_y\check\rho\) of \((T'^{\theta_y})^\circ\) satisfies \(\langle\lambda,\beta\rangle=\langle\check\rho,\beta\rangle+\langle\check\rho,\theta_y\beta\rangle>0\). So all weights lie in an open half-space.

Remark (where \(p\ne2\) and smoothness enter). For \(p=2\), step 3 and the splitting in step 6 fail (\(H\) can be non-reduced and \(B\cap H\) can contain infinitesimal or étale 2-groups). Then \(B(F)\)-orbits can be infinite in number (Lemma 1.4).