Galois cohomology of reductive algebraic groups over the field of real numbers

Abstract

We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$ of a connected reductive algebraic group $G$ over the field $\mathbb R$ of real numbers in terms of a certain action of the Weyl group on the real points of order dividing 2 of the maximal torus containing a maximal compact torus. This result was announced with a sketch of proof in the author's 1988 note. Here we give a detailed proof.

Computing Galois cohomology of a real linear algebraic group

“…\end{equation*}$$Then, $$D^k\circ D^{k-1}=0$$. We define the $$k$$ ‐th Tate hypercohomology group (see [16, section 3]) $$$$\begin{equation*} {\mathbb {H}}^k(A_{1}\xrightarrow {\;\; \partial \;\; } A_0)={\rm Z}^k(A_{1}\xrightarrow {\;\; \partial \;\; } A_0)\hspace{0.8pt}/\hspace{0.8pt}{\rm B}^k(A_{1}\xrightarrow {\;\; \partial \;\; } A_0), \end{equation*}$$$$where …”

“…Consider the right action of $$N_0$$ on $${\rm Z}^1\hspace{0.8pt}{\bf T}$$: $$$$\begin{equation*} t*n=n^{-1}\cdot t\cdot \hspace{-0.8pt}\hspace{0.8pt}^\gamma \hspace{-0.8pt}n\quad \text{for}\ n\in N_0, \ t\in {\rm Z}^1\hspace{0.8pt}{\bf T}. \end{equation*}$$$$This action induces a well‐defined right action of $$W_0$$ on $${\rm H}^1\hspace{0.8pt}{\bf T}$$ (which might not preserve the group structure in $${\rm H}^1\hspace{0.8pt}{\bf T}$$); see [11, Section 3]. We know that there exists $$w_1=n_1\cdot T$$ such that $$[z]*w_1=[1]\in {\rm H}^1 {\bf T}$$; we can find such $$n_1$$ by brute force, by computing $${\rm H}^1 {\bf T}$$, $$W_0$$ and the right action of $$W_0$$ on …”

“…Now set $$C={\mathcal {Z}}_G(s)$$. Then, $$C$$ is reductive, defined over $${\mathbb {R}}$$, and $$s$$ lies in the identity component $$C^0$$ of $$C$$ (see [11, proof of Theorem 3.1]). We write $${\bf C}$$ for the corresponding $${\mathbb {R}}$$‐group.…”

“…This action induces a well-defined right action of 𝑊 0 on H 1 𝐓 (which might not preserve the group structure in H 1 𝐓); see [11,Section 3]. We know that there exists 𝑤 1 = 𝑛 1 ⋅ 𝑇 such that [𝑧] * 𝑤 1 = [1] ∈ H 1 𝐓; we can find such 𝑛 1 by brute force, by computing H 1 𝐓, 𝑊 0 and the right action of 𝑊 0 on H 1 𝐓.…”

Galois cohomology of reductive groups over global fields

“…1.3. In the case F = R, the set H 1 (F, G) was computed in [Bor88] (see also [Bor22a]) using the method of Borel and Serre [BS64, Theorem 6.8], and by the first-named author and Timashev in [BT21, Theorem 7.14] using the method of Kac [Kac69]. Note that over F = R, the set H 1 (F, G) cannot be computed in terms of π 1 (G) only.…”