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$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=0$. From this it is not difficult to deduce that $H^1(\Gal(K/k),\SL_n(K))=0$.

In Serre's Galois cohomology book (III.3.1), there is a conjecture asserting that $H^1(\Gal(k^{\mathrm{sep}}/k),G(k^{\mathrm{sep}}))=0$ for any simply-connected semisimple $G$ over $k$ providing $k$ fulfills a cohomological dimension condition.

In any case, my question concerns with a particular choice of $K/k$:

Question: Letting $K/k=\mathbb{C}/\mathbb{R}$, do we have $H^1(\Gal(\mathbb{C}/\mathbb{R}),G(\mathbb{C}))=0$ for any simply-connected semisimple algebraic group $G$ defined over $\mathbb{R}$?

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    $\begingroup$ I'm not sure whether I misunderstood your question, but ⓐ to be clear, $\mathbb{R}$ does not satisfy the condition of having cohomological dimension $≤2$ (since its Galois group is finite cyclic nontrivial, and such have periodic cohomology), and ⓑ if $G$ is, for example, the simply-connected semisimple algebraic group $G_2$, then $H^1(k,G_2)$ classifies octonion algebras over $k$ (Serre, Galois Cohomology, III, appendix 2, 3.3), and for $k=\mathbb{R}$ there are two of them (the split octonions and usual octonions), so $H^1(\mathbb{R},G_2)$ is nontrivial. $\endgroup$
    – Gro-Tsen
    Commented May 28 at 8:32
  • $\begingroup$ PS: If you want a good introduction to the Galois cohomology of semisimple groups over the reals, Mikhail Borovoi has some nice papers (like this one) and talks about this topic. I'm a bit confused between the similarly-named ones, but if you write to him, I'm sure he can recommend where to start. $\endgroup$
    – Gro-Tsen
    Commented May 28 at 8:47
  • $\begingroup$ @Gro-Tsen Thank you! This answers my question. $\endgroup$
    – user148212
    Commented May 28 at 8:58
  • $\begingroup$ OK, then I'll copy it as an actual answer, then. $\endgroup$
    – Gro-Tsen
    Commented May 28 at 9:19

2 Answers 2

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[Copied and extended from comments.]

First, it needs to be clarified in relation to the question that $\mathbb{R}$ does not satisfy the cohomological condition (viꝫ., having cohomological dimension $\leq 2$) in the hypothesis of the cited conjecture by Serre. Indeed, $\mathbb{R}$ has absolute Galois group cyclic of order $2$, and finite cyclic groups have periodic cohomology (so not at all finite cohomological dimension). So the following negative answer is not a counterexample to Serre's conjecture (😅).

Now a negative answer to the question posted is provided, for example, by the simply-connected semisimple algebraic group $G_2$ (the exceptional group of dimension $14$). In this case, the nonabelian Galois cohomology set $H^1(k, G_2)$ classifies octonion algebras over $k$ (Serre, Galois Cohomology, III, appendix 2, 3.3), as is to be expected since $G_2$ is the automorphism group of the octonions; but there are (exactly) two octonion algebras over the real numbers: the split octonions (which are the distinguished element of $H^1(\mathbb{R}, G_2)$) and the usual octonions. (Equivalently, since $G_2$ is also adjoint, $H^1(\mathbb{R}, G_2)$ classifies real forms of $G_2$ itself, of which there are exactly two: the split form, which is the automorphisms of the split octonions, and the compact form, which is the automorphisms of the usual octonions.) So $H^1(\mathbb{R}, G_2)$ has two elements: it is not trivial.

For more about the description of the Galois cohomology of semisimple groups over the reals, I recommend papers and talks by Mikhail Borovoi such as Borovoi & Timashev, Galois cohomology of real semisimple groups. For more about the Galois cohomology of $G_2$ and its relation to octonion algebras, see Springer & Veldkamp, Octonions, Jordan Algebras and Exceptional Groups (2000).

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In addition to the answer of Gro-Tsen: Let $G=\operatorname{SU}(3)$. Then $H^1({\mathbb R},G)$ classifies matrices of Hermitian forms in 3 variables with determinant 1. There are two equivalence classes of such matrices: with representatives $\operatorname{diag}(1,1,1)$ and $\operatorname{diag}(-1,-1,1)$. Thus $\#H^1({\mathbb R},G)=2$. Similarly, $\#H^1({\mathbb R},\operatorname{SU}(2n+1))=n+1$.

In general, if $G$ is a compact connected $\mathbb R$-group with maximal torus $T$ and Weyl group $W={\mathcal N}_G(T)/T$, then there is a canonical isomorphism $$T({\mathbb R})_2/W\cong H^1({\mathbb R},G).$$ This is a result of Borel and Serre of 1964, repeated in Section III.4.5 of Serre's book "Galois cohomology". For a generalization to connected reductive $\mathbb R$-groups that are not necessarily compact, see my paper Galois cohomology of reductive algebraic groups over the field of real numbers, Commun. Math. 30 (2022), Issue 3, 191–201.

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