# Hilbert's Satz 90 for real simply-connected groups?

$$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$$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 $$\mathbb{R}$$?

• 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. Commented May 28 at 8:32
• 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. Commented May 28 at 8:47
• @Gro-Tsen Thank you! This answers my question. Commented May 28 at 8:58
• OK, then I'll copy it as an actual answer, then. Commented May 28 at 9:19

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).
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.