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Given a complex simple Lie algebra $ \mathfrak g $ of type $E_7$, Cartan subalgebra $ \mathfrak h $ and simple roots $\alpha_1,…\alpha_n $, suppose $\pi $ is an involution of the extended Dynkin diagram. I would like to know whether $\pi $ must be induced from an involution of $ \mathfrak g $. Writing $\Delta $ for the root system and W for the Weyl group, since $ Aut(\Delta)/W $ is isomorphic to the group of automorphisms of the Dynkin diagram and this is trivial for $ E_7 $, the answer is "yes" if the map $ Aut(\mathfrak g,\mathfrak h)\rightarrow Aut(\Delta) $ splits over the Weyl group.

That is, it would suffice if there is a subgroup of the automorphism group of $Aut(\mathfrak g, \mathfrak h)$ which is isomorphic to the Weyl group under this mapping.

Is this true? Or do you otherwise know whether every involution of the extended Dynkin diagram for $ E_7 $ must arise from an involution of $ \mathfrak g $?

Thanks very much!

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Obviously what one would like is a functorial way of constructing Lie algebras from root systems, and one can't have it because the Weyl group acts on the root system and doesn't act on the group. Which is basically your splitting question. The closest I've seen to a functorial construction is in Jacob Lurie's undergrad thesis math.harvard.edu/~lurie/papers/thesis.pdf and a construction Richard Borcherds explained to me that doubles the root system in order to let the normalizer of the Z-torus act: math.berkeley.edu/~allenk/courses/spr02/261b/notes/… –  Allen Knutson Jul 22 '11 at 6:17
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Thank you for this comment Allen, it was very helpful in answering a question I hadn't been able to ask. –  B. Bischof Sep 7 '12 at 14:47

3 Answers 3

There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ contains a reflection subgroup of type $A_1 + D_6$ which acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\alpha|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.

As for the second part of the question, it seems likely to me that the involution of the extended Dynkin diagram of $\Delta$ can be lifted to an involution of $G$. It has the form $w_0w_1$ where $w_0$ is the longest element of $W$ and $w_1$ is that of the Weyl group of the $E_6$-subdiagram. Let $l$ be the corresponding Levi subalgebra of $g$. One just needs to find an involution in $N_G(l)$ which acts as the nontrivial graph automorphism on $[l,l]$ and as $-1$ on the centre of $l$.

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Sasha Premet has provided a concrete direct approach to the question, which at first sight looks convincing. But I'd like to follow up with more detail about one aspect of the question. First, the paper by Tits which he mentions is Normalisateurs de tores I, J. Algebra 4 (1966), 96-116 (alas there is no part II). Basically Tits is investigating the relationship between the normalizer of a maximal torus, in a split reductive group over some field, and the resulting Weyl group. But he doesn't quite state explicitly here in which simple groups the extension splits (in other words, when is the Weyl group naturally a subgroup of the algebraic group).

Later some topologists at Rice (Morton Curtis, Alan Wiederhold, Bruce Williams) wrote a couple of joint papers, the first one Normalizers of maximal tori appearing in Springer Lect. Notes in Math. 418 (1974), 31-47. Here their framework involves compact connected semisimple Lie groups, so a standard translation into the language of complex Lie algebras or corresponding algebraic groups is needed. But they do show directly that two such Lie groups are isomorphic if and only if the respective normalizers are isomorphic (Theorem 1). Along with this, they work out explicitly a table (Theorem 2) showing which compact simple Lie groups (simply connected or taken modulo center) have a natural subgroup isomorphic to the Weyl group. Here only type $G_2$ among the exceptional groups has such a splitting of the normalizer. The method is reasonable but too long to outline here.

Amusingly, they write at first: It seems strange that Theorems 1 and 2 do not seem to have been known, because their proofs use no techniques not known for many years. But then an added footnote states that they learned recently that in unpublished work Tits had earlier proved Theorem 2 in a more comprehensive way.

One other comment is that automorphisms of finite order of semisimple complex Lie algebras play a major role in the work of Victor Kac and are discussed in his book on infinite dimensional Lie algebras as well as in section X.5 of the classic book by Helgason Differential Geometry, Lie Groups, and Symmetric Spaces. But these sources don't seem to address the specific question asked.

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Jim, many thanks for pointing out the reference to the paper by topologists. Regarding Theorem 2, did the authors prove that the extension is non-split outside type $G_2$? The argument I gave can be carried out for other types as well (with some modfications). –  Alexander Premet Sep 8 '12 at 8:52
    
@Sasha: Yes. Theorem 2 covers each simple type including $E_7$. The authors consider only simply connected or adjoint groups (which is enough for this example), but in their footnote they mention that Tits has also treated the various intermediate groups in his unpublished work. I can't immediately compare their approach with yours. –  Jim Humphreys Sep 8 '12 at 18:30

Thanks very much all for these most helpful answers, much appreciated! Regarding the 2nd (and easier) part of the question, as you suspected this is true. In the preprint http://front.math.ucdavis.edu/1111.4028 Katharine Turner and I were needing this for some work on harmonic maps, and the form of the statement we prove there is below. It seems like something that would be known to folks working in this area, but we couldn't find a reference.

Every involution of the extended Dynkin diagram for a simple complex Lie algebra $\mathfrak {g} ^\mathbb {C} $ is induced by a Cartan involution of a real form of $\mathfrak {g} ^\mathbb {C} $.

More precisely, let $\mathfrak {g}^\mathbb {C} $ be a simple complex Lie algebra with Cartan subalgebra $\mathfrak {t} ^\mathbb {C} $ and choose simple roots $\alpha_1,\ldots,\alpha_N $ for the root system $\Delta (\mathfrak {g} ^\mathbb {C},\mathfrak {t} ^\mathbb {C}) $. Given an involution $\pi $ of the extended Dynkin diagram for $\Delta $, there exists a real form $\mathfrak {g} $ of $\mathfrak {g} ^\mathbb {C} $ and a Cartan involution $\theta $ of $\mathfrak {g} $ preserving $\mathfrak {t} =\mathfrak {g}\cap\mathfrak {t} ^\mathbb {C} $ such that $\theta $ induces $\pi $ and $\mathfrak {t} $ is a real form of $\mathfrak {t} ^\mathbb {C} $. The Coxeter automorphism $\sigma $ determined by $\alpha_1,\ldots,\alpha_N $ preserves the real form $\mathfrak {g} $.

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