Let $N$ be an even integer, $N>2$. Let $E$ be the set of all outer automorphisms $\phi$ of $G = SU(N)$ which are of order 2, i.e. $\phi \circ \phi = \mathrm{id}_G$.

Choose a particular element $\psi \in E$. Since $\mathrm{Out}(G) \simeq \mathbb{Z}_2$, for all $\phi \in E$ there exists a matrix $A_{\phi} \in G$ such that $$\forall g \in G , \quad \phi(g) = A_{\phi} \psi (g) A_{\phi}^{-1} \, . $$ A simple computation shows that $$A_{\phi} \psi (A_{\phi}) = \psi (A_{\phi}) A_{\phi} = \alpha_{\phi} \mathbf{1}_N$$ with $\alpha_{\phi} \in \{1,-1\}$.

So there is a partition of $E$ in terms of the value of $\alpha$. Does this fact have a name? Are there references discussing it? Is there a geometric interpretation of this partition?


I don't know that the partition has a name, so to speak, but it is well-understood and falls into the classification of the symmetric spaces of type A. Namely, those of type AI, which are $\mathrm{SU}(n)/\mathrm{SO}(n)$ and, when $n=2m>2$ is even, those of type AII, which are $\mathrm{SU}(2m)/\mathrm{Sp}(m)$.

$\mathrm{SU}(2)$ does not have any outer automorphisms.

When $n$ is odd, the set of outer automorphisms of order $2$ is a connected set, and they are all conjugate to simple conjugation in $\mathrm{SU}(n)$, i.e., the fixed subgroup of such an outer automorphism is a conjugate of $\mathrm{SO}(n)$.

When $n=2m>2$, there are two components of the space of outer automorphisms of order $2$ (i.e., outer involutions): One, which consists of the outer involutions whose fixed subgroup is a conjugate of $\mathrm{SO}(2m)$ and another, which consists of the space of outer involutions whose fixed subgroup is conjugate to $\mathrm{Sp}(m)$.

  • $\begingroup$ Thanks for the precise answer! Now I realize that having the precise key-word "involutive automorphism" gives me a lot of relevant references (e.g. the book by Helgason, or the one by Onishchik and Vinberg). $\endgroup$
    – Antoine
    Sep 6 '18 at 17:56

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