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? 



A partition of the set of order 2 outer automorphisms of $SU(N)$
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 $\mathrm{Sp}(m)$.


