# What is the name of the real form corresponding to the quaternionic symmetric space?

Let $$G$$ be a compact simple Lie group. Choose a system of positive roots, and let $$\mathrm{SU}(2) \subset G$$ correspond to the highest root, and $$\mathbb{Z}/2 \subset \mathrm{SU}(2)$$ the centre. The centralizer of this $$\mathbb{Z}/2$$ inside of $$G$$ is a subgroup $$H \subset G$$ of shape $$\mathrm{SU}(2) \circ K = (\mathrm{SU}(2) \times K) / (\mathbb{Z}/2)$$. The Dynkin diagram for $$H$$ can be found by drawing the affine dynkin diagram for $$G$$, and deleting the node(s) adjacent to the affine root. The now-isolated affine root is the copy of $$\mathrm{SU}(2)$$, and the rest of the Dynkin diagram for $$K$$. In the type-A case, $$H$$ is reductive but not simple, picking up a $$\mathrm{U}(1)$$ factor; this is because in that case the affine root had two neighbours, not just one. The list of $$H$$'s is available in the Wikipedia article Quaternion-Kähler symmetric space, because the quotient spaces $$G/H$$ are precisely the quaternionic symmetric spaces.

Standard arguments then say that $$G$$ has a real form with maximal compact $$H$$. It is not the compact form (except for $$G = H = \mathrm{SU}(2)$$), and it is usually not the split real form. Rather, it is a third canonical real form for any group. For the classical series, it is $$\mathrm{SU}(2,n-2)$$, $$\mathrm{SO}(4,n-4)$$, and $$\mathrm{Sp}(1,n-1)$$. If I am reading Wikipedia correctly, then, together with $$\mathrm{SO}(3,n-3)$$, these are the real forms that admit quaternionic discrete series representations.

Does this canonical real form have a standard name in the literature?

• This is a fairly trivial answer, but the term “quaternionic real form” is found by Google in a number of texts and seems to correspond to what you're describing. (Thanks for pointing out that this form is canonical, I had never realized this!) – Gro-Tsen May 3 at 17:20

There are other references to “quaternionic real form” found by Google, and all seem to refer to the same thing. I have seen it regularly at least to denote the form of $$E_8$$ with Cartan index $$-24$$. So I think we can say that “quaternionic real form” is a reasonably standard term.
(There is still at least some possibility of confusion as this question uses the term, albeit with quotes, to denote a different real form of the $$D_n$$ series.)
• @TheoJohnson-Freyd Regarding the form mentioned in the linked question (which, again, is not the one you were asking about), the notation $SO^*(n)$ is fairly standard, and it's the only real form of $SO(n)$ (for $n$ even) that's not a $SO(k,n-k)$. Its maximal compact subgroup is $U_n$ and it's called DIII in Cartan's notation. – Gro-Tsen May 3 at 19:17