Spin representation of Spin(9)

Definition

The compact spin group $\operatorname{Spin}(9)$ has a real irreducible spin representation

\Delta_9\cong\mathbb R^{16}\cong\mathbb O^2.

It is the restriction to the maximal compact subgroup of the defining 16-dimensional real action of $SL_2(\mathbb O)\cong \operatorname{Spin}(9,1)$ .

Transitive sphere action

The action is orthogonal and transitive on the unit sphere $S^{15}\subset\Delta_9$ ; the stabilizer of a unit spinor is isomorphic to $\operatorname{Spin}(7)$ . Hence

S^{15}\cong\operatorname{Spin}(9)/\operatorname{Spin}(7).

This is one of the exceptional transitive actions of a compact connected Lie group on a sphere.

Octonionic geometry

The action descends through the octonionic Hopf fibration to a transitive action on $\mathbb OP^1\cong S^8$ . It preserves octonionic-line geometry and is the symmetry responsible for the invariance of the octonionic pseudovolume.

Two appearances of Spin(9)

This spin action on $\mathbb O^2$ is different from the nine-dimensional vector representation obtained from $\operatorname{Spin}(9)\to SO(9)$ . It is also compatible with, but should not be conflated with, the appearance of $\operatorname{Spin}(9)$ as the stabilizer of a point in the Cayley plane.

References

John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §4.2.
Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$ -invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: §§1.3–1.4.