I would like to know the ring structure of $K(Q_n)$ explicitly where $Q_n \subset \mathbb{P}^{n+1}$ is the non-singular $n$-dimensional complex quadric and $K(Q_n) = K^0(Q_n)$ is the complex topological $K$-theory of $Q_n$ (with analytic topology). What is it? Can anyone provide a reference?

Ideally, I would like an expression for $K(Q_n)$ with generators for which I know how to compute the Chern character. Also, as it happens, I am most interested in the case where $n$ is odd.

To my great surprise, I have not been able to find this in the literature (with one qualification, see below). I am certain that such a basic calculation must be well-known to experts but I have been unable to find it. Hence this question.

While I have been aware of the basics of $K$-theory for years, this is the first time I have really had to work with it so I am very inexpert. In the last week I grokked relevant-seeming chunks of the books of Karoubi, Atiyah, Hatcher but I'm still quite green.

**Motivation**

My interest in this ring arose in the course of a problem I have been thinking about but I am hoping that the fact that $K(Q_n)$ is such a basic object will be sufficient motivation to justify this question.

**What I do know**

- Since the quadric has a cell decomposition with only even-dimensional cells, $K^1$ vanishes and $K = K^0$ is free Abelian with rank equal to the number of cells (and generators corresponding to the cells). For $n$ even this rank is $n+2$ (because there is an extra cell in middle dimension) for $n$ odd, it is $n+1$.
- The ordinary cohomology $H^* = H^{\rm even}$ is of course free Abelian of the same rank (Lefshetz tells us the restriction map from $H^*(\mathbb{P}^{n+1}, \mathbb{Z})$ is an isomorphism in all dimensions below $2n+2$ except middle dimension for $n$ even). The Chern character thus embeds $K$ as a maximal-rank lattice inside $H^*(Q_n, \mathbb{Q})$. However it is not the same as the lattice $H^*(Q_n, \mathbb{Z})$.
- Since we know $H^*(Q_n, \mathbb{Q})$ as a ring, it might be satisfactory to know the images of the Chern character on a set of generators of $K(Q_n)$.
- The cases $n=1, 2, 4$ are easy since $Q_n$ is respectively $S^2$, $S^2\times S^2$, $G(2, 4)$ (the complex Grassmannian).
- $Q_n$ is diffeomorphic to the real oriented Grassmannian $\tilde G(2, n+2)$ and so is a homogeneous space $SO(n+2)/SO(2)\times SO(n)$. There are tools for calculating $K$-theory for homogeneous spaces pioneered by Atiyah and Hirzebruch (I believe). Subsequently Hodgkin introduced a spectral sequence which seems to allow relatively straightforward (if lengthy) calculation in many cases, including $Q_n$.
- I managed to find a paper where the above technique is apparently used to calculate $K(\tilde G(k, n))$ for general $k, n$: Sankaran, Zvengrowski "K-theory of Oriented Grassmann Manifolds", Math. Slovaca 47(3). It looks right though I would probably be tempted to work from first principles myself than to specialize their results to my $k=2$ case.

**Bottom line**

Surely I am missing the obvious here? I find it astonishing that I should need to use the methods of Atiyah-Hirzebruch-Hodgkin for such a simple space. Perhaps if I thought more carefully about $\mathbb{P}^{n+1}/Q_n$ or $Q_{n+1}/Q_n$ (bearing in mind natural cell decompositions) then I could use the exact sequences either for the pairs $Q_n \subset \mathbb{P}^{n+1}$ or $Q_{n} \subset Q_{n+1}$ to work this out?

I am tempted to believe the reason I cannot find this in the literature is that it is so trivial. What am I missing?