Is the real KO-theory of real Stiefel $V\_k(\mathbb{R}^n)$ manifold known? Or have the real vector bundles over the real Steifel manifolds $V\_k(\mathbb{R}^n)$ been studied?



KO-theory of real Stiefel manifold
The case of $V\_1(\mathbb{R}^n) = \mathbb{R}P^{n-1}$ is in 


*Michikazu Fujii*, MR 219060 [**$K\_{O}$-groups of projective spaces**](http://www.ams.org/mathscinet-getitem?mr=219060), *Osaka J. Math.* **4** (1967), 141--149.


Perhaps some of the methods there will generalise to $k>1$? The complex case is also known, and is in 


*Daisuke Kishimoto*, MR 2103788 [**$KO$-theory of complex Stiefel manifolds**](http://www.ams.org/mathscinet-getitem?mr=2103788), *J. Math. Kyoto Univ.* **44** (2004), no. 3, 669--674.


