Morava K-theory of orthogonal groups and motives of projective quadrics.

For a generic quadric $Q$ we consider the category of Morava K-theory motives, and determine the indecomposable summands of the $\mathrm K(n)$-motive of $Q$. We also prove that the Morava K-theory of a group variety $\mathrm{SO}_m$ stabilizes for large $m$.