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

TL;DR

This paper computes the algebraic Morava K-theory of orthogonal groups and applies these results to analyze the motivic decompositions of quadrics, revealing new stabilization phenomena and indecomposable summands.

Contribution

It provides the first detailed computation of Morava K-theory rings for orthogonal and spin groups and explores their implications for motivic decompositions of quadrics.

Findings

Stabilization results for Morava K-theory of orthogonal groups

Determination of indecomposable summands of Morava motives of quadrics

Application of Morava K-theory to motivic decompositions

Abstract

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.