Morava $J$-invariant

TL;DR

This paper computes the algebraic Morava K-theory co-multiplication for split orthogonal groups, enabling the decomposition of Morava motives of orthogonal Grassmannians and defining a Morava K-theory analogue of the classical J-invariant.

Contribution

It introduces a Morava K-theory analogue of the J-invariant and computes the co-multiplication for algebraic Morava K-theory in the context of orthogonal groups.

Findings

Decomposition of Morava motives of orthogonal Grassmannians

Definition of Morava K-theory J-invariant

Explicit computation of co-multiplication in algebraic Morava K-theory

Abstract

We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the $J$-invariant in terms of the ordinary (Chow) $J$-invariant.