Invariants, exponents and formal group laws

TL;DR

This paper generalizes the concept of exponents related to Weyl group actions to various algebraic oriented cohomology theories, linking classical invariants to formal group laws and analyzing torsion in twisted flag varieties.

Contribution

It extends the notion of exponents to algebraic oriented cohomology theories and relates classical invariants like the Dynkin index to formal group laws.

Findings

Generalized exponents for Weyl group actions in cohomology theories

Connection between Dynkin index and formal group law deformation

Application to torsion in twisted flag varieties

Abstract

Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by reflections. In the present notes we extend the notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline, arXiv:1106.4332] to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel, Panin-Smirnov and the associated formal group law. From this point of view the classical Dynkin index of the associated Lie algebra will be the second exponent of the deformation map from the multiplicative to the additive formal group law. We apply this generalized exponent to study the torsion part of an arbitrary oriented cohomology theory of a twisted flag variety.