Rational equivariant cohomology theories with toral support

TL;DR

This paper develops a model for rational G-spectra with toral support for any compact Lie group G, enabling a convergent Adams spectral sequence based on geometric isotropy modules.

Contribution

It introduces a new algebraic model capturing geometric isotropy at subgroups of the maximal torus, extending rational equivariant cohomology theories.

Findings

Constructed a model for rational G-spectra with toral isotropy.

Established a convergent Adams spectral sequence for these spectra.

Linked geometric isotropy to modules over specific cohomology rings.

Abstract

For an arbitrary compact Lie group G, we describe a model for rational G-spectra with toral geometric isotropy and show that there is a convergent Adams spectral sequence based on it. The contribution from geometric isotropy at a subgroup K of the maximal torus of G is captured by a module over H^*(BW_G(K)_e) with an action of \pi_0(W_G(K)), where W_G(K)=N_G(K)/K and the subscript e denotes the identity component.