Factorization of Aut(BK)-actions through Aut(K)

Determine whether the Aut(BC_n)-action on the C_n-geometric fixed points of equivariant elliptic cohomology theories factors through Aut(C_n), and whether the Aut(BK)-action on the K-geometric fixed points of an arbitrary tempered cohomology theory factors through Aut(K).

Background

The paper distinguishes the canonical higher-categorical Aut(BK)-action on geometric fixed points from the truncated Aut(K)-action used to construct the principal Galois extensions. In general, the canonical action need not factor through Aut(K), and consequently ordinary Weyl group actions need not factor through global Weyl group actions.

For tempered cohomology theories associated with tori, and for elliptic cohomology theories when K is C_n times C_n, the paper proves specific factorization results. It leaves unresolved whether the same factorization holds for cyclic subgroups in equivariant elliptic cohomology and, more broadly, for arbitrary tempered cohomology theories.