Rational Mackey functors for compact lie groups I

Abstract

A Mackey functor $M$ is a structure analogous to the representation ring functor $H \mapsto R(H)$ encoding good formal behaviour under induction and restriction. More explicitly, $M$ associates an abelian group $M(H)$ to each closed subgroup $H$ of a fixed compact Lie group $G$, and to each inclusion $K \subseteq H$ it associates a restriction map ${\rm res}^H_K:M(H) \rightarrow M(K)$ and an induction map${\rm ind}^H_K:M(K) \rightarrow M(H)$. This paper gives an analysis of the category of Mackey functors $M$ whose values are rational vector spaces: such a Mackey functor may be specified by giving a suitably continuous family consisting of a ${\Bbb Q} \pi_0(W_G(H))$-module $V(H)$ for each closed subgroup $H$ with restriction maps $V(\hat{K}) \rightarrow V(K)$ whenever $K$ is normal in $\hat{K}$ and $\hat{K}/K$ is a torus (a ‘continuous Weyl-toral module’). We show that the category of rational Mackey functors is equivalent to the category of rational continuous Weyl-toral modules. In Part II this will be used to give an algebraic analysis of the category of rational Mackey functors, showing in particular that it has homological dimension equal to the rank of the group.

1991 Mathematics Subject Classification: 19A22, 20C99, 22E15, 55N91, 55P42, 55P91.