A remark on Baker operations on the elliptic cohomology of finite groups (Q1579038)

Let \(G\) be a finite group and \(BG\) its classifying space. It has been known for some time that every element of \(Ell^{\text{(even)}}(BG)\) gives a certain \(p\)-adic limit of a Thompson series by applying the elliptic character. Actually rather more is true -- up to taking denominators there is an isomorphism between \(Ell^{\ast}(BG)\) and a suitably completed `elliptic representation ring'. Indeed, if the Sylow structure of \(G\) is not too complicated we can even remove the denominators. This note makes clear that the maps above are natural in the sense that the Hecke operators on representations (arising ultimately from Hecke operators on modular forms) are compatible with certain stable (Baker) operations in elliptic cohomology. This is hardly surprising, given the way in which the Baker operations are defined. Exercise: try it on cyclic groups of prime power order.