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Power operations in elliptic cohomology and representations of loop groups. (English) Zbl 0958.55016

The best way to approach this paper is probably by way of the example of the elliptic cohomology associated with the Weierstrass model of an elliptic curve. Various authors have shown that, up to the introduction of denominators there is an Atiyah-like isomorphism between \( \text{Ell}^*(BG)\) and a completion of what one can call an elliptic representation ring for the finite group \(G\). One can either think of this as being defined by suitable infinite dimensional representations of the loop group \(LG\), or of compatible families of representations of the centralizers of elements representing conjugacy classes of elements in \(G\). The modularity condition on the characters of these representations allows one to introduce Hecke operators into the completed representation ring. Using the geometry of the Tate curve over \(Z[1/6]((q))\) Andrew Baker was able to construct analogous stable operations defined on \(\text{Ell}^*(X)\) and taking values in \(\text{Ell}^*(X)[1/p]\), and these two constructions are compatible with the elliptic character map inducing the near-isomorphism above. All this can be made explicit for particular groups \(G\), such as the Mathieu group \(M_{24}\).
The aim of the paper under review is to put these results in a more general setting. The first part uses isogenies of elliptic curves to describe unstable cohomology operations into a wide class of theories. These are complex-oriented and sometimes 2-periodic. The second part discusses the geometric description of such theories by means of loop group representations. Without being too precise there is an isomorphism between the \(Z((q))\)-module of representations of \(LG\) of level \(k\) and the \(Z((q))\)-module of Weyl group invariant global sections of the \(k\)th power of a line bundle over the tensor product of the Tate curve and \(\operatorname{Hom}[S^1,T]\). Here \(T\) is a maximal torus in the connected, compact Lie group \(G\). The third part ‘works backwards from the effect of operations in cohomology \((K_{\text{Tate}})\) to operations on representations of \(LG\). The extra operations reflect the interaction of tensor powers and the circle group acting on loop groups by rotation’.
This is undoubtedly an important and carefully written paper. The problem, at least for the reviewer, is that it is written in a very dense language, one known to a small circle of experts, but hard for the uninitiated to understand. However, this said, it is important to know that the foundations of the subject have been set down in this way, and that the results which looked as though they ought to hold in certain cases do actually form part of a general theory. But reader beware!

MSC:

55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N91 Equivariant homology and cohomology in algebraic topology
22E67 Loop groups and related constructions, group-theoretic treatment
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