A functorial description of the Morava K-theories of elementary abelian 2-groups

The aim of this article is to study, from a functorial viewpoint, the mod 2 Morava K-theories of elementary abelian 2-groups. Namely, we study the functors V bar right arrow K(n) * (BV) for the prime p = 2 and n a positive integer. They are graded over Z/(2(n+1) - 2), the odd terms of this graduation are trivial. The case n = 1, which follows directly from the work of Atiyah on topological K-theory, gives us a coanalytic functor which contains no non-constant polynomial sub-functor. This is very different from the case n > 1, where the above-mentioned functors are analytic. The case of K(2)* is very special: the functor is auto-dual.