The primary goal of this paper is to study SpanierโWhitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the LubinโTate spectrum ๐ธ๐, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that ๐ธ๐ is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group ๐พ๐ of the formal group in question. In this paper we find that the ๐พ๐-equivariant dual of ๐ธ๐ is in fact ๐ธ๐ twisted by a sphere with a non-trivial (when ๐>1) action by ๐พ๐. This sphere is a dualizing module for the group ๐พ๐, and we construct and study such an object ๐ผ๐ข for any compact p-adic analytic group ๐ข. If we restrict the action of ๐ข on ๐ผ๐ข to certain type of small subgroups, we identify ๐ผ๐ข with a specific representation sphere coming from the Lie algebra of ๐ข. This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local SpanierโWhitehead duals of ๐ธโ๐ป๐ for select choices of p and n and finite subgroups H of ๐พ๐.
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The principal series of p-adic groups with disconnected center: THE PRINCIPAL SERIES OF p-ADIC GROUPS WITH DISCONNECTED CENTER
- NSF-PAR ID:
- 10046684
- Publisher / Repository:
- DOI PREFIX: 10.1112
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 114
- Issue:
- 5
- ISSN:
- 0024-6115
- Page Range / eLocation ID:
- 798 to 854
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
