The classification of p-compact groups and homotopical group theory

We survey some recent advances in the homotopy theory of classifying spaces, and homotopical group theory. We focus on the classification of p-compact groups in terms of root data over the p-adic integers, and discuss some of its consequences e.g. for finite loop spaces and polynomial cohomology rings.Comment: To appear in Proceedings of the ICM 2010

Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin

With an explicit, algebraic indexing $(2,1)$-category, we develop an efficient homotopy theory of cyclonic objects: circle-equivariant objects relative to the family of finite subgroups. We construct an $\infty$-category of cyclotomic spectra as the homotopy fixed points of an action of the multiplicative monoid of the natural numbers on the category of cyclonic spectra. Finally, we elucidate and prove a conjecture of Kaledin on cyclotomic complexes.Comment: 28 pages. Comments very welcom

On generalised Deligne--Lusztig constructions

This thesis is on the representations of connected reductive groups over finite quotients of a complete discrete valuation ring. Several aspects of higher Deligne–Lusztig representations are studied. First we discuss some properties analogous to the finite field case; for example, we show that the higher Deligne–Lusztig inductions are compatible with the Harish-Chandra inductions. We then introduce certain subvarieties of higher Deligne–Lusztig varieties, by taking pre-images of lower level groups along reduction maps; their constructions are motivated by efforts on computing the representation dimensions. In special cases we show that their cohomologies are closely related to the higher Deligne–Lusztig representations. Then we turn to our main results. We show that, at even levels the higher Deligne–Lusztig representations of general linear groups coincide with certain explicitly induced representations; thus in this case we solved a problem raised by Lusztig. The generalisation of this result for a general reductive group is completed jointly with Stasinski; we also present this generalisation. Some discussions on the relations between this result and the invariant characters of finite Lie algebras are also presented. In the even level case, we give a construction of generic character sheaves on reductive groups over rings, which are certain complexes whose associated functions are higher Deligne–Lusztig characters; they are accompanied with induction and restriction functors. By assuming some properties concerning perverse sheaves, we show that the induction and restriction functors are transitive and admit a Frobenius reciprocity

Rational equivariant cohomology theories with toral support

For an arbitrary compact Lie group GG, we describe a model for rational GG–spectra with toral geometric isotropy and show that there is a convergent Adams spectral sequence based on it. The contribution from geometric isotropy at a subgroup KK of the maximal torus of GG is captured by a module over H∗(BWeG(K))H∗(BWGe(K)) with an action of π0(WG(K))π0(WG(K)), where WeG(K)WGe(K) is the identity component of WG(K)=NG(K)∕KWG(K)=NG(K)∕K