K-Theory, Hecke Algebras and Representation Theory 
marking Roger Plymen's 75th birthday

July 3-6, Sheffield

Organized by 
Nigel Higson, Bram Mesland, Haluk Sengun, Hang Wang 


All talks will take place in Lecture Theatre C of the Hicks Building.
See below for the titles and abstracts.

14:30 - 15:30 Mesland  [SLIDES]
16:00 - 17:00 Emerson  [NOTES]

9:30 - 10:30 Wang  [NOTES]
11:00 - 12:00 Afgoustidis  [SLIDES]

14:00 - 15:00 Clare  [SLIDES]
15:30 - 16:30 Higson  [SLIDES]


9:30 - 10:30 Hochs  [SLIDES]
11:00 - 12:00 Schlichtkrull  [NOTES]

14:00 - 15:00 Romano  [NOTES]
15:30 - 16:30 Crisp

9:30 - 10:30 Voigt
11:00 - 12:00 Solleveld  [SLIDES]

14:00 - 15:00 Mendes  [SLIDES]
15:30 - 16:30 Aubert   [SLIDES]



"A Hecke module structure on the KK-theory of arithmetic groups"

Let G be a locally compact group, H a discrete subgroup and C(G,H) the commensurator of H in G. The cohomology of H is a module over the Shimura Hecke ring of the pair (H,C(G,H)). This construction recovers the action of the Hecke operators on modular forms for SL(2,Z) as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair (H, C(G,H)) maps into the KK-ring associated to an arbitrary H-C*-algebra. From this we obtain a variety of K-theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of H-spaces. Examples include the Borel-Serre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the Bruhat-Tits tree of SL(2,Z). 

This is joint work with M.H. Sengun (Sheffield).


"Noncommutative Lefshetz fixed-point formulas via KK-theory

Using the idea of K-theoretic Poincaré duality, it is possible to formulate an analogue of the classical Lefschetz fixed-point formula from basic algebraic topology, which applies to KK-endomorphisms (e.g. ordinary C*-algebra endomorphisms) of a C*-algebra equipped with such a duality. We discuss the general methodology and apply it in the example of the C*-algebra crossed-product of a discrete group acting properly, smoothly and co-compactly on a smooth manifold. The result is an `orbifold' Lefschetz formula of some interest; our hope is that many other examples should exist. 


"Role of local Langlands correspondence in K-theory of group C*-algebras"

K-theory of C*-algebras associated to a Lie group can be understood both from the geometric point of view via Baum-Connes assembly map and from the representation theoretic point of view via harmonic analysis of Lie groups. Inspired by the local Langlangds correspondence and work by Plymen and collaborators, one can study relations between two groups, where their L-parameters are related in a nice way, from the aspects of K-theory and index theory of invariant elliptic operators. I will introduce two examples I investigated with Peter Hochs (when the two groups are inner forms to each other) and with Kuok Fai Chao (when there is a base change involved between the L-parameters of the two groups). 


"On the tempered dual of a real reductive group and that of its Cartan motion group"

Given a reductive Lie group G and a maximal compact subgroup K, one can consider the isometry group of the (flat) tangent space to G/K at the identity coset: this is a first-order approximation of G near K, called the Cartan motion group of G. George Mackey’s early work on semi-direct products describes its unitary representations in very simple and concrete terms. 

In the 1970s, Mackey noticed that his parametrization for the representations of the motion group showed unexpected similarities with Harish-Chandra’s more subtle parametrization for the tempered representations of G. Motivated by quantum-mechanical considerations related with the existence of a one-parameter family of Lie groups interpolating between both groups, he suggested that a kind of rigidity of representation theory along the deformation may be observed in general. Alain Connes and Nigel Higson later pointed out that the Baum-Connes-Kasparov isomorphism in operator K-theory can be viewed, for real reductive groups, as a cohomological reflection of Mackey’s ideas. For the special case of complex semisimple groups, Nigel Higson gave in 2008 a precise form to Mackey’s analogy and its relationship with the Baum-Connes-Kasparov isomorphism.

For real reductive groups, I will describe a natural one-to-one correspondence between the tempered and admissible duals of both groups, and discuss some geometrical (or topological) aspects of the rigidity revealed by the correspondence along the deformation from one group to the other.


"On the reduced C*-algebra of real reductive groups"

I will report on joint work with Nigel Higson regarding the description up to Morita equivalence of the reduced C*-algebra of a class of real reductive groups. The results build on previous work, joint with Tyrone Crisp, and are related to the approach to the Connes-Kasparov isomorphism promoted by Roger Plymen and others.


"On (some of) the work of Roger Plymen"

I shall give an appreciation of some of the fundamental contributions of Roger Plymen to the themes of this conference, focusing on his studies of the  C*-algebras of real and p-adic groups, and their K-theory groups.


"K-types of tempered representations and index theory"

Let G be a semisimple Lie group. Tempered representations of G are the ones occurring in the Plancherel decomposition of L^2(G). They are also relevant to the Langlands classification of the more general admissible representations. In joint work with Yanli Song and Shilin Yu, we realise the restriction of any tempered representation to a maximal compact subgroup K as an equivariant index. This is a concrete expression of Kirillov's orbit method. A consequence of this realisation is a geometric expression for the multiplicities of the irreducible representations of K in that restriction. (The irreducible representations that occur are the K-types of the representation.) This helps to study the general behaviour of those multiplicities. As an example, we show that admissible representations of SU(p,1) and SO_0(p,1) restrict multiplicity-freely to maximal compact subgroups. That was proved earlier by Koornwinder, but now illustrates our multiplicity formula.


"Harmonic analysis on real spherical spaces"

Let G be a real reductive Lie group. A homogeneous space Z of G is called real spherical if the minimal parabolic subgroups of G have only finitely many orbits on Z. For example, the Bruhat decomposition of G implies that Z=G is real spherical for the two-sided action of GxG. A survey will be given of some recent progress (by F. Knop, B. Krötz, and others) on the generalization of Harish-Chandra's harmonic analysis to such spaces.


"The local Langlands correspondence in small residue characteristic"

Through explicit examples, I'll discuss why the local Langlands correspondence becomes mysterious for small residue characteristic. I'll focus on examples and conjectures related to ``epipelagic" representations, which have minimal positive depth.


"Parabolic induction over the p-adic integers"

For p-adic reductive groups like GL(n,Q_p), the right-hand side of the Baum-Connes conjecture --- i.e., the K-theory of the group C*-algebra --- is in many respects better understood than the left-hand side. This unusual state of affairs is due to the extremely complicated representation theory of compact p-adic groups like GL(n,Z_p). In this talk I shall present an ongoing program, joint with Ehud Meir and Uri Onn, that aims to understand the representations of these compact groups in terms of parabolic induction from Levi subgroups, analogously to the way one usually studies representations of real, complex, p-adic, and finite reductive groups.


"Categorification and Hecke algebras"

The idea of categorification is to replace set theoretic constructions and theorems by category theoretic analogues, recovering the original constructions via taking isomorphism classes or K-groups. In this talk I’ll discuss some examples of this procedure related to Hecke algebras and connections to noncommutative geometry.


Topological K-theory of affine Hecke algebras

An affine Hecke algebra can be completed to a C*-algebra. These algebras appear in the theory of reductive p-adic groups, and they are of interest in representation theory and in relation with the Baum--Connes conjecture.They provide typical examples of C*-algebras which are close to commutative.
In this talk I will discuss results about the K-theory of such C*-algebras, and techniques used to study it. In particular, I will show that the K-theory does not depend on the deformation parameter of the Hecke algebra. In the end all calculations will be reduced to equivariant K-theory of topological spaces, with respect to certain nice actions of finite groups. I will show that under mild conditions these equivariant K-groups are torsion-free.


"On L-packets and depth for SL2(K) and its inner forms"

An invariant that makes sense on both sides of the local Langlands correspondence is depth. In this talk we survey the notion of depth and study depth-preservation under the local Langlands correspondence. Examples will be provided with special emphasis to the group SL2 over a local field with characteristic 2.

This talk is based on joint work with Anne-Marie Aubert, Roger Plymen and Maarten Solleveld.


"Affine Hecke algebras on the Galois side"

We will explain a way to attach affine Hecke algebras to certain Langlands parameters on Levi subgroups of a given p-adic reductive group in relation with the ABPS-conjecture. 

This is joint work with Ahmed Moussaoui and Maarten Solleveld.