K-Theory and Representation Theory

On certain components of the tempered dual of classical p-adic groups

 
Suppose G is a real or p-adic reductive group. The Fell topology on the tempered dual of G, can be studied by noncommutative-geometric methods: it naturally identifies with the spectrum of the C*-algebra of G, and its connected components identify with the spectra of certain `blocks’ in the C*-algebra.
 
For real reductive groups, A. Wassermann proved in 1987 that each `block’ has, up to Morita equivalence, a beautiful and simple structure. This was a crucial step in his proof of the Baum-Connes-Kasparov conjecture for G. For p-adic groups, it is not obvious that such a structure can exist, but important examples were given by R. Plymen and his students.
 
I will report on joint work with Anne-Marie Aubert which (1) for arbitrary G, gives a geometric condition for the existence of a Wassermann-type structure on a given block, and (2) when G is a quasi-split symplectic, orthogonal or unitary group, explicitly determines the connected components of the tempered dual for which the geometric assumption is satisfied.

Towards an explicit Langlands correspondence for $p$-adic groups: The example of G_2.

 
Let F be a non-archimedean local field and let G be the group of F-rational points of a connected reductive algebraic group defined over F. We will express the Langlands correspondence for irreducible non-supercuspidal smooth representations of G as a conjectural correspondence between certain (possibly twisted) extended quotients, a notion that originates in noncommutative geometry. As an illustration of this general picture, we will construct an explicit local Langlands correspondence for the exceptional group of type G_2.

A categorical Deligne-Langlands correspondence for split reductive groups

 
Let G be a split reductive group over a p-adic field F. We construct a natural coherent sheaf on the moduli stack of unipotent Langlands paramters for G, called the coherent Springer sheaf, whose self-Ext algebra is naturally isomorphic to the Iwahori Hecke algebra for G. As a consequence we deduce the existence of a fully faithful embedding of the Iwahori block of Rep(G) into the derived category of ind-coherent sheaves on the moduli stack of Langlands parameters. For G = GL_n we can go further and construct a fully faithful embedding of the category of all smooth representations of G into this derived category.

Equivariant analytic torsion for proper actions

 
Analytic torsion for compact manifolds was constructed by Ray and Singer in the 1970s, as a way to realise Reidemeister-Franz torsion analytically. Cheeger and Mueller proved independently that the two notions of torsion are indeed equal. In this way, analytic torsion is a link between analysis and topology. It is also related to Quillen metrics, and to dynamical systems via the Fried conjecture. In the 1990s, different approaches to a construction of equivariant analytic torsion were developed, incorporating group actions. The two main kinds of group actions considered were actions by finite or compact groups on compact manifolds, and actions by fundamental groups on universal covers of compact manifolds. With Hemanth Saratchandran, we construct a general notion of equivariant analytic torsion for proper group actions, and study its properties. This unifies and extends earlier work on equivariant analytic torsion.

Theta correspondence, Rieffel induction and Morita equivalence

 
Howe's local theta correspondence relates irreducible representations of a reductive dual pair (G,H) of subgroups of the metaplectic group over a local field. In joint work with Haluk Sengun (Sheffield) we have shown that various instances of Howe's correspondence can be realised as a Rieffel induction via a C*-correspondence between the full or reduced group C*-algebras of G and H. The C*-correspondence is constructed directly from the oscillator representation. In the equal rank case, it gives rise to a Morita equivalence between ideals of the reduced C*-algebras of G and H respectively. In this talk I will discuss our construction and its implications for K-theory and representation theory.

A Fourier transform for unipotent representations of p-adic groups

 
Representations of finite reductive groups have a rich, well-understood structure, first explored by Deligne--Lusztig. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we show a way to lift some of this structure to representations of p-adic groups. In particular, we work with unipotent representations of split p-adic groups and their inner twists. We consider the relation between Lusztig's nonabelian Fourier transform and a certain involution we define on the level of p-adic groups using Langlands parameters. This talk will be an introduction to these ideas with a focus on examples.

Conditions for discrete series

 
Let G be a real reductive Lie group with maximal compact subgroup K. A geometric criterion for the existence of discrete series representations of G was determined by Harish-Chandra: K has to have the same rank as G. A new proof (joint with Krötz, Kuit and Opdam) will be presented, and possible generalizations to homogeneous spaces of G will be briefly discussed.

The Fried conjecture for admissible twists

 
The relation between the spectrum of the Laplacian and the closed geodesics on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured that the analytic torsion, which is an alternating product of regularized determinants of the Laplacians, equals the zero value of the dynamical zeta function. In this talk, I will show the Fried conjecture on locally symetric spaces twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises the results of myself for unitarily twists, and the results of Brocker, Muller, and Wotzker on closed hyperbolic manifolds.

Hochschild homology of reductive p-adic groups

 
Hochschild homology is a subtle invariant of algebras. It is defined for all (possibly noncommutative) algebras and generalizes differential forms on smooth varieties.

In this talk we discuss the Hochschild homology of Hecke algebra H(G) of a reductive p-adic group G. It turns out that the Hochschild homology of a single Bernstein block H(G)^s of H(G) can be described with just three pieces of data:
 
- the group X_nr (L) of unramified characters of a Levi subgroup L of G,
 
- a finite group W(L,s) acting on X_nr (L),
 
- a 2-cocycle of W(L,s).
 
Our results can be regarded as a refinement of the ABPS conjectures about irreducible G-representations.

Connes-Kasparov theorem, lowest K-type and Dirac cohomology

 
The Connes-Kasparov theorem is about the computation of K-theory of group C*-algebra. The lowest K-type is a notion introduced by Vogan and is useful in the classification of tempered representation of a real reductive Lie group. And Dirac cohomology relates the Dirac operators and representation theory. In this talk, I will discuss some results on the connection between Connes-Kasparov theorem and Dirac cohomology. This is a joint work with Clare, Higson and Tang.

Representations and Hecke algebras for p-adic classical groups

 
I will try to give a survey of what is known about the structure of the category of smooth representations for p-adic classical groups: their block decomposition, the structure of blocks as modules over an algebra, and the structure of these algebras. Along the way, I will try to describe some of the techniques used in the proofs, including the theory of types, and some of the tools needed to get explicit descriptions of representations.

Twisted Ruelle zeta function on locally symmetric spaces, the Fried's conjecture and further applications

 
In this talk, we will present the twisted Ruelle zeta function, associated with representations that are not necessarily unitary, and how its special value at zero is related to the complex-valued analytic torsion. The relation between the twisted Ruelle zeta function and spectral (or topological) invariants is the so called Fried's conjecture. In addition, we will present results that are related to the Fried's conjecture for hyperbolic surfaces and orbisurfaces. If time allows, we will present some recent results concerning the investigation of the spectrum of the twisted Laplacians, as the representation varies in a suitable Teichmueller space.

The Whittaker Plancherel decomposition

 
Let G = KAN be a real semisimple Lie group, with the indicated Iwasawa decomposition. Fix a unitary character X of N with certain regularity properties. In the talk we will explain the structure of the Plancherel decomposition of the unitary representation of G obtained by inducing from the character X. The validity of this Whittaker Plancherel decomposition was announced by Harish-Chandra in 1980. Details of the proof appeared in the posthumous volume 5 of Harish-Chandra's collected work, edited by Gangolli and Varadarajan, with assistance of Kolk, 2018. However the proof given was not complete. In the talk we will explain two results that allow completion of the proof.

Complex semisimple groups, deformations, and the Baum-Connes assembly map

 
The Baum-Connes assembly map for groups like SL(n, C) can be interpreted geometrically in terms of a natural deformation of these groups to their associated Cartan motion groups. In this talk I will discuss a quantum version of this construction, and explain some related results from the representation theory of complex quantum groups. The “quantum assembly map” is an isomorphism which contains the “classical assembly map” as a direct summand, and I will argue that the former is somewhat easier to understand than the latter.

K-theory for Exceptional Extended Affine Weyl Groups

 
Affine Weyl groups are a class of groups for which the group C*-algebra can in principle be understood directly. I will give an overview of how Langlands duality gives a Poincare duality for these algebras, and discuss the problem of calculating the K-theory for the extended affine Weyl groups for the exceptional Lie group E_6.