TITLE: K-Theory and Representation Theory
DATES: 18-22 July 2022
LOCATION: University of Bath
Given its close connection to the “K-theory, algebraic cycles and motivic homotopy theory” (KAH) programme – which will also be running in July 2022 – INI would like to help promote this below event and to draw it to the particular attention of all KAH programme and workshop participants.
Further details follow.
This is a synergistic, intra-disciplinary symposium that will focus on new and developing links between operator K-theory and representation theory.
The meeting will consider the following threads:
Recent advances in operator K-theory and representation theory for affine Hecke algebras and p-adic groups (including the local Langlands correspondence). In a series of works, Plymen and collaborators have constructed a conjectural, and partly proven, bridge (known as the ABPS conjecture) between the K-theory of the reduced group C*-algebra of a p-adic group G and the parametrization (a la Bernstein) of the tempered dual of G. This conjecture relates intimately to the Langlands parameters and thus is of relevance to a wide group of researchers.
New constructions linking K-theory, trace formula and automorphic forms. In recent works, Mesland and Sengun constructed a KK-theoretic counterpart of the Hecke ring of Shimura. This contruction endowed many K-groups associated to arithmetic groups (such as group C*-algebras and boundary cross product algebras) with the action of the Hecke ring. Could these new K-theoretic Hecke modules play a role similar to that of the cohomology of arithmetic groups in the theory of automorphic forms?
New approaches to tempered representation theory for real Lie groups via operator algebras and noncommutative geometry. In recent works, Higson and collaborators have pioneered approaches to parabolic induction in the tempered representation theory of reductive groups via Hilbert C*-modules, and approaches to the tempered dual in full via the Mackey bijection.
Emerging approaches to the topological aspects of harmonic analysis on reductive symmetric spaces. Given a quotient X=G/H of a real reductive group by the fixed points of an involution on that group, the spectrum of L2(X) as a G-representation is well-understood by harmonic analysis as a measure space. The recent interactions indicated that there is a promising possibility to study the spectrum topologically, using C*-algebras and K-theory.
We plan to push further the synergistic aspects of the meeting by bringing together a balance of experts who work in representation theory and harmonic analysis, with experts in noncommutative geometry and operator K-theory. Speakers will be encouraged to present open problems accessible across boundaries, and touch on possible new points of contact between the above research threads.