Appears in collection : 2025 - T1 - WS2 - Tempered representations and K-theory

Let $G$ be a connected real semisimple Lie group, and $K<G$ maximal compact. For a discrete subgroup $\Gamma < G$, we have the locally symmetric space $X = \Gamma \backslash G/K$. If $X$ is smooth and compact, then Atiyah-Singer index theory is a source of useful and computable invariants of $X$. One then also has the higher index, with values in the $K$-theory of the $C^*$-algebra of $\Gamma$. In many relevant cases $X$ is noncompact, but still has finite volume. Then Moscovici showed in the 1980s that a relevant index of Dirac operators on $X$ can still be defined. Barbasch and Moscovici computed this index in terms of group- and representation-theoretic information in the case of real rank 1 groups. (Stern generalised this to groups of higher real rank.) With Hao Guo and Hang Wang, we construct a $K$-theoretic index, from which Moscovici’s index, and the individual terms in Barbasch and Moscovici’s index theorem, can be extracted and computed.