Differential Geometry Seminar: A higher index theorem on finite-volume locally symmetric spaces

Let G be a (connected, real, semisimple, real rank one) Lie group, and K a maximal compact subgroup. Let Gamma be a torsion-free, discrete subgroup of G.

If the double-coset space X = Gamma\G/K is compact, then we can do index theory on it, both in the classical Atiyah-Singer sense and in the sense of higher index theory with values in the K-theory of the C*-algebra of Gamma. But in many relevant cases, X has finite volume, but is noncompact. This includes the case where G = SL(2,R), K = SO(2) and Gamma = SL(2,Z). Then Moscovici constructed an index of Dirac operators on X, and Barbasch and Moscovici computed it using the (Arthur-)Selberg trace formula. In ongoing work with Hao Guo and Hang Wang, we upgrade this to a higher index with values in a relevant K-theory group. This talk is a relatively gentle introduction, focusing mainly on the classical case.

School of Computer and Mathematical Sciences