A higher index theorem on finite-volume locally symmetric spaces

Peter Hochs  (Radboud University Nijmegen)

13:30-15:00, March 20, 2024   Mathematics Building 401, Minhang Campus


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.

About the speaker:

Peter Hochs是荷兰Radboud大学的教授,专长指标理论,李群表示论和K-理论。他曾是欧盟的Marie Curie基金获得者。他最重要的研究工作是他与合作者们在几何量子化理论中得到的对于非紧李群的量子化与约化可交换原则,统一并发展了“麻-张”和“Paradan-Vergne”紧李群几何量子化的先驱性工作。他在指标理论与表示论之间联系中也有出色的结果。他的论文发表在《Duke Mathematical Journal》,《Advances in Mathematics》,《Journal of Functional Analysis》,《Journal of K-theory》等著名期刊上。