Abstract
Let G be a semisimple Lie group and Γ a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L 2( Γ∖G) associated to test functions f ∈ C c(G).
In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C ∗-algebras of G and Γ. As an application, we exploit the role of group C ∗-algebras as recipients of “higher indices” of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
In memory of Ronald G. Douglas
B. Mesland and M. H. Şengün gratefully acknowledge support from the Max Planck Institute for Mathematics in Bonn, Germany. H. Wang is supported by NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Lafforgue uses the notation 〈H, x〉 for our π ∗(x), where H is the representation space of π.
References
J. Arthur, An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., 4, 1–263, Amer. Math. Soc., Providence, RI, (2005).
M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32–33, 43–72 (1976).
M. F. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Inventions Math. 42 1–62 (1977).
P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory of group C ∗ -algebras. C ∗-algebras: 1943–1993 (San Antonio, TX, 1993), 240–291, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994.
D. Barbasch and H. Moscovici, L 2 -index and the Selberg trace formula. Journal of Functional Analysis 53 151–201 (1983).
U. Bunke, Orbifold index and equivariant K-homology, Math. Ann. 339 175–194 (2007).
A. Connes and H. Moscovici, L 2 -index theorem for homogeneous spaces of Lie groups. Ann. Math. 115.2 291–330 (1982).
A. Deitmar, On the index of Dirac operators on arithmetic quotients. Bull. Austral. Math. Soc. 56 489–497 (1997).
H. Donnelly, Asymptotic expansions for the compact quotients of properly discontinuous group actions. Illinois J. Math. 23, 485–496 (1979).
S. Echterhoff, Bivariant KK-theory and the Baum–Connes conjecture. https://arxiv.org/abs/1703.10912
C. Farsi, K-theoretical index theorems for good orbifolds. Proceedings of the American Mathematical Society, 115.3 769–773 (1992).
J. Fox and P. Haskell, K-theory and the spectrum of discrete subgroups of spin(4, 1). Operator algebras and topology (Craiova, 1989) 30–44, Pitman Res. Notes Math. Ser., 270, Longman Sci. Tech., Harlow, 1992.
J.-S. Huang and P. Pandzic, Dirac operators in representation theory. Birkhäuser, Mathematics: Theory & Applications 2006.
T. Kawasaki, The index of elliptic operators over V -manifold. Nagoya Math. J. 84 135–157 (1981).
J.-P. Labesse, Pseudo-coefficients trés cuspidaux et K-théorie. Math. Ann. 291 607–616 (1991).
V. Lafforgue, Banach KK-theory and the Baum–Connes conjecture. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 795–812, Higher Ed. Press, Beijing, 2002
E.C Lance, Hilbert C ∗ -modules, a toolkit for operator algebraists, Cambridge University Press, 1995.
R.P. Langlands, Dimension of automorphic forms. PSPM vol. 9, 1966, 253–257.
B. Mesland and M.H. Şengün, Hecke operators in KK-theory and the K-homology of Bianchi groups. Journal of Noncommutative Geometry 14.1 125–189 (2020).
B. Mesland and M.H. Şengün, Hecke modules for arithmetic groups via bivariant K-theory Annals of K-theory 4 631–656 (2018)
H. Moscovici, L 2 -index elliptic operators on locally symmetric spaces of finite volume. Contemporary Mathematics, 10 129–137 (1982).
F. Pierrot, K-théorie et multiplicités dans L 2(G∕ Γ). Mém. Soc. Math. Fr. (N.S.) 89 (2002), vi+ 85 pp.
M. Puschnigg, New holomorphically closed subalgebras of C ∗ -algebras of hyperbolic groups. Geom. Funct. Anal., 20.1 243–259 (2010).
M. A. Rieffel, Induced representations of C ∗ -algebras. Adv. Math. 13 176–257 (1974).
J. Rosenberg, Group C ∗ -algebras and topological invariants. Proc. Conf. on operator algebras and group representations (Neptun, Romania, 1980) Monographs and Studies in Math. 18, London 1984.
S. K. Samurkaş, Bounds for the rank of the finite part of operator K-theory. ArXiv:1705.07378.
N. R. Wallach, On the Selberg trace formula in the case of compact quotient. Bull. Amer. Math. Soc. 82.2 171–195 (1976).
S. P. Wang, On integrable representations. Math. Z. 147 201–203 (1976).
B.-L. Wang and H. Wang, Localized index and L 2 -Lefschetz fixed-point formula for orbifolds. J. Differential Geom. 102.2 285–349 (2016).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Mesland, B., Şengün, M.H., Wang, H. (2020). A K-Theoretic Selberg Trace Formula. In: Curto, R.E., Helton, W., Lin, H., Tang, X., Yang, R., Yu, G. (eds) Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology . Operator Theory: Advances and Applications, vol 278. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-43380-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-43380-2_19
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-43379-6
Online ISBN: 978-3-030-43380-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)