Math 123: Automorphic forms, representations and C*-algebras

Winter 2016

General information

Instructor: Pierre Clare, 316 Kemeny Hall
Lectures: Monday & Friday 11:15 - 12:20, Wednesday 8:45 - 9:50, in 201 Kemeny Hall
Office hours: Monday & Wednesday 4 - 5:30 and by appointment or by chance

Special considerations: students with disabilities who may need classroom accommodations are encouraged to make an appointment to see the instructor as soon as possible.


We shall mostly follow Chapter 2 of Automorphic forms and representations, by D. Bump. Other useful resources for the material are:

Syllabus of the course

A more detailed account of the course progress, with references, is available in the course diary.

11/04Topological groups, representations, the regular representation
1/06The unitary dual and the Plancherel formula: overview
1/08Quasi-regular representations, Selberg's $\frac 1 4$ Conjecture
21/11Examples of L-functions and applications
1/13Maass operators, introduction to unbounded operators
1/15Symmetric and self-adjoint operators, the weighted hyperbolic Laplacian
31/18MLK Day - No classes
1/21Maass forms and the spectral problem
1/22Iwasawa decomposition and Haar measures
41/25$K$-isotypical decompositions
1/27Maass and Laplace-Beltrami operators on $\mathrm{SL}(2,\mathbb{R})$
Introduction to Lie algebras
1/29Universal enveloping algebras, the Casimir element
52/01The Casimir element as Laplace operator
2/03The Cartan decomposition and $K$-bi-invariant functions
2/05Compact operators, the Spectral Theorem
62/08Hilbert-Schmidt operators, integration of unitary representations
2/10Guest lecture by J. Voight: Arithmetic significance of $\lambda=\frac 1 4$
2/12Semisimplicity of $\mathrm{L}^2(\Gamma\backslash G,\chi)$ and discreteness of the spectrum
72/15Representations of Lie groups and Lie algebras
2/17Elements of Peter-Weyl Theory, admissible representations
2/19Introduction to ($\mathfrak{g},K$)-modules
82/22Underlying ($\mathfrak{g},K$)-modules of admissible representations
2/24Irreducible ($\mathfrak{g},K$)-modules for $\mathrm{SL}(2,\mathbb{R})$
2/26Realizability of ($\mathfrak{g},K$)-modules
92/29Intertwining integrals and unitarity
3/01Aside: induced representations for finite groups and Frobenius reciprocity
3/02The unitary dual of $\mathrm{SL}(2,\mathbb{R})$, solution of the spectral problem
3/04C*-algebras, Hilbert modules and C*-correspondences
103/07Rieffel induction, C*-algebraic universal principal series