| 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.
| References |
|---|
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.
| Week | Date | Topics |
| 1 | 1/04 | Topological groups, representations, the regular representation |
| 1/06 | The unitary dual and the Plancherel formula: overview | |
| 1/08 | Quasi-regular representations, Selberg's $\frac 1 4$ Conjecture | |
| 2 | 1/11 | Examples of L-functions and applications |
| 1/13 | Maass operators, introduction to unbounded operators | |
| 1/15 | Symmetric and self-adjoint operators, the weighted hyperbolic Laplacian | |
| 3 | 1/18 | MLK Day - No classes |
| 1/21 | Maass forms and the spectral problem | |
| 1/22 | Iwasawa decomposition and Haar measures | |
| 4 | 1/25 | $K$-isotypical decompositions |
| 1/27 | Maass and Laplace-Beltrami operators on $\mathrm{SL}(2,\mathbb{R})$ Introduction to Lie algebras | |
| 1/29 | Universal enveloping algebras, the Casimir element | |
| 5 | 2/01 | The Casimir element as Laplace operator |
| 2/03 | The Cartan decomposition and $K$-bi-invariant functions | |
| 2/05 | Compact operators, the Spectral Theorem | |
| 6 | 2/08 | Hilbert-Schmidt operators, integration of unitary representations |
| 2/10 | Guest lecture by J. Voight: Arithmetic significance of $\lambda=\frac 1 4$ | |
| 2/12 | Semisimplicity of $\mathrm{L}^2(\Gamma\backslash G,\chi)$ and discreteness of the spectrum | |
| 7 | 2/15 | Representations of Lie groups and Lie algebras |
| 2/17 | Elements of Peter-Weyl Theory, admissible representations | |
| 2/19 | Introduction to ($\mathfrak{g},K$)-modules | |
| 8 | 2/22 | Underlying ($\mathfrak{g},K$)-modules of admissible representations |
| 2/24 | Irreducible ($\mathfrak{g},K$)-modules for $\mathrm{SL}(2,\mathbb{R})$ | |
| 2/26 | Realizability of ($\mathfrak{g},K$)-modules | |
| 9 | 2/29 | Intertwining integrals and unitarity |
| 3/01 | Aside: induced representations for finite groups and Frobenius reciprocity | |
| 3/02 | The unitary dual of $\mathrm{SL}(2,\mathbb{R})$, solution of the spectral problem | |
| 3/04 | C*-algebras, Hilbert modules and C*-correspondences | |
| 10 | 3/07 | Rieffel induction, C*-algebraic universal principal series |