# 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.

References

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

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