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

*An introduction to the representation theory of groups*, by E. Kowalski. See also these**course notes**.- N. Elkies's
**introduction to analytic number theory** - N. Wallach's
**notes**on automorphic forms. - P. Garett's notes on
**invariant differential operators**and**waveforms**. - S. Sternberg's notes on
**Lie algebras** - W. Casselman's
**essays**, especially regarding ($\mathfrak{g},K$)-modules.

Syllabus of the course |
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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 |