A course about the connections between Harmonic Analysis, Algebraic Geometry, Topology, and Representation Theory

Exam: PM4
Arun Ram
3 June 1996

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 27 November 2014

  1. State an important theorem about locally compact groups.
  2. Define unimodular.
  3. What is a Hecke algebra?
  4. In some sense the following concepts are equivalent (if the Hecke algebra H(G,K) is commutative): Zonal spherical functions,
    1-dimensional representations of the Hecke algebra,
    Spherical measures,
    Irreducible representations of G with a K-fixed vector.     Describe what these objects are and give some brief explanation of why they are equivalent concepts.
  5. State an important structure theorem for affine Weyl groups. Explain how this theorem generalizes to affine Hecke algebras.
  6. Explain what the following notations might mean. R W0 C R+ W C0 Q∨ Ha sa P∨ S(R)
  7. Define length in the Weyl group in a geometric way. One must make a choice to do this. What choice is this?
  8. What is Macdonald's explicit formula for the zonal spherical function for the pair (G,K) where G is a p-adic group and K is a maximal compact subgroup.
  9. What are the main ingredients in the definition of a p-adic group?
  10. State three important structure theorems for p-adic groups.
  11. What is (the name of) a standard technique for proving that a Hecke algebra is commutative and what is its main ingredient?
  12. Define convolution.
  13. Draw a picture of the hyperplane arrangement of a reduced irreducible root system and of the corresponding affine root system. Which root system did you choose to draw?
  14. Why are BN-pairs useful?

Notes and References

These are a typed copy of an exam for A course about the connections between Harmonic Analysis, Algebraic Geometry, Topology, and Representation Theory.

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