Lectures Harmonic Analysis (WISL 420, Wonder/ MasterMath)

See also: exercises


Spring semester 2015
 
(01) Week 6, Feb 5
  • Sect 1: locally convex spaces
  • Sect 2: integration

    • (02) Week 7, Feb 12
  • Sect. 3: $K$-finite vectors
  • Sect. 4: Ring of representative functions.
  • Sect. 5: Schur orthogonality

    • (03) Week 8, Feb 19
  • Sect 6: The Peter-Weyl theorem
  • The Plancherel theorem for a compact group.

    • (04) Week 9, Feb 26
  • Sect. 7: Decomposition of $K$-modules
  • Homogeneous spaces and multiplicity
  • Relation to the convolution algebra
  • Sect. 8: Compact symmetric spaces

    • (05) Week 10, March 5
  • Sect. 9: Universal enveloping algebra
  • Multiplicity one for compact symmetric spaces

    • (06) Week 11, March 12
  • Sect 10: Symmetrizer map
  • Sect 11: Invariant differential operators on a group

    • (07) Week 12, March 19
  • Sect 12: Invariant diff ops on homogeneous space
  • Laplace operator on a Riemannian manifold
  • Sect 13: The center of the universal enveloping algebra
  • The Harish-Chnadra homomorphism, beginning

    • (08) Week 13, March 26
  • The Harish-Chandra isomorphism, proof of Weyl invariance
  • Read on your own: proof of bijectivity, based on Chevalley's theorem.
  • Read on you own: proof of Chevalley's theorem, Sect 14.

    • (09) Week 14, April 2
  • Sect 15: Cartan involution and infinitesimal Cartan decomposition
  • Duality of compact and non-compact symmetric spaces
  • Global Cartan decomposition: local diffeomorphism
  • idem standard example of SL_n
  • global Cartan decomposition, notion of maximal compact subgroup
  • algebra of invariant differential operators commutative

    • (10) Week 15, April 9, planning:
  • G/K is simply connected
  • introduction of a in p, restricted roots
  • notion of possibly non-reduced root system, proportional roots
  • special sl 2 algebras
  • reflection preserves roots
  • infinitesimal Iwasawa decomposition
  • formulation of global Iwasawa deco
  • Flag manifold and global Iwasawa deco for SL(n,R)

    • (11) Week 16, April 16, planning:
  • nilpotent groups
  • global Iwasawa deco: local diffeomorphism
  • reduction to adjoint case, main idea.
  • the notion of induced representation

    • (*) Week 17, April 23: no lecture this week

    (12) Week 18, April 30
  • The principal series of representations

    • (13) Week 19, May 7
  • The compact picture and admissibility of the principal series
  • Smooth and $K$-finite vectors of a representation
  • Admissible representations, Harish-Chandra modules
  • Formulation of the subrepresentation theorem

    • (*) Week 20, May 14: no lecture: Hemelvaart

    (14) Week 21, May 21: last lecture (3 x 45 minutes)
  • A description of Harish-Chandra’s Plancherel theorem, including:
  • Case of the Riemannian symmetric space
  • Discrete series, Generalized principal series, Intertwining operators
  • Plancherel measure
  • (possibly) a description of the Langlands classification of irreducible admissible representations

    • Last update: 6/5-2015