Introduction to Representation Theory, Lie Theory, Harmonic Analysis
by examples

( See also: [ vignettes ] ... [ functional analysis ] ... [ intro to modular forms ] ... [ representation theory ] ... [ Lie theory, symmetric spaces ] ... [ buildings notes ] ... [ number theory ] ... [ algebra ] ... [ complex analysis ] ... [ real analysis ] ... [ homological algebra ] )

Math 8300, 2014-15, (Fall: 207 Vincent Hall) 1:25-2:15, MWF

• Representations of finite abelian groups [ updated 08:10, Oct 04, 2014]
• Representations of dihedral groups [ updated 11:06, Jun 24, 2024]
• Representations of finite 2x2 mirabolic subgroups
• Generalities
  • Some generalities about finite groups [ updated 19:10, Oct 06, 2014]
  • More generalities about finite groups [ updated 11:10, Oct 25, 2014]
• Representations of GL₂ over a finite field [ updated 11:11, Nov 24, 2014]
  Supplement:
  • Left/right-exactness of right/left-adjoint functors [ updated 08:10, Oct 24, 2014]
• Representations of finite Heisenberg groups, theta correspondences on finite SO(2)xSL₂ [ updated 11:11, Nov 24, 2014]
  • Representations of finite Heisenberg groups
  • Finite Segal-Shale-Weil (oscillator) representations
  • Segal-Shale-Weil (oscillator) representation construction of cuspidal representations for SL₂(F_q)
• ...
  • Representations of the circle group T
  • unitary repns of topological groups [ updated 14:08, Aug 03, 2014]
  • Representations of the rotation group SO(3) of the two-sphere S² and spherical harmonics
  • Verma modules of sl₂, highest-weight representations
  • Highest-weight classification of irreducibles for unitary groups U(n)
  • Finite-dimensional representations of matrix groups SL₂(R) and SL₂(C) and their Lie algebras sl₂(R) and sl₂(C)
• Quantum harmonic oscillator, oscillator/Segal-Shale-Weil representation of the Lie algebra sl₂(R)
• Weyl character formula for unitary groups U(n)
• Unitary representations of SL₂(R) and SL₂(C), principal series representations, holomorphic discrete series
• Intertwining operators among principal series for SL₂(R) [ updated 16:01, Jan 03, 2009]
• Representations of profinite groups: p-adic group GL₂(Z_p)
• Representations of p-adic groups GL₂(Q_p) principal series, supercuspidal representations
• The tree attached to SL₂(Q_p) and the buildings attached to SL_n(Q_p)
• Borel-Casselman-Matsumoto theorem on representations with Iwahori-fixed vectors
• Stone-vonNeumann theorem for real and p-adic Heisenberg groups [ updated 13:04, Apr 22, 2015]
• Segal-Shale-Weil repns, theta/Howe correspondences, liftings of automorphic forms
• ...