M559 - Harmonic Analysis on Compact Groups

Course No: 
M205, M304, M401
Review of General Theory: Locally compact groups, Computation of Haar measure on $\mathbb R$, $\mathbb T$, $SU(2)$, $SO(3)$ and some simple matrix groups, Convolution, the Banach algebra $L_1 (G)$. Representation Theory: General properties of representations of a locally compact group, Complete reducibility, Basic operations on representations, Irreducible representations. Representations of Compact groups: Unitarilzibality of representations, Matrix coefficients, Schur’s orthogonality relations, Finite dimensionality of irreducible representations of compact groups. Various forms of Peter-Weyl theorem, Fourier analysis on Compact groups, Character of a representation. Schur’s orthogonality relations among characters. Weyl’s Chracter formula, Computing the Unitary dual of $SU(2)$, $SO(3)$; Fourier analysis on $SO(n)$.
Reference Books: 
  1. T. Brocker, T. Dieck, “Representations of Compact Lie Groups”, Springer-Verlag, 1985.
  2. J. L. Clerc, “Les Repr ́esentatios des Groupes Compacts, Analyse Harmonique” (J.L. Clerc et. al., ed.), C.I.M.P.A., 1982.
  3. G. B. Folland, “A Course in Abstract Harmonic Analysis”, CRC Press, 2000. 
  4. M. Sugiura, “Unitary Representations and Harmonic Analysis”, John Wiley &Sons, 1975.
  5. E. B. Vinberg, “Linear Representations of Groups”, Birkh ̈ auser/Springer, 2010. 
  6. A. Wawrzy ́ nczyk, “Group Representations and Special Functions”, PWN–Polish Scientific Publishers, 1984.

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