Folland: Find Irreducible Representations of $SU(2)$ and decomposition of $L^2$ via Fourier analysis on compact groups.

functional-analysisharmonic-analysislocally-compact-groupsrepresentation-theory

I am working through the following textbook:

Folland, Gerald B., A course in abstract harmonic analysis, Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-2713-6/hbk; 978-1-4987-2715-0/ebook). xiii, 305 p. (2016). ZBL1342.43001.

At the start of Section 5.4 (pages 149-156), our aim is to describe the irreducible representations of $SU(2)$ and its decomposition of $L^2$, and I have a few questions about this section:

  1. I want to show that the subrepresentation $\pi_m$ of $SU(2)$ on $\mathcal{P}_m\subset\mathcal{P}$ is unitary with respect to the inner product on $L^2(\sigma)$, ($\sigma$ is the surface measure on the unit circle $S^3$) where the inner product is given by
    $\langle P,Q \rangle :=\int_{S^3} P\overline{Q}\,d\sigma$.

My attempt: for $x\in SU(2)$,
$\langle \pi_m(x)P,\pi_m(x)Q \rangle =\int_{S^3}\pi_m(x)P\overline{\pi_m(x)Q}\,d\sigma=\int_{S^3}\pi_m(x)P\overline{\pi_m(x)}\overline{Q}\,d\sigma=\int_{S^3}(\pi_m \cdot\overline{\pi_m})(x)P\overline{Q}\,d\sigma=\int_{S^3}P\overline{Q}\,d\sigma$
i.e. $\pi_m$ is unitary. However, Folland simply states that $\sigma$ is rotation invariant. Is my working still correct?

  1. I would like to prove the 'orthogonality' property of the monomials $w^j z^k$, however there is notation used which I cannot find anywhere else in the textbook: "$\Gamma(p+1)$". It is used in the previous result but again there is no mention of what this actually denotes.

  2. When describing the decomposition of $L^2(S^3)$ into direct sum of $\mathcal{E}_{\pi_m}$'s, Folland says this is given by "the decomposition of functions of the unit sphere S^3 into spherical harmonics". There is a reference to a textbook to elaborate on this however I cannot get access to it. Can anyone expand on this? I'm not familiar with spherical harmonics.

Many thanks!

Best Answer

  1. No, your derivation is not correct (e.g. what is $(\pi_m\cdot\overline{\pi_m})(x)$ supposed to mean?) Note that you haven't even used (or stated) the definition of $\pi_m$ anywhere. You need to use the definition with the polynomials $P, Q$, make a change of variables in the integral, and use the fact the measure $\sigma$ is invariant under this change.

  2. $\Gamma$ is the Gamma function.

  3. Higher-dimensional spherical harmonics are discussed here (you're looking at the $n=4$ case). They form a basis for homogeneous polynomials restricted to $S^3$ that is orthonormal with respect to the inner product defined previously.

Related Question