Spinor harmonics and the Dirac operator spectrum

Newman and Penrose (1966) introduced a family of special functions on $\bS^2$ that yield an orthonormal basis of spinors, in the same way that the conventional spherical harmonics $Y_{lm}$ yield an orthonormal basis of $L^2$-functions. For functions, $l$ and $m$ are integers, but the spinors are labelled by ``half-odd-integers'' in $\bZ + \half$. When expressed in our coordinates $(z,\bar z)$, they are given as follows.

For $l \in \{\half, \frac{3}{2}, \frac{5}{2},\dots\} = \bN + \half$, and $m \in \{-l, -l+1, \dots, l-1, l\}$, write
$$\begin{align*} Y^+_{lm}(z, \bar z) &:= C_{lm} q^{-l} \sum_{r-s=m-\shalf} \bi... ...shalf} \binom{l+\shalf}{r} \binom{l-\shalf}{s} z^r (-\bar z)^s, \end{align*}$$
where $r,s$ are integers with $0 \leq r \leq l \mp \half$ and $0 \leq s \leq l \pm \half$ respectively; and

$$C_{lm} = (-1)^{l-m} \sqrt{\frac{2l+1}{4\pi}} \sqrt{\frac{(l+m)!(l-m)!}{(l+\half)!(l-\half)!}}.$$

Then define pairs of full spinors by

$$Y'_{lm} := \frac{1}{\sqrt{2}} \begin{pmatrix}Y^+_{lm} \ [... ...\begin{pmatrix}- Y^+_{lm} \ [\jot] i Y^-_{lm} \end{pmatrix}.$$

These turn out to be eigenspinors for the Dirac operator.

Goldberg et al (1967) showed that these half-spinors are special cases of matrix elements $\sD_{nm}^l$ of the irreducible group representations for $SU(2)$, namely,

$$Y^\pm_{lm}(z,\bar z) = \sqrt{\frac{2l+1}{4\pi}} \sD^l_{\mp\shalf, m}(-\phi, \th, -\phi),$$

By setting $h^\pm_{lm}(\th, \phi, \psi) := e^{\mp\shalf(\phi+\psi)} Y^\pm_{lm}(z, \bar z)$, we get an orthonormal set of elements of $L^2(\SU(2))$, such that $\int_{\SU(2)} \vert h^\pm_{lm}(g)\vert^2 dg = (1/4\pi) \int_{\bS^2} \vert Y^\pm_{lm}\vert^2 \nu$. The Plancherel formula for $SU(2)$ can then be used to show that these are a complete set of eigenvalues for $\Dslash$. Thus we have obtained the spectrum:

$$\spec(\Dslash) = \set{\pm(l + \half) : l \in \bN + \half} = \{\pm 1, \pm 2, \pm 3, \dots\} = \bN \setminus \{0\},$$

with respectively multiplicities $(2l + 1)$ in each case, since the index $m$ in $Y^\pm_{lm}$ takes $(2l + 1)$ distinct values.