Spherical Harmonics and Linear Representations of Lie Groups

Abstract

This chapter and the next focus on topics that are somewhat different from the more geometric and algebraic topics discussed in the previous chapters. Indeed, the focus of this chapter is on the types of functions that can be defined on a manifold, the sphere S ⁿ in particular, and this involves some analysis. A main theme of this chapter is to generalize Fourier analysis on the circle to higher dimensional spheres. One of our goals is to understand the structure of the space L ²(S ⁿ) of real-valued square integrable functions on the sphere S ⁿ, and its complex analog $L^2_{\mathbb {C}}(S^n)$. Both are Hilbert spaces if we equip them with the inner product

$$\displaystyle \langle f, g\rangle _{S^n} = \int _{S^n} f(t)g(t) \, dt = \int _{S^n}fg\,\mathrm {Vol}_{S^n}, $$

and in the complex case with the Hermitian inner product

$$\displaystyle \langle f, g\rangle _{S^n} = \int _{S^n} f(t)\overline {g(t)} \, dt = \int _{S^n}f\overline {g}\,\mathrm {Vol}_{S^n}. $$

This means that if we define the L ²-norm associated with the above inner product as $\left \|f\right \| = \sqrt {\langle f, f\rangle }$, then L ²(S ⁿ) and $L^2_{\mathbb {C}}(S^n)$ are complete normed vector spaces (see Section 7.1 for a review of Hilbert spaces). It turns out that each of L ²(S ⁿ) and $L^2_{\mathbb {C}}(S^n)$ contains a countable family of very nice finite-dimensional subspaces $\mathcal {H}_k(S^n)$ (and $\mathcal {H}^{\mathbb {C}}_k(S^n)$), where $\mathcal {H}_k(S^n)$ is the space of (real) spherical harmonics on S ⁿ, that is, the restrictions of the harmonic homogeneous polynomials of degree k (in n + 1 real variables) to S ⁿ (and similarly for $\mathcal {H}^{\mathbb {C}}_k(S^n)$); these polynomials satisfy the Laplace equation

$$\displaystyle \Delta P = 0, $$

where the operator Δ is the (Euclidean) Laplacian,

$$\displaystyle \Delta = \frac {\partial ^2}{\partial x_1^2} + \cdots + \frac {\partial ^2}{\partial x_{n+1}^2}. $$