How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?



On eigenfunctions of the Laplace Beltrami operator
For $\mathrm{SU}(2)$, with the scale for the biïnvariant metric so that it becomes isometric to the unit $3$-sphere $S^3$ in Euclidean $4$-space, it is well-known what the eigenvalues of the Laplace-Beltrami operator are and, indeed, what the eigenfunctions are as well. The eigenvalues are $\lambda\_k = k(k{+}2)$ for $k=0,1,2,\ldots$, and the $k$-th eigenspace consists of the restrictions to $S^3$ of the harmonic polynomials homogenous of degree $k$ on $\mathbb{R}^4$. Hence, the multiplicity of $\lambda\_k$ is $(k{+}1)^2$. For example, see:


* the classic reference: Berger, M., Gauduchon, P., Mazet, E.: *Le spectre d'une variété riemannienne*, Springer Lecture Notes No.194, 1971.
* these lecture notes from MIT (theorem 2.9): [https://math.mit.edu/~dav/spheres.pdf](https://math.mit.edu/%7Edav/spheres.pdf)


More generally, for the biïnvariant metric on a compact, simply-connected simple Lie group $G$ corresponding to the (negative of the) Cartan-Killing form on the Lie algebra, the computation of the eigenvalues of its Laplace-Beltrami operator and their multiplicities can be reduced to a combinatorial problem by means of the Peter-Weyl Theorem, Cartan's classification of the irreducible representations of $G$, the Casamir operator, and various representation-theoretic formulas such as Weyl's character formula, Steinberg's formula, and formulae of Kostant. It is not a trivial combinatorial problem, though. I would suggest getting a reference such as A. Knapp's *Lie Groups: Beyond an Introduction* and then searching the internet for phrases such as *the spectrum of compact simple groups* to get started.


In principle, one could work out the spectrum for any left-invariant metric on a compact Lie group, but the combinatorial problems that one runs into in carrying this out explicitly can be thorny for all but the simplest cases.


**Addendum:** Because of [IGT's question/comment below from June 24, 2022](https://mathoverflow.net/questions/415485/on-eigenfunctions-of-the-laplace-beltrami-operator#comment1093465_415595), I decided to include the example of a left-invariant metric on $\mathrm{SU}(2)\simeq S^3$ in the first few cases to illustrate what's going on. If we let $g\_0$ be the biïnvariant metric on $\mathrm{SU}(2)$ with sectional curvature $1$, and let $g$ be any other left-invariant metric on $\mathrm{SU}(2)$, then there are left-invariant $1$-forms $\omega\_i$ and positive constants $c\_i$ such that $g\_0 = {\omega\_1}^2+{\omega\_2}^2+{\omega\_3}^2$ while $g = {\omega\_1}^2/c\_1+{\omega\_2}^2/c\_2+{\omega\_3}^2/c\_3$. If $X\_i$ are the dual left-invariant vector fields, then the Laplacian with respect to the metric $g$ is easily seen to be the second-order operator $\Delta = -c\_1\,{X\_1}^2 -c\_2\, {X\_2}^2 -c\_3\, {X\_3}^2$. Since differentiation with respect to $X\_i$ (whose flow is an isometric rotation of the sphere) clearly preserves the $(k{+}1)^2$ dimensional subspace $H\_k$ that consists of restrictions to $S^3$ of the harmonic polynomials homogeneous of degree $k$ on $\mathbb{R}^4$, it follows that $\Delta$ preserves this space as well, i.e., $\Delta(H\_k)=H\_k$ for $k\ge1$.


On $H\_1$, $\Delta$ has a single eigenvalue: $c\_1 + c\_2 + c\_3$, of multiplicity $4$.


on $H\_2$, $\Delta$ has three eigenvalues:
$4(c\_i + c\_j)$ ($i\not=j$), each of multiplicity $3$.


on $H\_3$, $\Delta$ has two eigenvalues: $5(c\_1+c\_2+c\_3)\pm 4\delta$, where
$$
\delta = \sqrt{{c\_1}^{2}+{c\_2}^{2}+{c\_3}^{2}-c\_1c\_2-c\_2c\_3-c\_3c\_1},
$$
and each eigenvalue has multiplicity $8$. (Note that $\delta=0$ if and only if $c\_1=c\_2=c\_3$.)


Finally, on $H\_4$, $\Delta$ has five eigenvalues:
Three of them are $4(c\_1+c\_2+c\_3)+12\,c\_i$ for $i=1,2,3$, and
the other two are $8(c\_1+c\_2+c\_3\pm\delta)$.
Each of these $5$ eigenvalues has multiplicity $5$.


As one can see, the eigenvalues of $g$ need not all be integer multiples of a single value, or even all rational if the $c\_i$ are rational. Also, note that if one of the $c\_i$ is much larger than the other two (for example, when $c\_1 > 3(c\_2{+}c\_3)$), the eigenfunction belonging to the lowest eigenvalue of $\Delta$ may not be the restriction of a linear function on $\mathbb{R}^4$ to $S^3$.


