Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question here. The gist of the theory is as follows:

The general set up here is that we have a group acting on a space $X$, and we look at the space of functions on $X$ (let me write it $\mathcal F(X)$). Then there is a natural representation of $G$ on $\mathcal F(X)$.

(a) If $ X = G = S^1$ (the circle group, say thought of as $\mathbb R/\mathbb Z$) acting on itself by addition, then the solution to the problem of decomposing $\mathcal F(S^1)$ is the theory of Fourier series. (Note that a function on $S^1$ is the same as a periodic function on $\mathbb R$.)

(b) If $ X = G = \mathbb R$, with $G$ acting on itself by addition, then the solution to the above question (how does $\mathcal{F}(\mathbb{R})$ decompose under the action of $\mathbb R$) is the theory of the Fourier transform.

(c) If $ X = S^2$ and $G = SO(3)$ acting on $X$ via rotations, then decomposing $\mathcal F(S^2)$ into irreducible representations gives the theory of spherical harmonics. (This is an important example in quantum mechanics; it comes up for example in the theory of the hydrogen atom, when one has a spherical symmetry because the electron orbits the nucleus, which one thinks of as the centre of the sphere.)

(d) If $ X = SL_2(\mathbb R)/SL_2(\mathbb Z)$ (this is the quotient of a Lie group by a discrete subgroup, so is naturally a manifold, in this case of dimension 3), with $G = SL_2(\mathbb R)$ acting by left multiplication, then the problem of decomposing $\mathcal F(X)$ leads to the theory of modular forms and Maass forms, and is the first example in the more general theory of automorphic forms.

The same idea can explain how Bessel, Hypergeometric functions etc... arise geometrically. Again my question has an example of this. This table:

lists 7 groups and the differential equation each is related to.

**My question is asking for an intuitive way to think about the 7 groups listed and what they geometrically mean, and why they relate to some invariance in Laplace's equation $\nabla^2 u = 0$.** I even think there are a few different interpretations, e.g. in terms of homogeneous spaces, hyperbolic spaces, etc... so I'm kind of lost.

To give an example of what I'm hoping for, take the Euclidean Group $E_2$, it can be represented by matrices of the form

$$ g(x,y,\theta) = \left( \begin{array}{ccc} \cos(\theta) & - \sin(\theta) & x \\ \sin(\theta) & \cos(\theta) & y \\ 0 & 0 & 1 \end{array} \right)$$

Intuitively this is the group of plane motions, the group of all translations and rotations in the plane. If Laplace's equation models a problem with cylindrical symmetry, i.e. Laplace's equation is invariant under the group of plane motions, classically we end up having to solve Bessel's equation, but group-theoretically we can solve the problem with representation theory this magical way instead. That shows how Bessel's equation is just another way to say: "find a function in the plane such that when we shift it right, then shift it back left again, all locally (i.e. differentially) in polar coordinates, we get the same function". The same model holds for all equations in that table, but I don't know the geometry or pictures allowing me to see it.

**Some present confusion of mine:**

When I look at $SL(2,R)$ and it's matrix representation I have no intuitive picture of the group it's representing or any geometric picture (like a cylinder) to think of that tells me when solving $\nabla^2 u = 0$ will result in the Hypergeometric equation, for example.

If I try to think of $SO(3)$ as the group of rotations of a sphere I get Gegenbauer functions, but Gegenbauer polynomials give Legendre polynomials as a special case. However they also arise from the $SU(2)$ spinor representation of $SO(3)$. My guess is that this all links together because rotations can be decomposed into products of reflections, and spinor representations arise from this simple idea. But is it okay to think of $SO(3)$ as a group of 3-D Rotations and $SU(2)$ as the reflections that generate those rotations?