This question is heavily related to one of my favorite relations between geometry and representation theory. Consider simple Lie algebras of the following types:
- $A_n$
- $D_n$
- $E_6$
- $E_7$
- $E_8$
Then these Dynkin diagrams correspond to all possible finite subgroups of $SL_2.$ The relation is given by special classes of isolated surface singularities known as Kleinian or Du Val singularities. These arise as follows. Given a finite subgroup $G \subset SL_2,$ we have an action of $G$ on $\mathbb{C}^2$ with no fixed points other than the origin. If we then look at the geometric quotient $\mathbb{C}^2/G$ corresponding to $G$-invariant polynomials in $\mathbb{C}[x,y],$ it is generated by three polynomials $f_1, f_2, f_3$ which are related by a homogeneous polynomial $g$ of degree 3 such that $g(f_1, f_2, f_3) = 0.$ We can then identify $\mathbb{C}^2/G$ with the hypersurface $\{ f = 0 \} \subset \mathbb{C}^3.$
The resulting hypersurfaces have the following equations (with corresponding subgroup):
- $A_n: x^{n+1} + y^2 + z^2$ (cyclic)
- $D_n: x^{n-1} + xy^2 + z^2$ (dihedral)
- $E_6: x^4 + y^3 + z^2$ (tetrahedral)
- $E_7: x^3y + y^3 + z^2$ (octahedral)
- $E_8: x^5 + y^3 + z^2$ (icosahedral)
The Dynkin diagram enter as follows. Each of these surfaces can be resolved through a finite number of blow-ups, and the exceptional fiber in the resolution will consist of a copy of $\mathbb{P}^1$ for each node of the Dynkin diagram, each of which is joined to the point of another $\mathbb{P}^1$ if there is a corresponding edge in the Dynkin diagram connecting the two nodes (so in the cyclic case, it's just a chain of $\mathbb{P}^1$'s).
Lastly, there is a neat connection between this and Springer theory that goes as follows. Let $\mathcal{N}$ denote the nilpotent cone of a Lie algebra of one of the types listed above, and let $\mathcal{O}$ denote the subregular orbit. Then $\mathcal{O}$ has codimension two in $\mathcal{N}$ and hence the corresponding Kostant/Slodowy slice is a surface in $\mathcal{N}.$ It then turns out that this surface is one of the surface singularities listed above, and that the corresponding Springer fiber of a subregular element is isomorphic to the exceptional fiber in the resolution of the surface mentioned above. So the Springer resolution encodes the information of the successive blow-ups of these surfaces.
A few good references:
Milnor, Singular Points of Complex Hypersurfaces
Dimca, Singularities and Topology of Hypersurfaces
Slodowy, Simple Singularities and Simple Algebraic Groups