Fun example of representations of $sl_2(\mathbb C)$

Question

What are fun occurence of $sl_2(\mathbb C)$ representations "in nature"? Ideally elementary examples would be the best.

Examples I know :

The cohomology ring $H^*(X,\mathbb C)$ for $X \subset \mathbb P^n$ smooth projective is such an example but definitely not elementary. I think some variant should work as well (e.g $X$ compact symplectic).

The double coinvariant ring $\mathbb C[\underline{x},\underline{y}]/ (\mathbb C[\underline{x},\underline{y}]^W_+)$ (where $\underline x = (x_1, \dots, x_n)$, same for $y$ and $W = S_n$) is another example but again it's not that elementary.

The ring $\mathbb C[x,y]$, where the action comes from $SL_2(\mathbb C)$ acting on $\mathbb C^2$ (so $sl_2(\mathbb C)$ acts as vector fields).

Any other examples are welcome, especially if it's outside representation theory (combinatorics, geometry, topology,...)

Answer 1

One classic is the Segal-Shale/oscillator representation of the Lie algebra $\mathfrak{sl}_2$ on various spaces of functions on $\mathbb R^n$ by sending $$ \pmatrix{0 & 1 \cr 0 & 0} \to (\hbox{multiplication by}) {|x|^2\over 2} \hskip30pt \pmatrix{0 & 0 \cr 1 & 0} \to {\Delta\over 2} \hskip30pt \pmatrix{1 & 0 \cr 0 & -1} \to {n\over 2} + \sum_{i=1}^n x_i{\partial\over \partial x_i} $$

Answer 2

The Fractional quantum Hall effect in physcis is strongly related to the Langlands program in automorphic forms within number theory, with the group $G=SL_2(\Bbb C)$ and its Langlands dual $G^L=PSL_2(\Bbb C)$ with their Lie algebras $\mathfrak{g}\cong \mathfrak{g}^L\cong \mathfrak{sl}_2(\Bbb C)$ and their irreducible representations. This is, for example, explained in the article Topological Aspects of Matters and Langlands Program by Kazuki Ikeda.

Answer 3

Question: "What are fun occurence of sl2(C) representations "in nature"? Ideally elementary examples would be the best."

Answer: If $k$ is a field and $V:=k\{e_0,e_1\}, V^*:=k\{x_0,x_1\}$, let $C:=\mathbb{P}(V^*)$ be the projective line on $V$. The tangent sheaf $T_C$ has global sections

$$H^0(C,T_C) \cong H^0(C,\mathcal{O}_C(2)) \cong \mathfrak{sl}(2,k).$$

The tangent sheaf is a "sheaf of Lie algebras" and its global sections is a $k$-Lie algebra isomorphic to $\mathfrak{sl}(2,k)$.If $D_C$ is the sheaf of differential operators on $C$ and if $E$ is a left $D_C$-module you get a left $\mathfrak{sl}(2,k)$-module structure on $H^0(C,E)$. This gives a functor

$$F: D_C-mod \rightarrow U(\mathfrak{sl}(2,k))-mod.$$

The global sections $H^0(C, \mathcal{O}(d)) \cong Sym^d(V^*)$ is canonically an $U(\mathfrak{sl}(2,k))$-module. This comes from the fact that there is a parabolic subgroup $P \subseteq SL(V)$ and an isomorphism $SL(V)/P \cong C$. The linebundle $\mathcal{O}(d)$ has an $SL(V)$-linearization inducing the $SL(V)$-module structure $Sym^d(V^*)$ on the vector space of global sections. This is the Borel-Weil-Bott theorem: Any finite dimensional irreducible $SL(V)$-module arise as the global sections of some inverible sheaf on $C$.

All constructions are elementary and may be "written out in detail by hand".

Fun for students: This example is "fun" for students: They can interpret elements in $\mathfrak{sl}(2,k)$ as algebraic vector fields on $C$ and make explicit calculations.