Irreducible weight modules for sl2 which are not highest or lowest-weight

Question

The special linear Lie algebra $\mathfrak{g} = \mathfrak{sl}_2(\mathbb{C})$ is three dimensional, with basis $e, f, h$ and bracket relations $$ [h, e] = 2e, \quad [h, f] = -2f, \quad [e, f] = h. $$ A $\mathfrak{g}$-module $(V, \rho)$ is called a weight module if $V$ is diagonalised by $\rho(h)$, i.e. $V = \bigoplus_{\lambda \in \mathbb{C}} V_\lambda$, where $V_\lambda = \{v \in V \mid \rho(h)v = \lambda v\}$. A $\mathfrak{g}$-module $(V, \rho)$ is called highest weight if it is a weight module generated by $\ker \rho(e)$, and lowest weight if it is a weight module generated by $\ker \rho(f)$. The classification of highest weight and lowest weight irreducible representations (up to isomorphism) is:

1. For each $n \in \{0, 1, 2, \ldots\}$, the $(n+1)$-dimensional irreducible module $V(n)$.
2. For each $\lambda \in \mathbb{C} \setminus \{0, 1, 2, \ldots\}$, the infinite-dimensional irreducible highest-weight Verma module $M(\lambda)$.
3. For each $\lambda \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$, the infinite-dimensional irreducible lowest-weight Verma module $M^-(\lambda)$.

In the classification of irreducible weight modules of $\mathfrak{g}$, what is missing from the list above?

For example, for any $\lambda \in \mathbb{C}$ we can define the infinite-dimensional representation $P(\lambda) = \mathbb{C}\{\ldots, p_{-2}, p_{-1}, p_0, p_1, p_2, \ldots\}$ with $$ h \cdot p_i = \lambda + 2i, \quad e \cdot p_i = p_{i + 1}, \quad f \cdot p_i = -i(i-1+\lambda)p_{i-1},$$ and $P(\lambda)$ will be irreducible for all $\lambda \in \mathbb{C} \setminus \mathbb{Z}$. (It also seems to me that $P(\lambda)$ should be isomorphic to $P(\mu)$ whenever $\lambda - \mu \in 2\mathbb{Z}$, but I have not checked this).

I know that classifying all $\mathfrak{sl}_2(\mathbb{C})$ representations is meant to be a "wild" problem, but does it have a nice answer when we restrict to weight modules? Or perhaps weight modules with finite-dimensional weight spaces?

Answer 1

As Tobias Kildetoft metioned, you can find details and proofs in Mazorchuk's "Lectures on sl2(C)-modules", section 3.4.

Your $P(\lambda)$ is a very good guess. But actually you need one more parameter to cover all weight modules, since you can vary the set of weights, as well as the central character.

Take $\xi \in \mathbb{C} / 2 \mathbb{Z}$ and $\tau \in \mathbb{C}$, and define the module $V(\xi,\tau)$ on basis $\{v_\mu \colon \mu \in \xi\}$ as follows:

• $h \cdot v_\mu = \mu v_{\mu}$,
• $f \cdot v_\mu = v_{\mu-2}$,
• $e \cdot v_\mu = \frac{1}{4} (\tau - (\mu+1)^2) v_{\mu+2}$.

These modules are sometimes called the principal series, and you can find all finite-dimensionals, Vermas, duals Vermas, as subquotients of them.

Anyway, the module $V(\xi,\tau)$ is simple if and only if $\tau \neq (\mu+1)^2$ for all $\mu \in \xi$. And such are pairwise non-isomorphic.

So, to your list you can add:

4. $V(\xi,\tau)$, for $\xi \in \mathbb{C} / 2 \mathbb{Z}$ and $\tau \in \mathbb{C}$ such that $\tau \neq (\mu+1)^2$ for all $\mu \in \xi$.

And you get a complete list of mutually non-isomorphic simple weight $\mathfrak{sl}_2(\mathbb{C})$-modules.