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

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?

• I don't have it available, so I can't check, but at least part of this might be covered in Mazotchuk's "Lectures on $\mathfrak{sl}_2(\mathbb{C})$-modules". Commented Oct 18, 2019 at 5:38

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.

1. $$V(\xi,\tau)$$, 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.