Question about unusual highest weight modules for $U_q(sl(2))$

Question

Background

Let $U_q(sl(2))$ be the quantum group associated with $sl(2)$ i.e. the associative algebra with 1 over $Q(q)$ generated by $x^+,x^-,K,K^{-1}$ with relations $$KK^{-1}=K^{-1}K=1$$ $$Kx^+K^{-1}=q^2x^+,Kx^-K^{-1}=q^{-2}x^-$$ $$x^+x^{-}-x^{-}x^+=\frac{K-K^{-1}}{q-q^{-1}}$$ Here $q$ is indefinite, in particular not a root of unity. A $U_q(sl(2))$-module $V$ has highest weight $\omega \in Q(q)$ if there exists a vector $v \in V$ such that $$ U_q(sl(2))\cdot v=V $$ $$ x^+\cdot v=0 $$ $$ K\cdot v=\omega v$$ Most standard texts (e.g. Chari Pressley, etc.) focus mainly on the case where $\omega$ equals $q$ to some power (i.e. $K$ acts as $q^m$ for some integer $m$ or $-q^m$ though this case is equivalent to the first by tensoring with a 1-dimensional module). If $\omega$ is not of this form then the module in question is necessarily infinite dimensional.

Question

Has anyone considered what happens when $K$ acts by something other than a power of $q$, say $2q-1$? I am especially interested in what happens in the $q=1$ specialization of these modules. Any references would be appreciated.

Answer 1

When talking about modules for such a quantized enveloping algebra, you have to be explicit about whether the module is finite dimensional or not. (For affine or other infinite dimensional Lie algebras of interest, the parallel question is whether modules are integrable or not.) In Jantzen's introductory AMS text Lectures on Quantum Groups, for instance, he starts out with analogues of the traditional finite dimensional representations. Here the associated highest weights are (up to sign) given by powers of $q$ in the rank one case. The infinite dimensional modules with dominant integral highest weights then have reasonable properties as well.

It's possible in this special case to look at infinite dimensional modules with arbitrary highest weights, including simple modules, but typically these are far less understood. There is a lot of literature by now on quantum groups, but I'll mention one earlier paper by Lusztig: Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988). In Section 2 he defines very general highest weight modules, but then specializes to the integrable ones where more can be said.

I'm not at all sure where the general case leads. Keep in mind that even in the classical situation, infinite dimensional simple modules with arbitrary highest weights pose difficult problems.