**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.