Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am assuming that $q \in \mathbb{C}^\times$ is not a root of unity. I am interested in unitarizability of representations of $U_q(\mathfrak{g})$ with respect to various $*$-structures.

The $*$-structures on $U_q(\mathfrak{g})$ have been classified; this can be found, for example, in section 9.4 of A Guide to Quantum Groups by Chari and Pressley, for example. For the $*$-structure known as the compact real form of $U_q(\mathfrak{g})$, it is known that each finite-dimensional irreducible representation $V_\lambda$ admits an invariant inner product, i.e. a positive-definite Hermitian form such that $$ \langle av, w \rangle = \langle v, a^* w \rangle $$ for all $v,w \in V_\lambda$ and all $a \in U_q(\mathfrak{g})$.

Question: for an arbitrary $*$-structure on $U_q(\mathfrak{g})$, has anybody classified the set of dominant integral weights $\lambda$ for which $V_\lambda$ admits an invariant inner product?

Chari and Pressley mention in passing in their book that this is an open question, but that was in 1995 and I thought I'd see if anybody had resolved it in the meantime. I would expect that for an arbitrary $*$-structure, most irreps do not have such an invariant inner product, since in the classical situation the corresponding real form of the group is not compact, so you can't just average over Haar measure as you do in the compact case.


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As far as I know, there is no such classification or even a good attempt. There are some results about unitary representations of quantum groups with the suitable choice of involution. L.Vaksman and his collaborators studied unitary representations for quantum analogs of $SU(n,n)$ (Most of these papers are in arxiv). Also there are papers of Klimyk, Pakulyak etc. But the classification is done only for the trivial case of quantum $SU(1,1)$.


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