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I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy classes in the corresponding Lie group. For example, the representations of $U_q(\mathfrak s\mathfrak l_2)$ when $q=e^{2\pi i/k}$ correspond more or less to the "first $k$" irreducible representations of $SU(2)$. They also correspond (so I hear) to the conjugacy classes: $$\left(\begin{matrix}e^{2\pi ia/k}&0\cr0&e^{-2\pi ia/k}\end{matrix}\right)$$ in $SU(2)$ where $1\leq a\leq k-1$ (I may be off on some numbering). One thinks of this as a sort of deformed version of the orbit method (if instead one takes "integral" adjoint orbits in $\mathfrak s\mathfrak u(2)$, one gets all the representations of $SU(2)$).

The fact that representations of quantum groups at roots of unity correspond to "integral" conjugacy classes in the corresponding Lie group is, for example, reflected in the following common construction. When constructing the quantum hilbert space associated to a surface with marked points, one considers the subvariety of the $SU(n)$ character variety of the punctured surface, where we require the monodromy around each puncture to lie in some specified conjugacy class (possibly different for each point). This corresponds roughly to putting the corresponding representation of $U_q(\mathfrak s\mathfrak l_n)$ at that puncture on $\Sigma$. We require some integrality condition on the conjugacy class corresponding to the "level" $k$ of the quantization desired (this corresponds exactly to the "integrality" condition on the conjugacy class when constructing the $U_q(\mathfrak s\mathfrak l_n)$ representation for $q=e^{2\pi i/(k+n)}$).

I've looked for references about this perspective on the representations of quantum groups at a root of unity, but have been mostly unsuccessful. Does anyone here know about good references and/or results about exactly how this correspondence (between "integral" conjugacy classes in $G$ and irreps of $U_q(\mathfrak g)$ at $q^e=1$) works? If it makes things any easier, I'm happy only dealing with $SU(2)$.

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About 10 years ago, Allen Knutson suggested I write a thesis on exactly this question. Unfortunately, that didn't happen, and I don't know a good reference. Oh well. – Ben Webster Jan 24 '12 at 2:26
I'd still like you to write it, though, in case you had any doubt. – Allen Knutson Jan 30 '12 at 3:52

In as much as the OP wants references that relate representations of quantum groups $U_q(\mathfrak{sl}_n)$ to character varieties, this paper of Adam Sikora appears to be relevant:

Skein theory for SU(n)-quantum invariants

In that paper, representations of quantum groups are used to construct the skein modules (see the proofs in Sections 4-), which themselves are deformations of character varieties (Corollary 20).

See also his paper:

Quantizations of Character Varieties and Quantum Knot Invariants

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As far as I know those papers don't say anything about representations of quantum groups or about roots of unity. – Peter Samuelson May 18 at 13:40
I made some edits to clarify that you need to read the proofs to see where representations of quantum groups arise. – Sean Lawton May 18 at 15:52

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