## On a conjecture about cellular characters for the complex reflection group $$G(d,1,n)$$.(English)Zbl 1469.20038

Summary: We propose a conjecture relating two different sets of characters for the complex reflection group $$G(d,1,n)$$. From one side, the characters are afforded by Calogero-Moser cells, a conjectural generalisation of Kazhdan-Lusztig cells for a complex reflection group. From the other side, the characters arise from a level $$d$$ irreducible integrable representations of $$\mathcal{U}_q(\mathfrak{sl}_{\infty})$$. We prove this conjecture in some cases: in full generality for $$G(d,1,2)$$ and for generic parameters for $$G(d,1,n)$$.

### MSC:

 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20G42 Quantum groups (quantized function algebras) and their representations 17B37 Quantum groups (quantized enveloping algebras) and related deformations 20G05 Representation theory for linear algebraic groups

### Keywords:

cellular characters; complex reflection groups
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### References:

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