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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
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