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Useful operators on representations of the rational Cherednik algebra of type \(\mathfrak{s} \mathfrak{l} _n\). (English) Zbl 1423.16017

Summary: Let \(n\) denote an integer greater than 2 and let \(c\) denote a nonzero complex number. In this paper, we introduce a family of elements of the rational Cherednik algebra \(\mathbf{H}^{\mathfrak{sl}_n}(c)\) of type \(\mathfrak{sl}_n\), which are analogous to the Dunkl-Cherednik elements of the rational Cherednik algebra \(\mathbf{H}^{gl_n}(c)\) of type \(gl_n\). We also introduce the raising and lowering element of \(\mathbf{H}^{\mathfrak{sl}_n}(c)\) which are useful in the representation theory of the algebra \(\mathbf{H}^{\mathfrak{sl}_n}(c)\), and provide simple results related to these elements.

MSC:

16G99 Representation theory of associative rings and algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C08 Hecke algebras and their representations
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