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PBW filtration and bases for symplectic Lie algebras. (English) Zbl 1233.17007
Consider the Lie algebra $$\mathfrak{sp}_{2n}$$ with a fixed triangular decomposition $$\mathfrak{n}_-\oplus\mathfrak{h}\oplus\mathfrak{n}_+$$. For a dominant $$\lambda\in\mathfrak{h}^*$$ let $$V(\lambda)$$ be the simple $$\mathfrak{sp}_{2n}$$-module with highest weight $$\lambda$$. As an $$\mathfrak{n}_-$$-module, the module $$V(\lambda)$$ is a quotient of $$U(\mathfrak{n}_-)$$. The degree filtration on $$U(\mathfrak{n}_-)$$ gives rise to a filtration on $$V(\lambda)$$ and the main object of the study in the paper under review is the associated graded space $$\mathrm{gr} V(\lambda)$$, which is a quotient of $$S(\mathfrak{n}_-)$$ modulo some ideal $$I(\lambda)$$. The first main result of the paper gives an explicit finite dimensional subspace generating $$I(\lambda)$$ as an $$S(\mathfrak{n}_-)$$-module. The second main result provides an explicit basis for $$\mathrm{gr} V(\lambda)$$. As a corollary the authors derive a graded combinatorial formula for the character of $$V(\lambda)$$ and obtain a new class of bases for the latter module.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras
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