PBW filtration and bases for symplectic Lie algebras. (English) Zbl 1233.17007

Feigin, Evgeny; Fourier, Ghislain; Littelmann, Peter

Int. Math. Res. Not. 2011, No. 24, 5760-5784 (2011).

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.

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

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

Keywords:

PBW filtration; symplectic Lie algebra; basis; highest weight module; character