an:05994505
Zbl 1233.17007
Feigin, Evgeny; Fourier, Ghislain; Littelmann, Peter
PBW filtration and bases for symplectic Lie algebras
EN
Int. Math. Res. Not. 2011, No. 24, 5760-5784 (2011).
00293165
2011
j
17B10 17B20
PBW filtration; symplectic Lie algebra; basis; highest weight module; character
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.
Volodymyr Mazorchuk (Uppsala)