Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra \( \mathfrak{g} \) was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra \( {\mathfrak{g}}_0\subset \mathfrak{g} \). As an application, we produce new monomial bases for representations of the symplectic Lie algebra associated with a natural chain of subalgebras. One of our bases is related via a triangular transition matrix to a suitably modified version of the basis constructed earlier by the first author. In type A, our approach shows that the Gelfand–Tsetlin basis and the canonical basis of Lusztig have a common PBW-parameterisation. This implies that the transition matrix between them is triangular. We show also that a similar relationship holds for the Gelfand–Tsetlin and the Littelmann bases in type A.

跟随Vinberg的问题，Feigin，Fourier和Littelmann在一系列论文中提出了一种通用的方法，该方法为约化李代数\（\ mathfrak {g} \）的有限维不可约表示构造单项式基。依靠这种方法，我们构造了多重空间的单项式底基，该底基与将表示限制为归约子代数\（{\ mathfrak {g}} _ 0 \ subset \ mathfrak {g} \）有关。作为一种应用，我们为与自然子代数链相关的辛李代数的表示提供了新的单项式基础。我们的基础之一是通过三角过渡矩阵与第一作者先前构建的基础的适当修改版本相关。在类型A中，我们的方法表明Gelfand–Tsetlin基础和Lusztig的规范基础具有相同的PBW参数化。这意味着它们之间的过渡矩阵是三角形的。我们还表明，A型Gelfand-Tsetlin和Littelmann基地也存在类似的关系。