MONOMIAL BASES AND BRANCHING RULES

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{g} $$\end{document} 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 g0⊂g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathfrak{g}}_0\subset \mathfrak{g} $$\end{document}. 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.