Good bases for G-modules
Let G be a semi-simple algebraic group over \({\mathbb{C}}\). Let T be a maximal torus in G and B a Borel subgroup, \(B\supset T\). Let R be the root system of G relative to T and S be the system of simple roots of R relative to B. For \(\beta\in R\), let \(X_{\beta}\) denote the element of the Chevalley basis of Lie(G) corresponding to \(\beta\). Let V be a G-module and let \(\lambda\) be any dominant weight, say \(\lambda =\sum^{n}_{i=1}a_ i\omega_ i\), where \(n=rank(G)\) and \(\{\omega_ i\), \(1\leq i\leq n\}\) is the set of fundamental weights, and \(a_ i\in {\mathbb{Z}}^+\). Let \(V(\lambda)=\{v\in V|\) \(X_{\alpha_ i}^{(a_ i+1)}v=0\), \(1\leq i\leq n\}\). Let \({\mathcal B}\) be a basis of V. For a dominant weight \(\lambda\), let \({\mathcal B}_{\lambda}={\mathcal B}\cap V(\lambda)\). The basis \({\mathcal B}\) is said to be good if \({\mathcal B}_{\lambda}\) is a basis of V(\(\lambda\)), for any dominant weight \(\lambda\). It was conjectured independently by Baclawski and Gelfand- Zelevinsky that every G-module V has a good basis. The author proves the above conjecture. One of the motivations for considering good bases is to describe the Clebsch-Gordan coefficients (in the tensor product decomposition); for example, if \({\mathcal B}=\{b_ i\}\) and \({\mathcal D}=\{d_ j\}\) are good bases for two G-modules V, W respectively, then \(\{b_ i\otimes d_ j\}\) is a good basis for \(V\otimes W\). This paper makes an important contribution to the Representation Theory of semi- simple algebraic groups.
scientific article; zbMATH DE number 4154729