Modular maximal vectors and the Kostant cone
Let L be a complex semisimple Lie algebra with root system R and V(\(\lambda)\) the highest weight module of L with highest weight \(\lambda\). If U is the universal enveloping algebra of L, H a Cartan algebra of L and \(\{Y_ 1,...,Y_ m,H_ 1,...,H_ n,X_ 1,...,X_ m\}^ a \)Chevalley basis for L, then the Kostant \({\mathbb{Z}}\)-form \(U_{{\mathbb{Z}}}\) of U is the subring of U generated by 1, \((1/k!)Y_ j^ k\), \((1/k!)X_ j^ k\) with \(k\in {\mathbb{N}}.\) Moreover consider the ring \(U_{{\bar {\mathbb{Z}}}}\) generated by 1 and \((1/k!)Y_ j^ k\). If now \(v\in V(\lambda)\) is a maximal vector, i.e. satisfies \(U_{{\mathbb{Z}}}v=U_{{\bar {\mathbb{Z}}}}v\), we say that a vector \(w\in U_{{\bar {\mathbb{Z}}}}v\) is p-modular maximal (where p is a prime number) if p divides each coefficient of the \(X_ jw\), \(j=1,...,m\), with respect to a once and for all chosen (special) basis of V(\(\lambda)\) contained in \(U_{{\bar {\mathbb{Z}}}}v\). The presence of such vectors indicates the existence of non-trivial submodules in the characteristic p version of V(\(\lambda)\). The main result of the paper is: If \(<\lambda,\alpha_ j><p\) for \(\alpha\) in a basis for R then the p-modular maximal vectors cannot exist outside the Kostant cone (see the review above for the definition).
scientific article; zbMATH DE number 4108961
scientific article