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Modular maximal vectors and the Kostant cone. (English) Zbl 0677.17007

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).
Reviewer: J.Hilgert

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Citations:

Zbl 0677.17006
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References:

[1] Bourbaki, N., Groupes et Algèbres de Lie (1975), Diffusion: Diffusion Paris, Chaps. VII, VIII · Zbl 0329.17002
[2] Deckhart, R. W., On the combinatorics of Kostant’s partition function, J. Algebra, 96, 9-17 (1985) · Zbl 0603.17005
[3] R. W. DeckhartJ. Algebra; R. W. DeckhartJ. Algebra · Zbl 0677.17006
[4] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory (1972), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0254.17004
[5] Humphreys, J. E., Ordinary and Modular Representations of Chevalley Groups (1976), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0341.20037
[6] Steinberg, R., Lectures on Chevalley Groups (1967), Yale University: Yale University New Haven
[7] R. W. Deckhart\(A_n Comm. Alg. \); R. W. Deckhart\(A_n Comm. Alg. \) · Zbl 0677.17007
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