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Nilpotent pairs, dual pairs, and sheets. (English) Zbl 1136.17303
This paper continues the study begun by V. Ginzburg [Invent. Math. 140, 511–561 (2000; Zbl 0984.17007)] of pairs of commuting nilpotent elements in complex semisimple Lie algebras \(\mathfrak g\). The earlier work had focussed on principal nilpotent pairs, that is, pairs whose simultaneous centralizer has the least possible dimension (equal to the rank of \(\mathfrak g\)). Here the author shows that many of the results in that setting extend to almost principal pairs, that is, pairs whose simultaneous centralizer is one more than the rank of \(\mathfrak g\). Some new phenomena do however occur. For example, the elements in the semisimple pair \((h_1,h_2)\) corresponding to an almost principal pair \((e_1,e_2)\) need not have integral eigenvalues.
The author also develops some beautiful connections between nilpotent pairs and Howe’s reductive dual pairs. Given a nilpotent pair \((e_1,e_2)\) and the corresponding semisimple pair \((h_1,h_2)\), he shows that the centralizers \({\mathfrak z}(e_i,h_i)\) form a dual pair in \(\mathfrak g\) under certain conditions, which are always satisfied if the pair is principal or almost principal. The dual pair is reductive if and only if the original nilpotent pair can be embedded in a direct sum of commuting copies of \(\mathfrak{sl}_2\). A class of particularly nice nilpotent pairs called excellent is defined and classified for the simple Lie algebras. To such pairs one can attach smooth sheets with affine sections, generalizing a well-known construction of Kostant for principal nilpotent orbits in semisimple Lie algebras.

MSC:
17B20 Simple, semisimple, reductive (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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