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Variations on themes of Kostant. (English) Zbl 1169.17002
Let \(\mathfrak{g}\) be a complex semi-simple Lie algebra and \({}^LG\) be the Langlands dual group for \(\mathfrak{g}\). In this paper the author proves that the cohomology algebra (with coefficients in \(\mathbb{C}\)) of an arbitrary spherical Schubert variety in the loop Grassmannian for \({}^LG\) is isomorphic to the quotient of the universal enveloping algebra of \(\mathfrak{g}\) modulo an appropriate ideal. The proof uses geometric Satake equivalence. Furthermore, it is shown that the quotient of the universal enveloping algebra which is mentioned above is isomorphic to a certain quotient of the symmetric algebra of \(\mathfrak{g}\) modulo the centralizer of a principal nilpotent in \(\mathfrak{g}\). At the end, the author gives a topological proof of a result (due to Kostant) describing the structure of the algebra \(\mathbb{C}[\mathfrak{g}]\).

17B05 Structure theory for Lie algebras and superalgebras
22E46 Semisimple Lie groups and their representations
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