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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 $$\mathbb{C}[\mathfrak{g}]$$.

##### MSC:
 17B05 Structure theory for Lie algebras and superalgebras 22E46 Semisimple Lie groups and their representations
##### Keywords:
Lie algebra; cohomology; Schubert variety; Satake equivalence
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##### References:
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