Variations on Themes of Kostant
Let $ \mathfrak{g} $ be a complex semisimple Lie algebra, and let LG be a complex semisimple group with trivial center whose root system is dual to that of $ \mathfrak{g} $. We establish a graded algebra isomorphism $ H^{\raise0.145em\hbox{${\scriptscriptstyle \bullet}$}} {\left( {X_{\lambda } ,\mathbb{C}} \right)} \cong {{\text{S}}\mathfrak{g}^{e} } \mathord{\left/ {\vphantom {{{\text{S}}\mathfrak{g}^{e} } {I_{\lambda } }}} \right. \kern-\nulldelimiterspace} {I_{\lambda } } $, where Xλ is an arbitrary spherical Schubert variety in the loop Grassmannian for LG, and Iλ is an appropriate ideal in the symmetric algebra of $ \mathfrak{g}^{e} $, the centralizer of a principal nilpotent in $ \mathfrak{g} $. We also discuss a ‘topological’ proof of Kostant’s result on the structure of $ \mathbb{C}{\left[ \mathfrak{g} \right]} $.