Functions on the universal cover of the principal nilpotent orbit
Let \(G\) be a complex simply connected semisimple Lie group, \(T\) a maximal torus, \(\mathcal O\) the principal nilpotent orbit in the Lie algebra \(\mathfrak g\) of \(G\), and \(\widetilde{\mathcal O}\) the universal cover of \(\mathcal O\). This paper studies the ring \(\widetilde{R}\) of regular functions on \(\widetilde{\mathcal O}\), proving a formula for its \(G\)-module structure which had earlier been conjectured by the reviewer and \textit{R. Brylinski} [J. Am. Math. Soc. 2, 517-533 (1989; Zbl 0729.17005); Invent. Math. 97, 209-217 (1989; Zbl 0648.22004)]. The formula states that \(\widetilde{R}\) is obtained as a \(G\)-module by algebraically inducing the sum of the one- dimensional \(T\)-modules corresponding to minimal dominant weights (including 0) to \(G\). (By contrast, an old result of Kostant states that the ring \(R\) of regular functions on \(\mathcal O\) itself agrees with the \(G\)- module induced from the trivial \(T\)-module.) Thus the ``smallest'' nontrivial \(G\)-submodules of \(\widetilde{R}\) in some sense are those with minimal highest weight, together with the unique copy of the adjoint representation in lowest degree. The author makes this last assertion precise by introducing a \(G\)-equivariant grading on \(\widetilde{R}\) such that the augmentation ideal \(\mathfrak m\) of \(\widetilde{R}\), modulo its square, is spanned by exactly these submodules (if \(\mathfrak g\) has no simple component of type \(A_ 1\)). This description of \({\mathfrak m}/{\mathfrak m}^ 2\) is particularly interesting in the light of a recent result of \textit{R. Brylinski} and \textit{B. Kostant} [Bull. Am. Math. Soc., New Ser. 26, 269-275 (1992; Zbl 0762.22008)] stating that this quotient has a natural Lie algebra structure extending that of \(\mathfrak g\). The author also points out an important mistake in [\textit{H. H. Andersen} and \textit{J. C. Jantzen}, Math. Ann. 269, 487-525 (1984; Zbl 0529.20027)], which invalidates Proposition 5.4 in that paper, a cohomology vanishing result. Unfortunately, the author's techniques do not seem to apply to covers of nonprincipal orbits.
complex simply connected semisimple Lie group
scientific article; zbMATH DE number 149684