Kac, Victor G. Laplace operators of infinite-dimensional Lie algebras and theta functions. (English) Zbl 0532.17008 Proc. Natl. Acad. Sci. USA 81, 645-647 (1984). Let \({\mathfrak g}\) be the Kac-Moody Lie algebra corresponding to a generalized Cartan matrix \(A\). Let \({\mathfrak h}^*\) be the dual space of the Cartan algebra \({\mathfrak h}\). The author considers the ring \(F\) of the complex valued functions on the complement \({\mathfrak h}^*-L\), where \(L\) is a union of certain affine hyperplanes of \({\mathfrak h}\). He constructs a completion \(U'\) of the enveloping algebra \(U({\mathfrak g})\), and a Harish-Chandra isomorphism \(H: Z'\to F\), where \(Z'\) is the center of \(U'\). He proves that the image of \(H\) contains all theta functions defined on the interior of the complexified Tits cone. Therefore, this image separates the orbits of the Weyl group. As another application of the same method the author announces a Chevalley type restriction theorem for simple finite-dimensional Lie superalgebras. Reviewer: Wim H. Hesselink Cited in 4 ReviewsCited in 53 Documents MSC: 17B65 Infinite-dimensional Lie (super)algebras 17B35 Universal enveloping (super)algebras 17B20 Simple, semisimple, reductive (super)algebras 17B70 Graded Lie (super)algebras Keywords:Tits cone; contravariant form; completed enveloping algebra; Harish-Chandra homomorphism; theta functions; Chevalley type restriction theorem; simple finite-dimensional Lie superalgebras PDFBibTeX XMLCite \textit{V. G. Kac}, Proc. Natl. Acad. Sci. USA 81, 645--647 (1984; Zbl 0532.17008) Full Text: DOI