Kashiwara, Masaki The invariant holonomic system on a semisimple Lie group. (English) Zbl 0704.22008 Algebraic analysis, Pap. Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday, Vol. 1, 277-286 (1989). [For the entire collection see Zbl 0665.00008.] Let G be a connected reductive algebraic group defined over \({\mathbb{C}}\), and let \({\mathfrak G}\) be its Lie algebra. The center \({\mathfrak Z}({\mathfrak G})\) of the universal enveloping algebra is identified with the ring of bi- invariant differential operators on G. Denote by \({\mathcal D}_ G\) the ring of differential operators on G. If \(\chi\) : \({\mathfrak Z}({\mathfrak G})\to {\mathbb{C}}\) is a character, let \({\mathcal M}_{\chi}\) be the \({\mathcal D}_ G\)- module \({\mathcal D}_ G/({\mathcal D}_ GAd({\mathfrak G})+\sum_{P\in {\mathfrak Z}({\mathfrak G})}{\mathcal D}_ G(P-\chi (P)))\). The author proves various results on \({\mathcal M}_{\chi}\) and on the \({\mathcal D}_ G\)-module \({\mathcal D}_ G/{\mathcal D}_ GAd({\mathfrak G})\). Reviewer: P.Godin Cited in 3 Documents MSC: 22E30 Analysis on real and complex Lie groups 22E60 Lie algebras of Lie groups 32C38 Sheaves of differential operators and their modules, \(D\)-modules 22E46 Semisimple Lie groups and their representations 32A45 Hyperfunctions 58J15 Relations of PDEs on manifolds with hyperfunctions Keywords:connected reductive algebraic group; Lie algebra; universal enveloping algebra; ring of bi-invariant differential operators; character Citations:Zbl 0665.00008 PDFBibTeX XML