Ramacher, Pablo Pseudodifferential operators on prehomogeneous vector spaces. (English) Zbl 1119.11068 Commun. Partial Differ. Equations 31, No. 4-6, 515-546 (2006). Let \(G\) be a connected linear algebraic group defined over \(\mathbb R\), acting regularly on a finite dimensional vector space \(V\) over \(\mathbb C\) with the \(\mathbb R\)-structure \(V_\mathbb R\). It is assumed that \(V\) possesses a Zariski-dense orbit (such a space \(V\) with the action of \(G\) is called a prehomogeneous vector space over \(\mathbb R\)).The author considers the left regular representation \(\pi\) of the group \(G_\mathbb R\) of \(\mathbb R\)-rational points on the Banach space \(C_0(V_\mathbb R)\) of continuous functions vanishing at infinity, and studies the convolution operators \(\pi (f)\) where \(f\) is a rapidly decreasing function on the identity component of \(V_\mathbb R\). Let \(S_\mathbb R=S\cap V_\mathbb R\) where \(S\) is the complement of the above dense orbit. It turns out that the restriction of \(\pi (f)\) to \(C_0^\infty (V_\mathbb R\setminus S_\mathbb R)\) is a pseudodifferential operator with a smooth Schwartz kernel. Under various assumptions, the author investigates the behavior of the Schwartz kernel near the singularities. If \(S_\mathbb R=\{ 0\}\), this gives (on the diagonal) a homogeneous distribution on \(V_\mathbb R\setminus \{ 0\}\). The case where \(G\) is reductive, \(S\) and \(S_\mathbb R\) are irreducible hypersurfaces, corresponds to some powers of a relative invariant of the prehomogeneous vector space (see T. Kimura, Introduction to prehomogeneous vector spaces. Providence, RI: American Mathematical Society (2003; Zbl 1035.11060)]). As an application, properties of the semigroup of operators on \(C_0(V_\mathbb R)\) generated by the Casimir element of the representation \(\pi\) are investigated. Reviewer: Anatoly N. Kochubei (Kyïv) Cited in 4 Documents MSC: 11S90 Prehomogeneous vector spaces 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 47G30 Pseudodifferential operators 47D03 Groups and semigroups of linear operators Keywords:prehomogeneous vector space; pseudodifferential operator; relative invariant; Casimir element; Schwartz kernel Citations:Zbl 1035.11060 PDFBibTeX XMLCite \textit{P. Ramacher}, Commun. Partial Differ. Equations 31, No. 4--6, 515--546 (2006; Zbl 1119.11068) Full Text: DOI arXiv References: [1] Elstrodt J., Ma{\(\beta\)}-und Integrationstheorie (1999) [2] Hille E., Functional Analysis and Semi-Groups (1957) · Zbl 0033.06501 [3] Hörmander L., The Analysis of Linear Partial Differential Operators (1983) [4] Hörmander L., The Analysis of Linear Partial Differential Operators (1985) [5] Kimura T., Introduction to Prehomogeneous Vector Spaces (2003) · Zbl 1035.11060 [6] Langlands , R. P. ( 1960 ). Semi-Groups and Representations of Lie-Groups . Ph.D. thesis , Yale University , unpublished . · Zbl 0105.31204 [7] DOI: 10.1007/BF02392873 · Zbl 0492.58023 [8] Milnor J. W., Morse Theory (1973) [9] Parshin A. N., Algebraic Geometry IV (1994) · Zbl 0832.14032 [10] Ramacher P., J. Lie Theory 15 pp 299– (2005) [11] Robinson D. W., Elliptic Operators and Lie Groups (1991) · Zbl 0747.47030 [12] DOI: 10.1007/978-3-642-56579-3 [13] Wallach N. R., Real Reductive Groups (1988) · Zbl 0666.22002 [14] Warner G., Harmonic Analysis on Semi-Simple Lie Groups (1972) · Zbl 0265.22020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.