Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups
type: Item
instance of: scholarly article
title: Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups
zbMATH Open document ID: 0848.43008
DOI: 10.1007/BF01446304
author: Fulvio Ricci
author: Detlef Müller
published in: Mathematische Annalen
publication date: 1996-10-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165414
review text: In Section 1 the authors extend earlier results on one-parameter subgroups in Howe's realization of the Shale-Weil representation of the metaplectic group \(M_p (d, \mathbb{R})\) to the setting of the affine metaplectic group \(M_p (d, \mathbb{R}) \ltimes H_d\), where \(H_d\) denotes the Heisenberg group of dimension \(2d + 1\). This constitutes the starting point for the discussion of solvability. -- In the subsequent sections they consider special cases in which the operator \(L\) turns out to be solvable; in all of these cases \(g\) is assumed to be \(MW\): \[ \begin{aligned} \text{(a)} \quad & S_\mu \text{ has generically a non-imaginary eigenvalue (Section 2)}; \\ \text{(b)} \quad & L \text{ is not real (Section 2)}; \\ \text{(c)} \quad & S_\mu \text{ is generically non-semisimple (Section 3)}; \\ \text{(d)} \quad & S_\mu \text{ is generically semisimple and } L \text{ is not reductible to } G^p \text{ (Section 4)}; \\ \text{(e)} \quad & S_\mu \text{ is generically semisimple, } A \text{ is non-degenerate and } B = 0,\;\beta = 0 \text{ (Section 4)}. \end{aligned} \] In Section 5 they show how to reduce the general case either to one of the previous cases or to the case of homogeneous operators. Finally in Section 6 they discuss the possibility of interpreting the conditions in terms of invariants attached to the symbol of \(L\), and indicate various open problems.
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 35A08
zbMATH DE Number: 861923
zbMATH Keywords: eigenvalue
zbMATH Keywords: generically semisimple
zbMATH Keywords: Shale-Weil representation
zbMATH Keywords: metaplectic group
zbMATH Keywords: Heisenberg group
zbMATH Keywords: solvability
zbMATH Keywords: homogeneous operators
reviewed by: Stefan Balint
MaRDI profile type: Publication
cites work: Star exponentials of the elements of the inhomogeneous symplectic Lie algebra
cites work: Q4205967
cites work: Analysis of second order differential operators on Heisenberg groups. I
cites work: Analysis of second order differential operators on Heisenberg groups. II
cites work: Solvability of second-order PDOs on nilpotent groups -- a survey of recent results
cites work: An example on the Heisenberg group related to the Lewy operator
OpenAlex ID: W2069234275
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description: scientific article; zbMATH DE number 861923