Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups
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.
scientific article; zbMATH DE number 861923
scientific article