Invariant fundamental solutions and solvability for $\mathrm{GL} (n, \mathsf{C})/U(p,q)$.

Abstract

where is the Dirac measure at the origin of G=H. Consider now the reductive symmetric space G=H ˆ GL n;C†=U p; q†. Let aq be a fundamental Cartan subspace for G=H (the `most compact' Cartan subspace) and let Aq be the associated Cartan subset of G=H, identified with a real abelian subgroup of G. For every non-trivial G-invariant differential operator D we let q D† be the differential operator with constant coefficients on Aq defined via the Harish-Chandra isomorphism. We use the Plancherel formula for GL n;C†=U p; q†, obtained by Bopp and Harinck in [4], to construct invariant fundamental solutions for G-invariant differential operators D on G=H for which the differential operator q D† has a fundamental solution, i.e. a distribution Tq on Aq solving the differential equation: q D†Tq ˆ q; where q is the Dirac measure at the origin of Aq. This result is similar to the results obtained by Benabdallah and Rouvie© re for semisimple Lie groups, see [2, Theore© me 1]. Their and our approach can be seen as a generalization of the method used by Ho« rmander to find fundamental solutions for non-zero differential operators with constant coefficients on R, see [7, p.189f].