Résolubilité d'équations différentielles et représentation induites. (Solubility of differential equations and induced representations) (Q1111219)

Let G be a connected Lie group, H a closed Lie subgroup of G, U a unitary representation of H on a separable Hilbert space E, \(C^{\infty}(V,H,U)\) the space of smooth E-valued functions u defined on an open set V of G such that for each (x,h)\(\in V\times H\), the following relation holds: \(u(x\cdot h)=\Delta^{1/2}_{H/G}(h)\cdot U^{-1}(h)\cdot u(x),\) and P: \(C^{\infty}(V,H,U)\to C^{\infty}(V,H,U)\) a smooth differential operator. In the paper under review the author gives some sufficient conditions such that the equation \(P(u)=v\) has solutions in \(C^{\infty}(V,H,U)\) or in some associated distribution spaces.