Invariant Theory of singular Riemannian foliations.

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The starting point of this project is the recent Algebraicity Theorem due to A. Lytchak and M. Radeschi. It says that every infinitesimal singular Riemannian foliation F on the vector space V with closed leaves is determined by its algebra A of basic polynomials, that is, polynomials on V that are constant on the leaves of F.If F is homogeneous, that is, if it coincides with the orbit decomposition under the linear action of a compact Lie group, then A is known as the algebra of invariants, and is the main object of study of Classical Invariant Theory. For any result in this branch of Mathematics, one may therefore ask whether it holds in the more general setting of (possibly inhomogeneous) singular Riemannian foliations.In collaboration with M. Radeschi I have generalized one such result by describing the algebra of smooth basic functions, which in the homogeneous case is due to G. Schwarz.The goal of the present project (in collaboration with A.Lytchak and M.Radeschi) is to generalize other results from Classical Invariant Theory. More specifically: Classify F for which A is generated in "small" degrees; generalize the notion of real, complex and quaternionic type of a representation; describe the module of vertical vector fields; and find an algorithm that computes a finite set of generators for the algebra A.