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Consider the transverse isometric action of a finite dimensional Lie algebrag on a Riemannian foliation. This paper studies the equivariant Morse-Botttheory on the leaf space of the Riemannian foliations in this setting. Amongother things, we establish a foliated version of the Morse-Bott lemma for ag-invariant basic Morse-Bott function, and a foliated version of the usualhandle presentation theorem. In the non-equivariant case, we apply theseresults to present a new proof of the Morse inequalities on Riemannianfoliations. In the equivariant case, we apply these results to studyHamiltonian action of an abelian Lie algebra on a presymplectic manifold whoseunderlying foliation is also Riemannian, and extend the Kirwan surjectivity andinjectivity theorem in equivariant symplectic geometry to this situation. Amongother things, this implies the Kirwan surjectivity and injectivity hold forHamiltonian torus actions on symplectic orbifolds.

Equivariant Morse theory for Lie algebra actions on Riemannian foliations