Essential bases and toric degenerations arising from birational sequences

We present a new approach to construct T-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair (S,>) consisting of a birational sequence and a monomial order, we attach to the affine variety G//U a monoid Γ=Γ(S,>). As a side effect we get a vector space basis BΓ of C[G//U], the elements being indexed by Γ. The basis BΓ has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of Alexeev and Brion [1] to this more general setting, once one knows that the monoid Γ is finitely generated and saturated.