Restricted shifted Yangians and restricted finite W-algebras

We study the truncated shifted Yangian Yn,l(σ) over an algebraically closed field k of characteristic p > 0, which is known to be isomorphic to the finite W-algebra U(g, e) associated to a corresponding nilpotent element e ∈ g = glN (k). We obtain an explicit description of the centre of Yn,l(σ), showing that it is generated by its Harish-Chandra centre and its p-centre. We define Y [p] n,l (σ) to be the quotient of Yn,l(σ) by the ideal generated by the kernel of trivial character of its p-centre. Our main theorem states that Y [p] n,l (σ) is isomorphic to the restricted finite W-algebra U[p] (g, e). As a consequence we obtain an explicit presentation of this restricted W-algebra.