Restricted shifted Yangians and restricted finite 𝑊-algebras

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