Abstract
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.
Highlights
Let G be a reductive algebraic group over an algebraically closed field k of characteristic p > 0, with Lie algebra g = Lie G
For χ ∈ g∗ the reduced enveloping algebra Uχ(g), is defined to be the quotient of U (g) by the ideal generated by the maximal ideal of Zp(g) corresponding to χ
The most important aspects of the representation theory of g are understood by studying Uχ(g)-modules, and the early work of Kac– Weisfeiler, in [KW], shows that it suffices to consider the case χ nilpotent, meaning χ identifies with a nilpotent element e ∈ g under some choice of G-equivariant isomorphism g ∼= g∗
Summary
In [BK1], Brundan–Kleshchev made a breakthrough by providing a presentation of the complex finite W -algebra for the case g = glN (C) by defining an explicit isomorphism with a certain quotient of a shifted Yangian This allowed them to make an extensive study of the representation theory of these finite W -algebras in [BK2]. The main benefit of this slight simplification is that we may apply the same argument to the integral forms of the Yangian and truncated shifted Yangian YnZ(σ) and YnZ,l(σ) We remark that our proof does not show that φ : Zp(Yn,l(σ)) → Zp(g, e), and so it remains an interesting open problem to decide if these centres really do line up
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