Restricted shifted Yangians and restricted finite $W$-algebras

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


Introduction
Let G be a reductive algebraic group over an algebraically closed field of characteristic p > 0, with Lie algebra g = Lie G. The centre of U(g) admits a large p-centre Z p (g) which is G-equivariantly isomorphic to the coordinate ring of (the Frobenius twist of) g * . For χ ∈ g * the reduced enveloping algebra U χ (g), is defined to be the quotient of U(g) by the ideal generated by the maximal ideal of Z p (g) corresponding to χ. The most important aspects of the representation theory of g are understood by studying U χ (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 * (we assume the standard hypotheses). We refer to [Ja] for a survey of this theory up to 2004, and also to [BM] for major developments based on deep connections with the geometry of Springer fibres. In [Pr1] Premet made a significant breakthrough: he showed that any such U χ (g) is Morita equivalent to a certain algebra U [p] (g, e), now known as the restricted finite W -algebra.
In this paper, we consider the case G = GL N ( ), so that g = gl N ( ). Our main theorem provides an explicit presentation for the restricted finite W -algebra U [p] (g, e). This is achieved by exhibiting an isomorphism with a restricted version of a truncated shifted Yangian, as stated in Theorem 1.1 below.
Before we proceed, we recall some relevant history. In [Pr1, Section 4] Premet constructed finite W -algebras over fields of characteristic zero, and since then these algebras have found many deep applications to classical problems surrounding the representations of complex semisimple Lie algebras; see [Pr3] and [Lo] for surveys on this theory.
In [BK1], Brundan-Kleshchev made a breakthrough by providing a presentation of the complex finite W -algebra for the case g = gl N (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].
Building on Premet's seminal work using the method of modular reduction of finite Walgebras, first considered in [Pr2] and exploited further in [Pr4], the authors developed a direct approach to theory of finite W -algebras U(g, e) over in [GT1]. Very briefly, for a choice of nilpotent e ∈ g corresponding to χ ∈ g * , the algebra U(g, e) is a filtered deformation of a good transverse slice χ +v to the coadjoint orbit G · χ. Further, U(g, e) admits a natural p-centre Z p (g, e) isomorphic to the coordinate algebra of (the Frobenius twist of) χ+v. Then the restricted W -algebra U [p] (g, e) is the quotient of U(g, e) by the ideal generated by the ideal of Z p (g, e) corresponding to χ.
In joint work with Brundan [BT] the second author developed the theory of shifted Yangians Y n (σ) over . One of the key features which differs from characteristic zero is the existence of a large central subalgebra Z p (Y n (σ)), called the p-centre, which is constructed using some very natural power series formulas.
In subsequent work [GT2], the authors showed that Brundan-Kleshchev's isomorphism descends to positive characteristic. To explain this, we require a little notation, and from now on we take g = gl N ( ). To each nilpotent element e ∈ g with Jordan type p = (p 1 ≤ · · · ≤ p n ), we may associate a choice of shift matrix σ = (s i,j ) 1≤i,j≤n , and thus a shifted Yangian Y n (σ), which is a subalgebra of the Yangian Y n . The beautiful formulas introduced in [BK1] lead to a surjective algebra homomorphismφ : Y n (σ) → U(g, e). Unsurprisingly the kernel ofφ has the same description as in characteristic zero, and so there is a natural isomorphism φ : Y n,l (σ) ∼ −→ U(g, e), where Y n,l (σ) is the truncated shifted Yangian of level l, first defined over the complex numbers in [BK1,Section 6].
Making use of the explicit presentation of U(g, e) obtained through the isomorphism φ, it was proved in [GT2] that every U χ (g)-module of minimal dimension is parabolically induced. This result is a modular analogue of Moeglin's famous theorem on completely prime primitive ideals, see [Moe], and some of our methods adapt those in the proof given by Brundan in [Br].
In this paper we define the p-centre Z p (Y n,l (σ)) of Y n,l (σ) to be the image of Z p (Y n (σ)) under the natural map Y n (σ) ։ Y n,l (σ), and this leads to a restricted truncated shifted Yangian Y [p] n,l (σ) by taking the quotient of Y n,l (σ) by the ideal generated by the natural generators of Z p (Y n,l (σ)).
We emphasise here that the origin of Z p (Y n (σ)) is totally distinct from the construction of Z p (g, e). Nevertheless, our main theorem states that the isomorphism φ factors through the restricted quotients.
of the centres of Y n,l (σ) and U(g, e) explicitly. Furthermore, we expect the development of highest weight modules in Section 5 will play an important role in future work.
Below we give an outline of the paper, in which we point out the most important steps.
In Section 2, we recall some relevant preliminaries, and introduce the combinatorial notation that we require. There are new results in §2.6, where we consider the centre Z(g e ) of the universal enveloping algebra of the centralizer of e. In particular, we use [BB] to give precise formulas for the generators of Z(g e ), sharpening the main results of [To]. Also in §2.7, we observe that g e is isomorphic to a truncated shifted current Lie algebra, which is useful later in this paper.
In Section 3, we recall the structural features of the shifted Yangian Y n (σ) and the finite W -algebra U(g, e), drawing on [BT], [GT1] and [GT2]. The key tools introduced here are the various filtrations on these algebras, and a precise description of their associated graded algebras. We also recall the definition of the mapφ lying at the core of our main theorem. In §3.3 we introduce the truncation Y n,l (σ) at level l, and use the shifted current algebra to simplify the proof of the PBW theorem for Y n,l (σ), see Theorem 3.1. The main benefit of this slight simplification is that we may then apply the same argument to the integral forms of the Yangian and truncated shifted Yangian Y Z n (σ) and Y Z n,l (σ). These integral forms, introduced in §3.4, are useful tools in some of our later proofs as they allow us to reduce modulo p certain formulas from the characteristic zero case, see Corollary 3.4. We expect these forms to find some independent interest, beyond the purposes of the present article.
Section 4 is devoted to describing the centres of Y n,l (σ) and U(g, e). Our results are perfect analogues of Veldkamp's classical description of the centre Z(g) of U(g); see for example [BG,Theorem 3.5] and the references there. We give definitions of the Harish-Chandra centres of Y n,l (σ) and U(g, e); these are denoted by Z HC (Y n,l (σ)) and Z HC (g, e), and they are defined so that they "lift" the centre in characteristic zero. The p-centres Z p (Y n,l (σ)) and Z p (U(g, e)) of Y n,l (σ) and U(g, e) are also introduced here. In Theorem 4.2 we give a detailed description of the centre of Y n,l (σ), in particular showing that is generated by Z p (Y n,l (σ)) and Z HC (Y n,l (σ)). The next significant result is Theorem 4.7 in which we deduce an analogous result for the centre Z(g, e) of U(g, e). We mention that in recent work, Shu-Zeng have stated a more general result about the centre of modular finite W -algebras associated to arbitrary connected reductive groups, under certain hypotheses, see [SZ,Theorem 1]. The more detailed description we give here is a necessary step in the proof of our main theorem, and will play an role in future work. A precise description of a set of generators for Z p (g, e) is given in §4.4, and this is important in the sequel. We also draw attention to Corollary 4.5 which shows that φ preserves the Harish-Chandra centres. In §4.3 and §4.5 we discuss the restricted quotients Y [p] n,l (σ) and U [p] (g, e) and their PBW bases. In Section 5 we develop some highest weight theory for Y n,l (σ) and study the action of U(g, e) on highest weight modules through the Miura map. One of the key ingredients of this theory is the use of a certain torus acting by automorphisms on both algebras, which is explained in detail in §5.1. The key results after that are Lemmas 5.4 and 5.6(c) which describe how the generators of the p-centres Z p (Y n,l (σ)) and Z p (g, e) act on highest weight modules. Other important results for us are Corollaries 5.5 and 5.7, which concern analogues of Harish-Chandra homomorphisms for Y n,l (σ) and U(g, e).
Finally, in Section 6, we combine our results to observe that the natural generators of φ(Z p (Y n,l (σ))) act on highest weight vectors in precisely the same manner as the generators for Z p (g, e). Using the Harish-Chandra restriction homomorphisms in U(g, e) we deduce that the ideal of Y n,l (σ) generated by the kernel of the trivial character of Z p (Y n,l (σ)) is mapped to the ideal of U(g, e) generated by the kernel of the trivial character of Z p (g, e), and the main theorem follows quickly.
We remark that our proof does not show that φ : Z p (Y n,l (σ)) → Z p (g, e), and so it remains an interesting open problem to decide if these centres really do line up. the second author is supported by EPSRC grant EP/N034449/1. We thank R. Tange for providing the idea for part of the proof of Corollary 5.5. We are also grateful to J. Brundan for helpful comments.

Preliminaries and recollection
Throughout this paper, let p ∈ Z ≥1 be a prime number, let F p be the field of p elements and let be an algebraically closed field of characteristic p.
2.1. A useful identity. We require a standard identity in the polynomial ring [t] for the proof of Lemma 5.4, and we recall it here. Each x ∈ F p satisfies x p − x = 0, so for an indeterminate t, we deduce that . More generally, for any a ∈ , we have the following equality in in [t] Observe that for 1 ≤ r ≤ p the coefficient of t p−r in the left hand side of (2.1) is (−1) r e r (0, 1, . . . , p−1), where e r (t 1 , . . . , t p ) denotes the rth elementary symmetric polynomial in indeterminates t 1 , . . . , t p . It follows that e r (0, 1, . . . , p − 1) = 0 in F p for r = 1, . . . , p − 2; this gives a short alternative proof of [BT,Lemma 2.7].
2.2. Some standard results on algebras and modules. We require a few elementary results from commutative and non-commutative algebra, which we state and prove for the reader's convenience. The first lemma is well-known. Let A be a commutative -algebra and B, C ⊆ A subalgebras. If A is generated by B ∪ C then it follows that there is a surjective homomorphism φ : B ⊗ B∩C C ։ A.
Lemma 2.1. Suppose that there exist elements c 1 , . . . , c m ∈ C such that: (a) the B-module generated by c 1 , . . . , c m is free on c 1 , . . . , c m ; and (b) C is generated by c 1 , . . . , c m as a B ∩ C-module.
Proof. We just have to prove that φ is injective, so we let y ∈ ker φ. It follows from (b) that Then we have 0 = φ(y) = m i=1 b i c i , and this implies b i = 0 for all i = 1, . . . , m by (a), so that y = 0. The next result concerns free modules for a commutative -algebra A. It is well-known that a surjective endomorphism of a finitely generated A-module is an isomorphism; this can be proved using Nakayama's lemma, see for example [Ma,Theorem 2.4].
Lemma 2.2. Let M be a free A-module of rank n and let m 1 , . . . , m n ∈ M. Suppose that M is generated by m 1 , . . . , m n as an A-module. Then M is free on m 1 , . . . , m n .
Proof. Let x 1 , . . . , x n ∈ M be free generators of M as an A-module. Consider the endomorphism θ : M → M defined by θ(x i ) = m i . Since M is generated by m 1 , . . . , m n , we have that θ is surjective, and thus an isomorphism. Hence, M is free on m 1 , . . . , m n .
The final result in this subsection is required several times in the sequel, and included for convenience of reference. Let A be a non-negatively filtered (not necessarily commutative) -algebra with filtered pieces F i A for i ∈ Z ≥0 . Also let M be a non-negatively filtered A-module with filtered pieces F i M for i ∈ Z ≥0 . We write gr A for the associated graded algebra of A and gr M for the associated graded module of M. If m ∈ F i M then the notation gr i m := m + F i−1 M ∈ gr M will be used throughout the paper. The following lemma can be proved with a standard filtration argument.
Lemma 2.3. Suppose that gr M is free as a graded gr A-module with homogeneous basis 2.3. Algebraic groups and restricted Lie algebras. We introduce some standard notation for algebraic groups and their Lie algebras, which is used in the sequel. Let H be a linear algebraic group over , and let h = Lie H be the Lie algebra of H. We write U(h) for the universal enveloping algebra of h, and Z(h) for the centre of U(h). We denote the ith filtered piece of U(h) in the standard PBW filtration by F i U(h). The associated graded algebra gr U(h) is identified with S(h), the symmetric algebra of h.
The adjoint action of H on h extends to an action on U(h). Also S(h) has adjoint actions of H and h. We use the standard notation (h, u) → Ad(h)u and (x, u) → ad(x)u for these actions, where h ∈ H, x ∈ h, and u ∈ U(h) or u ∈ S(h). For a closed subgroup K of H and K-stable subspace A of U(h) or of S(h), we write A K for the invariants of K in A and A k for the invariants of k in A. Given x ∈ h, we write h x for the centralizer of x in h, and we write H x for the centralizer of x in H.
We have that h is a restricted Lie algebra and we write x → x [p] for the p-power map. The 2.4. Combinatorial notation. We require various pieces of combinatorial notation, which we set out below.
By a composition we simply mean a sequence q = (q 1 , q 2 , . . . ), where q i ∈ Z ≥0 and only finitely many are nonzero. When q is a composition, l ∈ Z ≥0 and q i = 0 for all i > l, we write q = (q 1 , . . . , q l ). Given a composition q, we define |q| = i≥1 q i and say that q is a composition of |q|. Also we define ℓ(q) = |{i ∈ Z ≥1 | q i > 0}|. In this paper a composition p is called a partition if 0 < p i ≤ p i+1 for all 1 ≤ i < ℓ(p). Given two compositions m and p, we say that m is a subcomposition of p if m i ≤ p i for all i ∈ Z ≥1 , and in this case we write m ⊆ p.
Let n ∈ Z ≥0 . By a shift matrix of size n we mean a n × n matrix σ = (s i,j ) with entries in Z ≥0 such that s i,j = s i,k + s k,j whenever i ≤ k ≤ j, or i ≥ k ≥ j. We note that this implies that s i,i = 0 for all i, and that σ is completely determined by the entries s i,i+1 and s i+1,i for i = 1, . . . , n − 1.
Let N ∈ Z ≥0 and let q = (q 1 , . . . , q l ) be a composition of N such that for some j we have 0 < q 1 ≤ · · · ≤ q j ≥ · · · ≥ q l > 0, and let n := q j = max i q i . We define the pyramid π = π(q) to be the diagram made up of N boxes stacked in columns of heights q 1 , . . . , q l . We let p = p(q) be the partition of N giving the row lengths of π from top to bottom; note that the number p n = l will often be referred to as the level. The boxes in π are labelled with 1, . . . , N along rows from left to right and from top to bottom. The columns of π are labelled 1, 2, . . . , l from left to right and the rows are labelled 1, 2, . . . , n from top to bottom. The box in π containing i is referred to as the ith box, and we write row(i) and col(i) for the row and column of the ith box respectively. We define the shift matrix σ = σ(q) from π by setting s j,i to be the left indentation of the ith row of π relative to the jth row, and s i,j to be the right indentation of the ith row of π relative to the jth row, for 1 ≤ i ≤ j ≤ n.
Then we obtain the partition p = (2, 3, 5), and the shift matrix Evidently the data encoded in the composition q is equivalent to the data given by the pyramid π. We have explained how to construct a shift matrix and a level (σ, l) from a pyramid. To complete the picture we observe that we can build the pyramid π from knowledge of (σ, l), by starting with a bottom row of length l, and indenting the higher rows according to σ. The partition p can be explicitly recovered from (σ, l) by the rule Therefore, the combinatorial data q, π and (σ, l) are all equivalent. Let π = π(q) be a pyramid. A π-tableau is a diagram obtained by filling the boxes of π with elements of . The set of all tableaux of shape π is denoted Tab (π). For A ∈ Tab (π), we write a i for the entry in the ith box of A; alternatively we sometimes write a i,1 , . . . , a i,p i for the entries in the ith row of A from left to right. Two π-tableaux are called row-equivalent if one can be obtained from the other by permuting the entries in the rows.
2.5. Nilpotent elements in gl N ( ) and their centralizers. Let π be a pyramid with partition p = (p 1 , . . . , p n ), such that |p| = N. Let G = GL N ( ), so g = gl N ( ), which is a restricted Lie algebra with p-power map given by the pth matrix power. We write {e i,j | 1 ≤ i, j ≤ N} for the standard basis of g consisting of matrix units.
The pyramid π is used to determine the nilpotent element which has Jordan type p. Note that e depends only on p and not the choice of pyramid π.
The centraliser g e of e in g has a basis This is stated for example in [GT2, Lemma 2.1], although we warn the reader that the notation used here and there differs by a shift by one in the superscripts. In [GT2,Lemma 2.1] it is also stated that the Lie brackets are given by It is straightforward to see that the p-power map on g e is given by To make sense of these formulas we adopt the convention, here and throughout, that c (r) i,j = 0 when r ≥ s i,j + p min(i,j) .
We note here that the labelling of the basis of g e given in (2.5) does depend on the choice of pyramid π. However, the elements in the basis only depends on the partition p, and relabelling between different choices of pyramids just involves shifting the superscripts.
2.6. The centre of the enveloping algebra of the centralizer. We now go on to describe the centre Z(g e ) of U(g e ). Such a description was first obtained by the second author in [To], however we will need a much more precise formulation of this result which is compatible with the theory of Yangians. As such we draw heavily on the description of Z(g e ) given by Brown-Brundan [BB, Main theorem] in characteristic zero. In loc. cit., the statement is given for the case that the pyramid π is left justified, which is equivalent to the condition s i,j = 0 when i > j on the shift matrix σ, i.e. that σ is upper triangular. From there it is easy to deduce a description of Z(g e ) in terms of the basis of g e corresponding to any pyramid, as this involves is a trivial change of notation. For this reason we assume that π is left justified up to and including Lemma 2.4, so that our notation is aligned with that of [BB].
We begin by stating some formulas for elements of U(g e ) which appeared in [BB,(1.3)] over C. Define the elementsc (2.8) This sequence is known to give the total degrees of a set of homogeneous generators of S(g e ) G e , as is explained in [To,Section 3].
For a subcomposition m of p such that ℓ(m) = d |m| , with nonzero entries m i 1 , . . . , m i d , where d = d |m| we define the m-column determinant of (c where S d denotes the symmetric group of degree d. It is shown in [BB,Lemma 3.8 Finally for s = 1, . . . , N we define (2.10) We move on to state Lemma 2.4, which can essentially be deduced from [To,Theorem 3]. As our statement is slightly different and more explicit, we include an outline of the proof.
(a) The elements z 1 , . . . , z N are algebraically independent generators of U(g e ) G e ; (b) Z(g e ) is a free Z p (g e )-module of rank p N with basis {z k 1 1 · · · z k N N | 0 ≤ k i < p}, and Z HC (g e ) is a free Z p (g e ) G e -module with the same basis. (c) The multiplication map Z p (g e ) ⊗ Zp(g e ) G e U(g e ) G e → Z(g e ) is an isomorphism.
Proof. We begin the proof by briefly considering the situation when has characteristic 0. In this case, using the fact that G e is connected, we have that Z(g e ) = U(g e ) G e . Since π is assumed to be left justified, the statement (a) in characteristic 0 is precisely [BB, Main Theorem]. Now a reduction modulo p argument, identical to that given in the proof of [To, Corollary 1], can be used to deduce that z s ∈ U(g e ) G e for of characteristic p.
By the definition given in (2.10) we have that z s ∈ F ds U(g e ) in the PBW filtration, for all s, and gr ds z s = m⊆p |m|=s, ℓ(m)=ds In [To,Theorem 9] it was demonstrated that {gr ds z s | s = 1, . . . , N} are algebraically independent generators of S(g e ) G e . We should warn the reader that the notation in loc. cit. was different: the partition p was denoted λ, the element c (r) i,j was denoted ξ i,r+p j −p i j , and the notation x s was used to denote the element determined by the formula for gr ds z s above. Now standard filtration arguments show that z 1 , . . . , z N are algebraically independent, and generate U(g e ) G e . This completes the proof of (a).
Taking associated graded algebras we have gr Z p (g e ) = S(g e ) p and gr Z(g e ) ⊆ S(g e ) g e , however this inclusion is actually an equality thanks to the proof of [To,Theorem 3]. It follows from [To,Theorem 9] that {(gr d 1 z 1 ) k 1 · · · (gr d N z N ) k N | 0 ≤ k i < p} generates S(g e ) g e as a S(g e ) p -module. and also that S(g e ) g e is free of rank p N over S(g e ) p . Therefore, {(gr d 1 z 1 ) k 1 · · · (gr d N z N ) k N | 0 ≤ k i < p} is in fact a basis of S(g e ) g e over S(g e ) p by Lemma 2.2. Now we can use that (gr d 1 z 1 ) k 1 · · · (gr d N z N ) k N = gr k 1 d 1 +···+k N d N (z k 1 1 · · · z k N N ) for any choice of k 1 , . . . , k N and apply Lemma 2.3 to obtain the first assertion in (b).
Next we observe that U(g e ) G e ∩Z p (g e ) = Z p (g e ) G e , and that gr Z p (g e ) G e = (S(g e ) G e ) p . It is clear that S(g e ) G e is free as an (S(g e ) G e ) p -module with basis {gr z k 1 1 · · · gr z k N N | 0 ≤ k i < p}. Therefore, using Lemma 2.3, we deduce that U(g) G e is free as a Z p (g e ) G e -module with basis {z k 1 1 · · · z k N N | 0 ≤ k i < p} giving the second assertion in (b). Now we can apply Lemma 2.1 to obtain (c).
We consider the special case where e = 0, i.e. when p = (1, . . . , 1). Here we can be more explicit about the generators of U(g) G as we explain below, where we observe that these generators arise from the Capelli identity. These generators of U(g) G are well-known in characteristic zero, see for example [BK2,§3.8], and we expect it is also known in positive characteristic, so we just give a short justification for convenience.
Recall that the column determinant cdet(A) of a square N × N-matrix A = (a i,j ) 1≤i,j≤N with coefficients in an associative algebra is defined by (2.11) Let u be a formal variable and consider the determinant where the entries of the matrix are considered as elements of U(g) [u].
For m ⊆ p, we let z 0 m be the matrix formed by the rows and columns indexed by the set of i such that m i = 1 of the matrix appearing in (2.12), after replacing a diagonal entry e i,i + u − i + 1 by e i,i − i + 1. Using the formula for calculating the column determinant in (2.11) we can get the decomposition Now we observe that cdet z 0 m is equal to cdet m (c (r) i,j ) as defined in (2.9), noting that in the present case where p = (1, . . . , 1), we havec Putting this all together, we can deduce that z r as defined in (2.10) is equal to Z (r) as defined in (2.12). Consequently, the statements in Lemma 2.4 hold for U(g) with Z (r) in place of z r .
We return to the case of general e, and we record an important technical lemma characterising the p-centre of U(g e ), which is crucial to our later arguments. Before this is stated in Lemma 2.5, we need to give some more notation. We let π be the pyramid obtained from π by adding an extra row on the bottom with p n boxes. Then let g = gl N +pn ( ), and let e ∈ g be the nilpotent element corresponding to π. The centralizer g e has basis given by {c (r) i,j | 1 ≤ i, j ≤ n + 1, s i,j ≤ r < s i,j + p min(i,j) }, where we extend the notation used in (2.5), setting p n+1 := p n . Inspecting (2.6) and (2.7) we see that g e identifies naturally with a restricted subalgebra of g e . Lemma 2.5. Z p (g e ) = U(g e ) ∩ Z(U(g e )).
Proof. Clearly we have Z p (g e ) ⊆ U(g e ) ∩ Z(U(g e )). Suppose that this inclusion is strict and let z ∈ U(g e ) ∩ Z(U(g e )) \ Z p (g e ) such that z ∈ F d U(g e ) with d as small as possible. If gr d z = y p ∈ S(g e ) p where y ∈ S(g e ), then z − ξ g e (y) ∈ U(g e ) ∩ Z(U(g e )) \ Z p (g e ) and z − ξ g e (y) ∈ F d−1 U(g e ). Thus we have that gr d z ∈ S(g e ) ∩ S(g e ) g e \ S(g e ) p = ∅. Let for certain elements f m ∈ S(g e ) p , where the sum is taken over all maps m : I → {0, . . . , p−1}. Since y / ∈ S(g e ) p there exists an m 0 : I → {0, . . . , p − 1} and a tuple (i 0 , j 0 , r 0 ) ∈ I such that f m 0 = 0 and m 0 (i 0 , j 0 , r 0 ) = 0. Using (2.6) we can write ad c = 0 and so the summand occurring in the first line of the above expression for ad(c (s n+1,i 0 ) n+1,i 0 )y is non-zero. Now it remains to observe that the non-zero monomial summands occurring in the expressions are all distinct; this follows readily from (2.6). We conclude that ad(c (s n+1,i 0 ) n+1,i 0 )y = 0 which contradicts the assumption y ∈ S(g e ) g e . This contradiction confirms that the inclusion Z p (g e ) ⊆ U(g e ) ∩ Z(U(g e )) is actually an equality.
2.7. The truncated shifted current Lie algebra. Let n ∈ Z ≥0 . The current Lie algebra of gl n ( ) is the Lie algebra c n := gl n ( ) ⊗ [t]. For x ∈ gl n ( ) and f ∈ [t] we abbreviate our notation by writing xf for x ⊗ f ∈ c n , and observe that c n has a basis where we write {e i,j | 1 ≤ i, j ≤ n} for the standard basis of matrix units in gl n ( ). The commutator between elements in this basis is given by (2.13) We have that c n is a restricted Lie algebra with the p-power map defined by (xf ) [p] denotes the pth matrix power of x, see for example [BT,Lemma 3.3]. So in particular the p-power map is given on the basis of c n by (2.14) Now let σ = (s i,j ) be any shift matrix of size n. The shifted current Lie algebra is defined to be the subspace c n (σ) of c n spanned by ( 2.15) It is observed in [BT,Lemma 3.3] that c n (σ) is a restricted Lie subalgebra of c n . We fix an integer l > s 1,n + s n,1 which we call the level, following the terminology of §2.4. Then using (2.3) we define the partition p = (p 1 , . . . , p n ) from the data (σ, l), and we let N = n i=1 p i . We define the truncated shifted current Lie algebra c n,l (σ) to be the quotient of c n (σ) by the ideal i n,l generated by {e 1,1 t r | r ≥ p 1 }.
We recall from §2.4 that (σ, l) determines a pyramid π, which we can use to define e ∈ g = gl N ( ) as in (2.4). The next lemma shows that the truncated current Lie algebra is isomorphic to the centralizer g e . For the statement we recall that a basis is given in (2.5).
(a) A basis of i n,l is given by is a surjective homomorphism of restricted Lie algebras with kerθ = i n,l . In particular, θ induces an isomorphism θ : c n,l (σ) ∼ −→ g e of restricted Lie algebras, and a basis of c n,l (σ) is given by (2.16) Proof. Let j n,l denote the subspace of c n (σ) with basis {e i,j t r | 1 ≤ i, j ≤ n, r ≥ s i,j +p min(i,j) }. A straightforward calculation with the commutator relations in (2.13) shows that j n,l is in fact an ideal of c n (σ), and thus we have i n,l ⊆ j n,l . Since e 1,1 t p 1 ∈ i n,l and e 1,j t s 1,j +r ∈ c n (σ) for r ≥ 0 we have e 1,j t p 1 +s 1,j +r = [e 1,1 t p 1 , e 1,j t s 1,j +r ]. Similarly e i,1 t p 1 +s i,1 +r ∈ i n,l for r ≥ 0. Next we observe [e 1,2 t p 1 +s 1,2 , e 2,1 t s 2,1 +r ] = (e 1,1 − e 2,2 )t p 2 +r for r ≥ 0, where we use that p 2 = p 1 + s 1,2 + s 2,1 . Since p 1 ≤ p 2 we have e 1,1 t p 2 +r ∈ i n,l , so we can deduce that e 2,2 t p 2 +r ∈ i n,l for r ≥ 0.
By considering the shifted current Lie algebra spanned by {e i,j t r | 2 ≤ i, j ≤ n, r ≥ s i,j }, and applying an inductive argument, we obtain that j n,l ⊆ i n,l . Hence, i n,l = j n,l which proves (a).
The fact that the linear mapθ in (b) is a homomorphism of restricted Lie algebras may be seen by comparing the Lie bracket and p-power map for g e given in (2.6) and (2.7) with those for c n (σ) given in (2.13) and (2.14).
It is evident that e 1,1 t r lies in the kernel ofθ for r ≥ p 1 , so we have that i n,l is contained in kerθ. By (a) we see that (2.16) gives a spanning set of c n,l (σ). Moreover, by (2.5) the elements given in (2.16) are sent to a basis g e by the induced map θ : c n,l (σ) → g e . From this it follows that θ is an isomorphism, and that (2.16) is a basis of c n,l (σ).
Remark 2.7. For later use we observe that the shifted current algebra and its truncation can be defined over the integers. We write c n (σ) Z for the free Z-submodule of gl n (Z) ⊗ Z Z[t] spanned by the elements (2.15), equipped with its Lie ring structure. We define c n,l (σ) Z to be the quotient of c n (σ) Z by the ideal generated by {e 1,1 t r | r ≥ p 1 }. Then we observe that the proof of Lemma 2.6 can be applied verbatim to show that c n,l (σ) Z is a free Z-module spanned by the elements (2.16).

Shifted Yangians and W -algebras
In this section we fix n ∈ Z ≥1 , a shift matrix σ = (s i,j ) of size n and an integer l > s 1,n +s n,1 , which, as usual, we call the level. We define the pyramid π from (σ, l) as explained in §2.4. The partition p = (p 1 , . . . , p n ) is defined by (2.3), and we let N = n i=1 p i . We define e ∈ g = gl N ( ) as in (2.4), and let G = GL N ( ).
and relations given in [BT,Theorem 4.15]. The definition of the shifted Yangian was first given in [BK1] over a field of characteristic zero and then considered in positive characteristic in [BT].
In order to state the PBW theorem we define the PBW generators of Y n (σ) as follows.
The loop filtration on Y n (σ) is defined by placing the elements E in filtered degree r for all r ≥ 0. We write F r Y n (σ) for the filtered piece of degree r, so that Y n (σ) = r≥0 F r Y n (σ), and we write gr Y n (σ) for the associated graded algebra. Then [BT,Lemma 4.13] says that there is an isomorphism for i < j. It follows immediately that the monomials in the elements taken in any fixed order give a basis of Y n (σ), and this gives the PBW theorem for Y n (σ).

3.2.
Finite W -algebras. We move on to introduce of the W -algebra U(g, e), and begin with the definition stated in [GT1]. This is the positive characteristic analogue of the definition first given by Premet in [Pr1,Section 4].
Consider the cocharacter µ : × → G defined by µ(t) = diag(t col(1) , . . . , t col(n) ); here we use the notation diag(t 1 , . . . , t N ) to mean the diagonal N × N matrix with ith diagonal entry equal to t i . Using µ we define the Z-grading (3.5) Since the adjoint action of µ(t) on a matrix unit is given by We define the subalgebras of g. Then p is a parabolic subalgebra of g with Levi factor h and m is the nilradical of the opposite parabolic to p. Let M be the closed subgroup of G generated by the root subgroups We define χ ∈ g * to be the element dual to e via the trace form on g. Since e ∈ g(1), we have that χ vanishes on g(r) for r = −1. Therefore, χ restricts to a character of m. We define and thus a projection pr : U(g) → U(p) (3.7) onto the second factor. The twisted adjoint action of M on U(p) is defined by tw(g) · u := pr(g · u), (3.8) for g ∈ M and u ∈ U(p); a twisted adjoint action on S(p) can be defined analogously. Then the W -algebra associated to e is defined to be the invariant subalgebra We move on to recall a set of generators for U(g, e). These are elements in U(p) defined using the remarkable formulas, given in [BK1,Section 9]; see also [GT2,Section 4]. As is shown in the [GT2,Theorem 4.3] the elements in (3.9) are twisted Minvariants, thus elements of U(g, e), and moreover they generate U(g, e). We note there is an abuse of notation as these generators of U(g, e) have the same names as the generators for Y n (σ) given in (3.1); this overloading of notation will be justified in the next subsection. Below we state the formula for D (r) i in (3.11), and require some notation for this. Let t be the Lie algebra of T , and write {ε 1 , . . . , ε N } for the standard basis of t * . We define the weight η ∈ t * by where we recall that q i is the height of the ith column in the pyramid π, and we note that η extends to a character of p. For e i,j ∈ p definẽ e i,j := e i,j + η(e i,j ).
Then by definition where the sum is taken over The expressions for the elements E ∈ U(p) are given by similar formulas; see [BK1,Section 9] or [GT2,Section 4]. Then we can define E . As a consequence of the PBW theorem for U(g, e), the monomials in taken in any fixed order form a basis of U(g, e), see [GT2,Lemma 4.2].
There are two filtrations of the W -algebra that we recall here. First we consider the loop filtration, which is defined by taking the grading of U(p) given by the action of the cocharacter µ, and then the induced filtration ∞ r=0 F r U(g, e) of U(g, e). We write gr U(g, e) ⊆ U(p) for the associated graded algebra.
As a consequence of [GT2,Lemma 4 It follows that the shift automorphism on S −η : (3.16) Next we consider the Kazhdan filtration of U(g, e). This is first defined on U(g) by placing x ∈ g(r) in Kazhdan degree r + 1, and as explained in [GT1,Section 7], the associated graded algebra can be identified with S(p). We write ∞ r=0 F ′ r U(g, e) of U(g, e) for the induced filtration on U(g, e), and gr ′ U(g, e) for the associated graded algebra. Using [GT2,Lemma 7.1] we identify gr ′ U(g, e) = S(p) twM , where the twisted adjoint action of M on S(p) is defined in analogy with (3.8). Further [GT1,Lemma 7.1] along with [GT2,Lemma 4.2] imply that the PBW generators D i,j given in (3.12) lie in Kazhdan degree r and that gr ′ r D i,j are algebraically independent generators of gr ′ U(g, e). 3.3. The truncated shifted Yangian. Our next step is to recall an algebra isomorphism φ from a truncation of the shifted Yangian to the finite W -algebra. This is done in Theorem 3.1, which also includes the PBW theorem for the truncation. Although this was proved in [GT2,Theorem 4.3], drawing heavily on the results of [BK1], we repeat a few of the details here to demonstrate that the proof can be simplified slightly by using the shifted current algebra.
The algebra homomorphismφ : .17) is defined by sending the generators E with the same names. Then we have thatφ is surjective. The fact that this is a homomorphism justifies the abuse of notation in naming the generators of Y n (σ) and U(g, e).
The truncated shifted Yangian Y n,l (σ) is defined to be the quotient of Y n (σ) by the ideal I n,l generated by the elements D (r) 1 with r > p 1 . It follows directly from formula [GT2,(4.2)] that the element D (r) 1 of U(g, e) is equal to zero for r > p 1 , and soφ factors through the quotient to give a surjection φ : Y n,l (σ) → U(g, e). (3.18) i,j for its image in Y n,l (σ). The loop filtration on Y n (σ) descends to a loop filtration on Y n,l (σ). We denote the filtered pieces by F r Y n,l (σ) for r ≥ 0, and write gr Y n,l (σ) for the associated graded algebra.
We are now ready to show that φ is an isomorphism and deduce the PBW theorem for Y n,l (σ).
Consequently, the ordered monomials in the elements taken in any fixed order form a basis of Y n,l (σ).
Proof. Usingψ from (3.3) we identify U(c n (σ)) with gr Y n (σ). The associated graded ideal gr I n,l contains the ideal i n,l defined in §2.7, so there is a surjection U(c n,l (σ)) ։ gr Y n,l (σ). It follows from Lemma 2.6(a) and the PBW theorem for U(c n,l (σ)) that gr Y n,l (σ) is spanned by the ordered monomials in the elements (3.20). Now the PBW theorem for U(g, e), given in [GT1,Theorem 7.2], along with [GT2,Lemma 4.2] imply that the images under φ of these spanning elements are linearly independent, and so they form a basis. This proves (b).
We have seen that φ sends a basis of Y n,l (σ) to a basis of U(g, e), so that it is an isomorphism, and we get (a).
It is helpful for us to give some notation for the PBW basis of Y n,l (σ) given by Theorem 3.1(c). We fix an order on the sets where the products respect the orders which we have fixed on J F , J D and J E . So that is a basis of Y n,l (σ).
We can see that the isomorphism φ in (3.17) is filtered for the loop filtration, as D have the same degree, namely r, when considered as elements of Y n (σ) or as elements of U(g, e). Thus we obtain an isomorphism gr φ : gr Y n,l (σ) given by Lemma 2.6, and we have the isomorphism S −η : gr U(g, e) = U(g e ). We note however that as isomorphisms gr Y n,l (σ) To explain this we note the adjoint action of µ(−1) gives an automorphism U(g e ) → U(g e ), which is determined by c i,j ; here we recall that µ is the cocharacter defining the good grading on g. Then we have (3.22) For Theorem 4.2, we need to fix an isomorphism between gr Y n,l (σ) ∼ −→ U(g e ). For consistency with [BB], we use S −η • gr φ : gr Y n,l (σ) ∼ −→ U(g e ) (3.23) which is determined by its effect on the generators as follows 3.4. The integral forms of Y n (σ) and Y n,l (σ). We introduce and study the integral (truncated) shifted Yangian. There are two natural approaches: we can consider the subring of the complex (truncated) shifted Yangian generated by the elements listed in (3.1); or we can consider the ring determined by these generators and the relations in [BT,Theorem 4.15] (along with the relations D (r) 1 = 0 for r > p 1 ). Lemmas 3.2 and 3.3 say that these two approaches lead to isomorphic rings. As explained by Corollary 3.4 this allows us to apply reduction modulo p to certain formulas in the complex truncated shifted Yangian, which will be useful later on.
Let A be a commutative ring. We define the shifted A-Yangian Y A n (σ) to be the A-algebra with generators given in (3.1) subject to the relations in [BT,Theorem 4.15]. Here we are only concerned with the cases where A = Z, C or . We note that Y n (σ) = Y n (σ), and that Y C n (σ) is the usual complex shifted Yangian, as considered in [BK1] and [BK2]. We mildly abuse notation by viewing the elements in (3.1) simultaneously as elements of Y Z n (σ), Y C n (σ) and Y n (σ).
There is a ring homomorphism Y Z n (σ) → Y C n (σ) sending a generator of Y Z n (σ) to the element of Y C n (σ) with the same name. This induces a ring homomorphism (a) Y Z n (σ) is a free Z-module with basis given by ordered monomials in the elements given in (3.4).
Proof. As introduced in Remark 2.7 we write the c n (σ) Z for Z-form of c n (σ). The argument in the penultimate paragraph of the proof of [BT,Theorem 4.3] can be applied verbatim to show that there is a surjection U(c n (σ) Z ) ։ Y Z n (σ): to apply this argument it is necessary to define a loop filtration on Y Z n (σ), which can be done by placing E in degree r. We can now deduce that the PBW monomials in the elements in (3.4) form a spanning set of Y Z n (σ) over Z. By [BK1, Theorem 2.1] these monomials are sent to C-linearly independent elements of Y C n (σ) under the map Y Z n (σ) → Y C n (σ). Therefore, they are certainly Z-linearly independent in Y Z n (σ). This proves (a). Also we have shown that Y Z n (σ) → Y C n (σ) sends a Z-basis to a C-basis, which implies (b).
Thanks to [BT,Theorem 4.14], ordered monomials in the elements in (3.4) form a -basis of Y n (σ). Thus the map Y Z n (σ) ⊗ Z → Y n (σ) sends to a -basis of Y Z n (σ) ⊗ Z to a -basis of Y n (σ), and we obtain (c).
We also want an analogue of Lemma 3.2 in the context of truncated shifted Yangians. To do this we first define the truncated shifted A-Yangian Y A n,l (σ) to be the quotient of Y A n (σ) by the ideal generated by {D (r) 1 | r > p 1 }. Similarly to the non-truncated case we have maps (a) Y Z n,l (σ) is a free Z-module with basis given in (3.21).
Proof. Recall that c n,l (σ) Z is defined in Remark 2.7. The argument at the start of the proof of Theorem 3.1 can be applied to show that U(c n,l (σ) Z ) surjects onto gr Y Z n,l (σ). Now we can complete the proof of the current lemma using the same steps as in the proof of Lemma 3.2, employing the PBW theorems for Y C n,l (σ) and Y n,l (σ), which are given in [BK1,Corollary 6.3] and Theorem 3.1.
Thanks to the previous lemma we can employ reduction modulo p to deduce formulas in Y n,l (σ) from certain types of formulas in Y C n,l (σ), as explained by the following corollary.
Corollary 3.4. Let h be a polynomial with coefficients in Z in the non-commuting indeterminates {f i | 1 ≤ i < n, r > s i,i+1 }, and let A be a ring. Write H A for the element of Y A n,l (σ) obtained by specialising h via d n,l (σ), so that H Z = 0. Further, under the identification Y n,l (σ) ∼ = Y Z n,l (σ) ⊗ Z , given by Lemma 3.3(c), we have H = H Z ⊗ 1. Hence, H = 0, and this proves (a) Using Lemma 3.3(a) may write H Z ∈ Y Z n,l (σ) as a Z-linear combination of the PBW basis given in (3.21) given by monomials in the elements from (3.20) in some fixed order. This also gives the expression for H C in terms of this PBW basis in Y C n,l (σ) and for H in terms of this PBW basis in Y n,l (σ). Furthermore, the filtered degree for the loop filtration can be read off directly from these expressions, which implies (b).
The observations of the previous lemma will be convenient for us at several places later in this paper. However we should mention that we expect that the formulas which we verify using this approach can also be established over by repeating the known methods over C. Thus the reduction modulo p procedure may be viewed as a convenient alternative to reciting certain technical arguments from characteristic zero.
We end this subsection by explaining that some parts of the the theory of U(g, e) from §3.2 can be carried out over Z. Let p Z be the parabolic subalgebra g Z = gl n (Z) such that p = p Z ⊗ Z . The element χ ∈ g * , can be viewed as a function from g Z → Z, and then we can define a projection pr Z : U(g Z ) → U(p Z ) (3.24) in analogy with pr as defined in (3.7).
The isomorphism φ : Y n,l (σ) ∼ −→ U(g, e) from (3.17) can be thought of as an embedding φ : Y n,l (σ) ֒→ U(p). By considering the formulas for of the twisted M-invariants D [BK1,Section 9], see also [GT2,Section 4], we see that they can be viewed as an elements of U(p Z ). Therefore, we can consider the ring homomorphism defined in the obvious manner φ Z : Y Z n,l (σ) → U(p Z ).
(3.25) We also note here that the procedure of the reduction modulo p given by Corollary 3.4 has an obvious analogue with U(p C ), U(p) and U(p Z ) in place of Y C n,l (σ), Y n,l (σ) and Y Z n,l (σ); these observations will be vital in the proof of Lemma 4.4.
3.5. The T (r) i,j generators for Y n,l (σ). We introduce some alternative PBW generators, which will be important later. They were described in [BK2, Section 2.2] over C. We recap the details for the readers convenience.
Let u be an indeterminate, and consider the power series ring Y n (σ) [[u −1 ]]. We adopt the convention D (0) i = 1 for all i and define the power series i,j u −r . By convention we also set E i,i (u) = F i,i (u) = 1.
Next we define the following n × n matrices with coefficients in Y n (σ)[[u −1 ]]: , whose (i, j)-entry can be written as a power series i,j . By direct calculation we easily see that T i,j = 0 for 0 < r ≤ s i,j . Also we can see that T (r+1) i,j ∈ F r Y n (σ) and then using the isomorphismψ from (3.3) to identify gr Y n (σ) ∼ = U(c n (σ)) we have This allows us to deduce that the T i,j | 1 ≤ i, j ≤ n, s i,j < r ≤ s i,j + p min(i,j) } form a basis for Y n,l (σ).
The next result is obtained as application of reduction modulo p, using Corollary 3.4.
Corollary 3.6.Ṫ (r) i,j = 0 in Y n,l (σ) for r > p min(i,j) + s i,j Proof. The version of this statement in Y C n,l (σ) is [BK2,Theorem 3.5]. Now we can apply Corollary 3.4.

Centres and restricted versions
In this section we study the centres of Y n,l (σ) and U(g, e). Both algebras admit a natural definition of a Harish-Chandra centre and a p-centre arising in different ways, and we show that in either case the centre is generated by these subalgebras. We continue to use the notation from Section 3. 4.1. The centre of Y n (σ). We proceed to recall the description of the centre of Y n (σ) given in [BT]. The power series D i (u) are defined in (3.26). From these we define By [BT,Theorem 5.11(1)], the elements in {C (r) | r > 0} are algebraically independent and lie in the centre Z(Y n (σ)) of Y n (σ). The subalgebra they generate is called the Harish-Chandra centre of Y n (σ), and is denoted Z HC (Y n (σ)).
For i = 1, . . . , n, we define By [BT,Theorem 5.11(2)] the elements in are algebraically independent, and lie in Z(Y n (σ)). The subalgebra they generate is called the p-centre of Y n (σ) and is denoted Z p (Y n (σ)). We note that by [BT,Theorem 5.8], the elements B (s) i can be written in terms of B (rp) i for 0 ≤ r ≤ s p , so in particular they lie in the p-centre of Y n (σ). Furthermore by [BT,Theorems 5.1,5.4,5.8 and under the identification of gr Y n (σ) ∼ = U(c n (σ)) given by the isomorphismψ from (3.3) we have From this it follows that Z p (Y n (σ)) is a polynomial algebra over the generators given in (4.2). Though we do not require it in this paper we remark that [BT,Theorem 5.11] contains more information about the centre of Y n (σ). In particular, it is generated by Z HC (Y n (σ)) and Z p (Y n (σ)). We also mention that [BT,Corollary 5.13] states that Y n (σ) is a free module over Z p (Y n (σ)) with basis given by the by the ordered monomials in the generators in (3.4) in which no exponent is p or more; we refer to such monomials as p-restricted monomials.

4.2.
The centre of the truncated shifted Yangian. In this subsection we prove Theorem 4.2, giving a precise description of the centre of Y n,l (σ). As in [BK2,Lemma 3.7] we define the Laurent series (4.5) Following the convention established in §3.1 we use the notationḂ (r) i ,Ċ (r) ,Ż r to denote the images of B (r) i , C (r) , Z r ∈ Y n (σ) in the quotient Y n,l (σ); similarly we use the power series notationĊ(u),Ż(u).
is a polynomial in u of degree N. Proof. We may viewŻ(u) as a Laurent series in u −1 with coefficients in the complex truncated shifted Yangian Y C n,l (σ), which can be expressed as an integral linear combination of products of the generators of Y n,l (σ). In this setting [BK2, Lemma 3.7] implies thatŻ(u) is in fact a polynomial in u of degree N. Now viewingŻ(u) as a Laurent series in u −1 with coefficients in Y n,l (σ) and using Corollary 3.4, we deduce thatŻ(u) is a polynomial in u of degree N.
Similarly we define the p-centre of Y n,l (σ) to be the image of the p-centre of Y n (σ) in Y n,l (σ), and denote it by Z p (Y n,l (σ)).
We are now ready to state and prove our description of the centre of Y n,l (σ).
For the next step we claim thatŻ r ∈ F r−dr Y n,l (σ) and that gr r−drŻr = (−1) r−dr z r ∈ gr Y n,l (σ) = U(g e ), under the identification gr Y n,l (σ) ∼ = U(g e ) given by (3.23). Thanks to (4.8), the definition ofŻ(u) given in (4.5) is the same as that given in [BB,(3.2)]. Next we observe that the formula given in [BB,Lemma 3.5] which expressesŻ r in terms of the elementsṪ (r) i,j can be expressed as an integral linear combination of products of the generators of of Y C n,l (σ) in (3.1). Applying Corollary 3.4 we conclude that the same formula holds foṙ Z r ∈ Y n,l (σ). Now the argument used to complete the proof of [BB,Theorem 3.4] can be repeated verbatim to deduce the claims made at the beginning of this paragraph. Now we may combine Lemma 2.4(a) with a standard filtration argument to deduce thaṫ Z 1 , . . . ,Ż N are algebraically independent, proving (a).
A similar argument shows that the elements (Ė j,i ) p , under the identification gr Y n,l (σ) ∼ = U(g e ). Since these elements are algebraically independent generators for Z p (g e ), it follows that the elements in (4.6) are algebraically independent in Z p (Y n,l (σ)).
We next show that Z p (Y n,l (σ)) coincides with the algebra generated by the elements in (4.6); we denote this latter algebra by Z p (Y n,l (σ)).
From the pyramid π associated to (σ, l) we construct the pyramid π by adding another row to the bottom of length p n , as we did in §2.6. This gives a new shift matrix σ with s n,n+1 = s n+1,n = 0 and s i,i+1 = s i,i+1 , s i+1,i = s i+1,i for i = 1, . . . , n. The defining relations of the truncated shifted Yangian, along with the PBW theorem given in Theorem 3.1(b) imply that there is an embedding Y n,l (σ) ֒→ Y n+1,l (σ). Since the elements (Ė are sent to the elements of Y n+1,l (σ) with the same names, it follows that these elements are central in Y n+1,l (σ). We conclude that every element of Z p (Y n,l (σ)) commutes with every element of Y n+1,l (σ). Following the notation of Lemma 2.5 we identify gr Y n+1,l (σ) with U(g e ) using the analogue of the isomorphism given in (3.23).
We will show that the inclusion Z p (Y n,l (σ)) ⊆ Z p (Y n,l (σ)) is an equality by considering the associated graded algebras. Thanks to our previous observations we have gr Z p (Y n,l (σ)) = Z p (g e ). Suppose that Z p (Y n,l (σ)) \ Z p (Y n,l (σ)) = ∅ and choose an element u of minimal loop degree, say d. By the remarks of the previous paragraph we see that gr d u commutes with everything in U(g e ) and applying Lemma 2.5 we see that gr d u ∈ Z p (g e ). As we observed above the generators of Z p (g e ) are all of the form gr (r−1)p B where the indexes i, j, r are restricted in accordance with (4.6). Consequently there exists u ′ ∈ Z p (Y n,l (σ)) of loop degree d such that gr d u = gr d u ′ . Since u / ∈ Z p (Y n,l (σ)) we deduce that u − u ′ ∈ Z p (Y n,l (σ)) \ Z p (Y n,l (σ)) is of strictly lower loop degree. Since the degree of Z was assumed to be minimal, we have reached a contradiction. This confirms that Z p (Y n,l (σ)) = Z p (Y n,l (σ)), and thus completes the proof of (b).
To prove (c), we start by observing that we have shown gr rpḂ generate both gr Z p (Y n,l (σ)) and Z p (g e ). Hence, gr Z p (Y n,l (σ)) = Z p (g e ).
We have seen that gr r−dr Z r = (−1) r z r , and we have also have (4.9). Thus Lemma 2.4 along with a standard filtration argument implies that Z(Y n,l (σ)) is generated by Z HC (Y n,l (σ)) and Z p (Y n,l (σ)). Now we can deduce (d) from Lemma 2.3 and Lemma 2.4(b).
We have now completed the proof in case σ is upper-triangular and it remains to explain how to deduce the theorem for arbitrary σ. First we note that our proof of (b) and (c) does not actually require the assumption that σ is upper triangular. So we are left to deal with (a) and (d).
It follows from [BT,4.5,(4)] that there exists an upper-triangular shift matrix σ u and an isomorphism ι : Y n (σ) ∼ −→ Y n (σ u ). Each of these algebras has a commutative subalgebra generated by {D (r) i | 1 ≤ i ≤ n, r ≥ 0}, and the isomorphism ι fixes this subalgebra pointwise. Consequently, there is an induced isomorphism Y n,l (σ) ∼ −→ Y n,l (σ u ). This same fact also shows that the coefficients of the series C(u) are fixed by ι which implies that ι : Z HC (Y n,l (σ)) ∼ −→ Z HC (Y n,l (σ u )) and that the elements denoted Z 1 , . . . , Z N in Y n,l (σ) are sent to the elements with the same names in Y n,l (σ u ). Furthermore it follows from the definition of ι that the generators of Z p (Y n (σ)) are sent bijectively to the generators of Z p (Y n (σ u )), and we conclude that ι : Z p (Y n,l (σ)) ∼ −→ Z p (Y n,l (σ u )). Now we can deduce (a) and (d) for Y n,l (σ) from the same statements for Y n,l (σ u ).
In the left-justified case, we saw in the proof above that gr Z HC (Y n,l (σ)) identifies with U(g e ) G e ⊆ U(g e ) ∼ = gr Y n,l (σ). It would be possible to prove this in general by using a reduction modulo p argument, but this fact is not required in the sequel.
For later use we record an immediate consequence of Theorem 4.2, which describes a basis for Z(Y n,l (σ)). To do this we use some notation introduced §3.3. For u = (u (4.10) Corollary 4.3. A basis for Z(Y n,l (σ)) is given by the ordered monomials 4.3. Restricted (truncated) shifted Yangians. It is well-known that U(g) is a free module over its p-centre with a basis given by PBW monomials in the standard basis of g in which every exponent is less than p; we refer to such monomials as p-restricted monomials. It follows that the restricted enveloping algebra U [p] (g) is spanned by the image of the p-restricted monomials. Analogous statements hold for Y n (σ) and Y n,l (σ), as we now explain. As explained at the end of §4.1, we have that Y n (σ) is a free Z p (Y n (σ))-module with basis given by the p-restricted monomials in the PBW generators of Y n (σ) given in (3.4). We define Z p (Y n (σ)) + to be the maximal ideal of Z p (Y n (σ)) generated by the elements given in (4.2). Now we can define the restricted shifted Yangian Y [p] n (σ) := Y n (σ)/Y n (σ)Z p (Y n (σ)) + .
The images in Y [p] n (σ) of the p-restricted monomials in the PBW generators of Y n (σ) given in (3.4) form a basis of Y [p] n (σ). As a consequence of Lemma 2.3 and Theorem 4.2(c), we see that Y n,l (σ) is free as an Z p (Y n,l (σ))-module. To give a basis for this module we recall that from (3.21) we have the basis {Ḟ uḊtĖv | (u, t, v) ∈ I F × I D × I E } of Y n,l (σ). We let I p be the set of all tuples (u, t, v) where all entries of u, t and v are less than p. Then the p-restricted monomials {Ḟ uḊtĖv | (u, t, v) ∈ I p } form a basis of Y n,l (σ) as a free Z p (Y n,l (σ))-module. We define Z p (Y n,l (σ)) + to be the ideal of Z p (Y n (σ)) generated by the elements (4.6) of Z p (Y n,l (σ)) and define the restricted truncated shifted Yangian Y [p] n,l (σ) := Y n,l (σ)/Y n,l (σ)Z p (Y n,l (σ)) + . Then a basis of Y [p] n,l (σ) is given by In particular, we note that dim Y n,l (σ) = p dim g e . We let I [p] n,l be the ideal of Y 1 + J p (Y n (σ)) | r > p 1 }. Then using Theorem 4.2(b) we can see that there is a natural isomorphism n,l .
(4.13) 4.4. The centre of U(g, e). We use the description of Z(Y n,l (σ)) given in Theorem 4.2 along with the isomorphism φ : Y n,l (σ) ∼ −→ U(g, e) to provide an explicit description of the centre Z(g, e) of U(g, e) as stated in Theorem 4.7 below.
We recall the map pr : U(g) → U(p) is defined in (3.7) and define the Harish-Chandra centre Z HC (g, e) of U(g, e) to be the image of U(g) G under pr. It is evident that Z HC (g, e) is invariant under the twisted adjoint action of M, and that these elements are central in U(g, e). Our first objective is to show that the isomorphism φ : Y n,l (σ) ∼ −→ U(g, e) from (3.18) preserves the Harish-Chandra centres.
Recall that Z HC (Y n,l (σ)) is generated by the coefficients of the polynomialŻ(u) ∈ Y n,l (σ)[u] defined in §4.2 whilst U(g) G is generated by the coefficients of the Capelli determinant Z * (u) = N r=0 Z (r) u N −r ∈ U(g) [u] given in (2.12). The following lemma relates these polynomials.
Lemma 4.4. We have the following equality in U(g, e) [u] pr(Z * (u)) = φ(Ż(u)). (4.14) Proof. Recall that pr Z : U(g Z ) → U(p Z ) is given in (3.24). If we view Z * (u) as a polynomial with coefficients in U(g Z ) then pr Z (Z * (u)) is a polynomial with coefficients in U(p Z ). Using Lemma 3.3(b) we view Y Z n,l (σ) as a subalgebra of Y C n,l (σ), and thus viewŻ(u) as a polynomial with coefficients in Y Z n,l (σ). Recalling the map φ Z from (3.25) we obtain two polynomials pr Z (Z * (u)) and φ Z (Ż(u)) with coefficients in U(p Z ). Using the natural inclusion U(p Z ) ֒→ U(p C ), [BK2,Lemma 3.7] implies that the equality pr Z (Z * (u)) = φ Z (Ż(u)) holds in U(p Z ) [u]. Now, by taking the image of this equality under the natural map U , we obtain (4.14).
We introduce the notation Z r := pr(Z (r) ) ∈ U(g, e) for r = 1, . . . , N; by the previous lemma we have that Z r = φ(Ż r ) too.
Proof. In §2.6 we demonstrated that U(g) G is generated by the coefficients of Z * (u), and it follows that Z HC (g, e) is generated by the coefficients of pr Z * (u), i.e. by Z 1 , . . . , Z n . Now Lemma 4.4 implies that the generatorsŻ 1 , . . . ,Ż N of Z HC (Y n,l (σ)) are sent bijectively to those of Z HC (g, e).
The p-centre of U(g, e) is defined to be In the general setting of finite W -algebras associated to reductive groups, this subalgebra was studied in some detail in [GT1,Section 8]. Using the explicit formulas for the generators (3.9) of U(g, e) given in §3.2 we now introduce an explicit generating set for Z p (g, e). Recall that the Kazhdan filtration of U(p) and U(g, e) was discussed at the end of §3.2; in particular, we identify gr ′ U(g, e) ∼ = S(p) twM . Also we remind the reader that ξ p : Lemma 4.6.
(a) Z p (g, e) is a polynomial algebra of rank dim g e generated by i 1 ,j 1 ) · · · (e p is,js − e is,js ) (4.16) where the sum is taken over the index set described in (3.11).
Proof. As remarked at the end of §3.2, S(p) tw(M ) is a polynomial algebra of rank dim g e generated by i,j }. Using [GT1,Lemma 7.6] we note that the restriction of pr : U(g) → U(p) to Z p (g) is the projection Z p (g) → Z p (p) along the decomposition . It follows that ξ p : S(p) (1) → Z p (p) is equivariant for the twisted action of M, so we can deduce (a).
Part (b) now follows easily from (a), because the formula for gr ′ r D (r) i is obtained from (3.11) by replacing each occurrence ofẽ i l ,j l with e i l ,j l .
Using the explicit formulas for E i ) analogous to that given for ξ p (gr ′ r D (r) i ). In principle, it is also possible, though more complicated, to provide expressions for the generators ξ p (gr ′ r E (r) i,j ) and ξ p (gr ′ r F (r) i,j ) when i < j + 1. We are now ready to prove our main result regarding the centre of U(g, e). For the statement of this theorem, we consider the intersection Z HC,p (g, e) := Z HC (g, e) ∩ Z p (g, e). It is a direct consequence of the definitions that this intersection is equal to pr(Z p (g) G ).
(a) The centre Z(g, e) of U(g, e) is free of rank p N over Z p (g, e) with basis We have a tensor product decomposition Z(g, e) = Z p (g, e) ⊗ Z HC,p (g,e) Z HC (g, e).
Proof. By Lemma 4.6 and the formulas given in (3.13), we have gr Z p (g, e) = Z p (g e ). Further, by Theorem 4.2(b) we have gr Z p (Y n,l (σ)) = Z p (g e ). The isomorphism φ : Y n,l (σ) is filtered with respect to the loop filtration by Theorem 3.1, and sendsŻ r ∈ Z HC (Y n,l (σ)) to Z r ∈ Z HC (g, e). Now (a) follows from Theorem 4.2(c).
To prove (b), we apply Lemma 2.1, with B = Z p (g, e) and C = Z HC (g, e), and the set of generators {c 1 , . . . , c m } = {Z k 1 1 · · · Z k N N | 0 ≤ k i < p}. The first condition that we need to verify is given in (a), so we are left to verify that Z HC (g, e) is generated as a Z HC,p (g, e) by {Z k 1 1 · · · Z k N N | 0 ≤ k i < p}. As explained after Lemma 2.4, in the case e = 0, we have z r = Z (r) . Thus from this lemma we obtain that Z HC (g) is generated as a Z p (g) G -module by {(Z (1) ) k 1 · · · (Z (N ) ) k N | 0 ≤ k i < p}. Since pr sends Z (r) to Z r by Lemma 4.4 we deduce the desired result. We set up some notation for a basis of Z(g, e). For (u, t, v) ∈ I F × I D × I E and w ∈ {0, 1, . . . , p − 1} N , we define

Then the ordered monomials
form a basis for Z(g, e).

4.5.
Restricted finite W -algebras. We move on to recall the definition of the restricted W -algebra U [p] (g, e). We write Z p (p) + for the ideal of Z p (p) generated by {x p − x [p] | x ∈ p}, so the restricted enveloping algebra of p is U [p] (p) = U(p)/U(p)Z p (p) + . Then the restricted W -algebra is defined as U [p] (g, e) := U(g, e)/(U(g, e) ∩ U(p)Z p (p) + ).
Since, the kernel of the restriction of the projection U(p) ։ U 0 (p) to U(g, e) is U(g, e) ∩ U(p)Z p (p) + , we can identify U [p] (g, e) with the image of U(g, e) in U [p] (p).
By [GT1,Theorem 8.4], we have that U(g, e) is free of rank p dim g e over Z p (g, e), and thus that dim U [p] (g, e) = p dim g e . We note that each of the elements in (4.15) lies in U(g, e) ∩ Z p (p) + , and we let Z p (g, e) + be the ideal of Z p (g, e) generated by these elements. By Lemma 4.6(a), we have that Z p (g, e) + is a maximal ideal of Z p (g, e), and it follows that Z p (g, e) + = U(g, e) ∩ Z p (p) + . By using the formulas given in (3.13), and a filtration argument we see that U(g, e)/U(g, e)Z p (g, e) + is spanned by the p-restricted monomials in the elements in (4.15). Hence, we see that U(g, e) ∩ U(p)Z p (p) + = U(g, e)Z p (g, e) + , and obtain the basis {F u D t E v + U(g, e)Z p (g, e) + | (u, t, v) ∈ I p } (4.18) of U [p] (g, e).

5.
Highest weight modules for Y n,l (σ) and U(g, e) For our proof of Theorem 1.1, we require some results about highest weight vectors in modules for Y n,l (σ) and U(g, e). In this section we cover the required material, with the key results being Lemmas 5.4 and 5.6. We continue to use the notation from Sections 3 and 4. 5.1. Torus actions. Before discussing highest weight theory we have to introduce the underlying torus actions.
Let T n be the maximal torus of GL n ( ) of diagonal matrices. We write {ε 1 , . . . , ε n } for the standard basis of the character group X * (T n ) of T n , i.e. ε i : T n → × is defined by ε i (diag(t 1 , . . . , t n )) = t i . The positive weights in X * (T n ) are X * for all i and a 1 > a n }. Now let T be the maximal torus of G of diagonal matrices, and let T e be the centralizer of e in T . We can describe T e explicitly in terms of certain cocharacters. Define τ 1 , . . . , τ n : × → T , where τ i (t) is the diagonal matrix with jth entry equal to t if row(j) = i and entry 1 otherwise. Then we have . From now on we use the above isomorphism to identify T e with T n . It is a straightforward to see that the basis element c (r) i,j of g e is a T n -weight vector with weight ε i − ε j .
We note that the adjoint action of T e on U(g) restricts to an adjoint action on U(g, e), so we have an action of T n on U(g, e). By inspection of the formula for D (r) i in (3.11), we see that it is fixed by T n . Similarly, by considering the formula for E (r) i given in [GT2,Section 4], we see that E (r) i is a T n -weight vector with weight ε i − ε i+1 ; and then deduce, using (3.2) that E (r) i,j has T n -weight ε i − ε j . Similarly, we see that F (r) i,j has T n -weight ε j − ε i . Further, we note that the action on T n on U(g, e) is filtered for the loop filtration, so there is an action of T n on gr U(g, e). Under the identification gr U(g, e) ∼ = U(g e ) given by S −η in (3.16), this action coincides with the natural action of T n ∼ = T e on U(g e ).
By considering the relations for Y n (σ) given in [BT,Theorem 4.15] and the definitions of E (r) i,j and F (r) i,j given in (3.2), we see that there is an action of T n on Y n (σ) by algebra automorphisms, such that D (r) i is fixed by T n , the weight of E (r) i,j is ε i − ε j , and the weight of F (r) i,j is ε j − ε i . Further, this action of T n is filtered for the loop filtration, and through the isomorphism ψ : U(c n (σ)) ∼ −→ gr Y n (σ) in (3.3) it corresponds to the natural action of T n on U(c n (σ)).
We note that the ideal I n,l is T n -stable, so that there is an induced action of T n on Y n,l (σ). From the description of the action of T n on Y n (σ) and on U(g, e) above, we see that the isomorphism φ : Y n,l (σ) ∼ −→ U(g, e) in (3.17) is T n -equivariant. 5.2. Highest weight modules for Y n,l (σ). For our proof of Theorem 1.1, we require some theory of highest weight modules for Y n,l (σ). We outline what we need below, much of which is a modular analogue of some results in [BK2,Chapter 6], though here we take a more elementary approach to some of the results we require. The key result in this subsection is Lemma 5.4, which tells us how the elementsḂ (rp) i act on highest weight vectors.
We recall that a PBW basis {Ḟ uḊtĖv | (u, t, v) ∈ I F × I D × I E } of Y n,l (σ) is given in (3.21). In the discussion below we also require the ordered sets J F , J D and J E , which are defined before (3.21), and used to fix the order in the PBW monomials. Since each F i,j is a T n -weight vector we see that the elements of the above PBW basis of Y n,l (σ) are also T n -weights. In order to define Verma modules for Y n,l (σ) we fix a = (a (r) We use this tuple to modify the basis given in (3.21) by settinġ Then we see that forms a basis for Y n,l (σ). Also we note that these basis elements are T n -weight vectors, and that the T n -weight ofḞ u (Ḋ − a) tĖv is the same as that ofḞ uḊtĖv . We define M(a) to be the set of monomials (5.2) for which t = 0 or v = 0, and write I(a) for the subspace of Y n,l (σ), which has these as a basis.
Lemma 5.1. Let a = (a (r) i | 1 ≤ i ≤ n, 1 ≤ r ≤ p i ) ∈ N , and define I(a) as above. Then: (a) any T n -weight vector in Y n,l (σ) with weight in X * + (T n ) lies in I(a); and (b) I(a) is a left ideal of Y n,l (σ).
Proof. For a monomialḞ u (Ḋ − a) tĖv to have a positive weight, it must have v = 0, and thus lies in I(a). From this we can deduce (a) as these monomials give a basis of Y n,l (σ).
For the proof of (b), we require another filtration of Y n,l (σ), known as the canonical filtration. First we recall that the canonical filtration is defined on Y n (σ) by placing E in filtered degree r, then we get the induced filtration on Y n,l (σ). We write gr ′ Y n (σ) and gr ′ Y n,l (σ) for the associated graded algebras for the canonical filtrations. As is remarked in [BT,§4.2], gr ′ Y n (σ) is commutative, and thus gr ′ Y n,l (σ) is also commutative.
Let X ′ =Ḟ u ′ (Ḋ − a) t ′Ė v ′ be in the basis given in (5.2) and X =Ḟ u (Ḋ − a) tĖv ∈ M(a). We write deg ′ (X ′ X) for the canonical degree of X ′ X and proceed to prove that X ′ X ∈ I(a) by induction on deg ′ (X ′ X). It is clear that (b) will follow immediately from this.
For our fixed value of deg ′ (X ′ X), we see that we can reduce to the case where u = 0, by writingḞ u ′Ḋ t ′Ė v ′Ḟ u asḞ u ′ +uḊt ′Ė v ′ plus a sum of the PBW monomials in the basis given in (5.2) of strictly lower canonical degree; here we use that gr ′ Y n,l (σ) is commutative. Thus we assume that X = (Ḋ − a) tĖv . We define the length ℓ(X ′ ) to be the sum of the entries of all three tuples u ′ , t ′ , v ′ , and now work by induction on ℓ(X ′ ), under the assumption that X is of the form (Ḋ − a) tĖv ∈ M(a).
Next suppose that v ′ = 0 and t ′ = 0. Let (i 0 , r 0 ) be largest with respect to our fixed order on J D such that t all commute with each other. Since ℓ(Ḟ u ′ (Ḋ − a) t ′ −s ) < ℓ(X ′ ) and (Ḋ − a) t+sĖv ∈ M(a) is of the required form, we conclude that X ′ X ∈ I(a) by induction on ℓ(X ′ ).
Last we consider the case v ′ = 0. Let (i 0 , j 0 , r 0 ) be largest with respect to our fixed order on J E such that v i,j for (i, j, r) up to (i 0 , j 0 , r 0 ) in our fixed order of J E , andĖ v ≥ is the remaining submonomial. We have is of the required form. So we conclude that this term lies in I(a) by induction on ℓ(X ′ ). We are left to consider the term Y As the associated graded algebra of Y n,l (σ) for the canonical filtration is commutative, we have that deg Ė v ≥ has positive T n -weight, so lies in I(a) by (a). Therefore, it can be rewritten as a linear combination of monomials in M(a). LetX be a monomial from M(a) occurring in this sum. Then we know that (Ḟ u ′ (Ḋ − a) t ′Ė v ′ −s )X ∈ I(a), by induction on deg ′ (X ′ X). Putting this all together we obtain that X ′ X ∈ I(a) as required, which completes the double induction. i,j annihilates 1 + I(a) for all (i, j, r) ∈ J E . In fact, something much stronger is true.
Lemma 5.2. The following elements of Y n,l (σ) annihilate 1 + I(a) ∈ M(a): i,j with a positive weight for all 1 ≤ i < j ≤ n, r > s i,j , part (a) follows from Lemma 5.1(a).
Using (3.27) we calculate thaṫ By Corollary 3.6 we know thatṪ (r) i,i = 0 for r > p i . Also by (a), we know that eachĖ (c) k,j on the righthand side of the above equation annihilates 1 + I(a). Hence, we deduce thatḊ (r) i also annihilates 1 + I(a) for r > p i .
Let M be a Y n,l (σ)-module and let v + ∈ M. We say that v + is a highest weight vector of weight a if I(a) annihilates v + . We say that M is a highest weight module of weight a if M is generated by some highest weight vector of weight a. The Verma modules {M(a) | a = (a (r) i ) 1≤r≤p i 1≤i≤n ∈ N } are defined to be the universal highest weight modules. Thus if v + ∈ M is a highest weight vector of weight a, then there is a unique map M(a) → M sending 1 + I(a) to v + .
It is helpful for us to relabel the Verma modules, following the approach of [BK2, Section 6.1]. Suppose we have a highest weight vector v + with weight a in some Y n,l (σ)-module. Then by Lemma 5.2 we know that By factorising and introducing a shift, we have that These are the formulas given in [BK2,3)].
We let A be the π-tableau with entries {a i,j | j = 1, . . . , p i } on the ith row, and note that A is only defined up to row equivalence. We denote the row equivalence class of A by A, and from now on we refer to A as the weight of v + , rather than a. This allows an alternative parametrization of the Verma modules, where we write M(A) instead of M(a); we use the notation v A,+ for the highest weight vector of M(A).
We define Y n,l (σ) 0 to be the (commutative) subalgebra of Y n,l (σ) generated by {Ḋ (r) i | 1 ≤ i ≤ n, 0 < r ≤ p i } and note that Y n,l (σ) 0 is in fact a polynomials algebra on these generators. The next lemma is a direct consequence of the Nullstellensatz, but we record it for convenience of reference.
(5.4) Therefore, we see thatŻ r acts as e r (a i,j | 1 ≤ i ≤ n, 1 ≤ j ≤ p i ), where we recall that e r denotes the rth elementary symmetric polynomial. Now we want to calculate the scalar by which B (rp) i acts on the highest weight vector v A,+ .
Lemma 5.4. Let A ∈ Tab(π), let 1 ≤ k ≤ n and 1 ≤ r ≤ p i . Set s := s(r) = r + ⌊ r−1 p−1 ⌋ and let D i,r be the set of all sequences d = (d 0 , d 1 , d 2 , . . . , d s ) of non-negative integers such that j≥0 d j = p i and rp = j≥1 d j (jp − j + 1). Theṅ by a unitriangular change of variables, such that We explain some of the steps in the above calculation. The first equality just uses the definition ofḂ i (u). To go from the first line to the second we use the definition of the action ofḊ i (u) in (5.3). Then to go from the third line to the fourth we use (2.2). Also we have that p−1 j=0 (u − j) p i = (u p − u) p i by (2.1), so we obtain The action ofḂ on v + is determined by the coefficient of u −rp in the above expression. Let d ∈ D i,r and let be the multinomial coefficient. By choosing the summand 1 in d 0 of the multiplicands in (5.6) and choosing a summand (a p i,k − a i,k )u pj−j+1 in d j of the multiplicands for each 1 ≤ j ≤ r, we obtain a term which contributes to the coefficient of u −rp . The contribution from all such terms will be a multiple of e j≥1 d j (a p k,1 − a k,1 , . . . , a p k,p k − a k,p k ) and a straightforward counting argument shows that the coefficient on We deduce that each d ∈ D i,r contributes to the coefficient of u −pr in (5.6). We note that our definition of s is chosen precisely so that all sequences d = (d 0 , d 1 , d 2 , . . . ) of non-negative integers such that rp = j≥1 d j (jp − j + 1), have d i = 0 for i > s. So the considerations above give all coefficients of u −rp . Therefore, the coefficient of u −rp in (5.6) is the sum over all d ∈ D i,r of the terms given in (5.7), which proves the first claim of the lemma. Now we observe that (p i − r, r, 0, . . . , 0) ∈ D i,r is the unique element which maximises j≥1 d j . It is easily verified that (p i −r, r, 0, . . . , 0) ∈ D i,r . To see that j≥1 d j is maximised we observe that for d ∈ D i,r we have pr = j≥1 d j (j(p − 1) + 1) ≥ p j≥1 d j . We now show that this is the unique element of D i,r with j≥1 d j = r. Let d ∈ D i,r . From the equation j≥1 d j (j − 1) = p j≥1 d j j − pr we deduce that p is a factor of j≥1 d j (j − 1), say mp = j≥1 d j (j − 1) = j≥2 d j (j − 1). Substituting back into rp = j≥1 d j (jp − j + 1) we have Finally we arrive at r = m(p − 1) + j≥1 d j , and we conclude that if r = j≥1 d j then m = 0, which forces d 2 = d 3 = · · · = d s = 0. Using j≥0 d j = p i we deduce that d = (p i − r, r, 0, . . . , 0). We have now proven that claim that (p i − r, r, 0, . . . , 0) uniquely maximises j≥1 d j in D i,r .
Since ( j≥1 d j )!/( j≥1 d j !) = 1 for d = (p i − r, r, 0, . . . , 0) it follows that for i fixed there is a upper unitriangular matrix C = (c s,r ) 1≤s,r≤p i such thaṫ If we take C −1 = (c s,r ) 1≤s,r≤n and defineB To finish the proof, we are left to show that if p > r, thenB , which will follow from showing that D i,r = {(p i − r, r, 0, . . . , 0)} under the assumption that p > r. So suppose that p > r and let d ∈ D i,r . From equation (5.8) we have Since j≥1 d j > 0 we have p( j≥1 d j + mp) > mp 2 = m(p − 1)p + mp. If m > 0, then the hypothesis p > r implies that m(p − 1) ≥ r and combining with the previous inequality we arrive at p( j≥1 d j + mp) > rp + mp, which contradicts (5.9). We conclude that m = 0 and, following the observations made after (5.8), we deduce that d = (p i − r, r, 0, . . . , 0). This completes the proof.
Our next corollary implies that certain elements of Z(Y n,l (σ)) are determined by their action on highest weight vectors. We need to set up some notation for its statement and proof.
Let Y n,l (σ) 0 be the subalgebra of Y n,l (σ) of all elements fixed by the action of T n . The PBW basis (3.21) is T n -stable, and Y n,l (σ) 0 has a basis consisting of those monomials such that (i,j,r) The subspace Y n,l (σ) 0,♯ of Y n,l (σ) 0 spanned by monomials with u = 0 is equal to the subspace spanned by monomials with v = 0, and thus this subspace is an ideal. Further, we have a direct sum decomposition Y n,l (σ) 0 = Y n,l (σ) 0 ⊕ Y n,l (σ) 0,♯ . We define ζ : Y n,l (σ) 0 → Y n,l (σ) 0 , to be the projection along this direct sum decomposition.
Recall the basis for Z(Y n,l (σ)) given in (4.11), and define Z(Y n,l (σ)) 0 to be the subspace of of Z(Y n,l (σ)) spanned by the monomials with u = v = 0. Clearly Z(Y n,l (σ)) 0 ⊆ Y n,l (σ) 0 . We write Z p (Y n,l (σ)) 0 for the subalgebra of Z(Y n,l (σ)) which is generated by {B it is a polynomial algebra on these generators thanks to Theorem 4.2(b) and Lemma 5.4. We note that Z(Y n,l (σ)) 0 is not a subalgebra of Z(Y n,l (σ)) but nonetheless, Z(Y n,l (σ)) 0 is a free Z p (Y n,l (σ)) 0 -module with basis given by the restricted monomials given in (4.7).
Proof. Thanks to Corollary 4.3 and Lemma 5.4 we know that Z(Y n,l (σ)) 0 has a basis consisting of ordered monomials be the polynomial ring in variables x i,j . We define a linear map ω : Z(Y n,l (σ)) 0 → R by setting . , x p i,p i − x i,p i ) and then extending multiplicatively.
Thanks to (5.4) and Lemma 5.4 we know that the action of any element of Z(Y n,l (σ)) 0 on the Verma module M(A) is given by the composition p A • ω where p A : R → is the homomorphism determined by x i,j → a i,j . In other words, for z ∈ Z(Y n,l (σ)) 0 and A ∈ Tab(π) we have zv A,+ = (p A • ω(z))v A,+ . Since we have zv A,+ = ζ(z)v A,+ for every z ∈ Y n,l (σ) 0 , and A∈Tab(π) ker p A = 0, we conclude by Lemma 5.3 that ker ζ| Z(Y n,l (σ)) 0 = ker ω. The rest of the proof is devoted to showing that ker ω = 0, which implies both (a) and (b).
Let S := ω(Z(Y n,l (σ)) 0 ) ⊆ R and S p := ω(Z p (Y n,l (σ)) 0 ). In order to show that ω is injective we show that it sends the basis of Z(Y n,l (σ)) 0 given in (5.10) to a basis of S. To this end we show that S p is a polynomial ring generated by {ω(B and that an S p -basis is given by We place a filtration on R with every x i,j in degree 1, and we have induced filtrations on S and S p . We identify the associated graded space of S with a subspace of R and we see that gr r e r (x i,j | 1 ≤ i ≤ n, 0 < j ≤ p i ) = e r (x i,j | 1 ≤ i ≤ n, 0 < j ≤ p i ) (as all the monomials lie in filtered degree r), whereas gr pr e r (x p i,1 − x i,1 , . . . , x p i,p i − x i,p i ) = e r (x p i,1 , . . . , x p i,p i ) = e r (x i,1 , . . . , x i,p i ) p ; in particular, we observe that gr S is in fact a subalgebra of R. Using Lemma 2.3 it suffices to show that the p-restricted monomials in {e r (x i,j | 1 ≤ i ≤ n, 0 < j ≤ p i ) | r = 1, . . . , N} form a basis for gr S over gr S p .
At this stage in the proof, we restrict to the case where p = (1 N ), because the other cases follow from this case, whilst the notation in this case is more transparent. Since n = N and p 1 = · · · = p n = 1 we use the notation x i instead of x i,1 for i = 1, . . . , N. and write e 1 , . . . , e N for the elementary symmetric polynomials in x 1 , . . . , x N . The subalgebra gr S of R is generated by {x p i | i = 1, . . . , N} ∪ {e r | r = 1, . . . , N}, and the subalgebra gr S p is generated by {x p i | i = 1, . . . , N}. The restricted monomials e w 1 1 · · · e w N N with w ∈ {0, . . . , p − 1} N clearly generate gr S over R p so it suffices to show that they are linearly independent. In turn it is enough to prove that e w 1 1 · · · e w N N are linearly independent over the fraction field of R p .
To achieve this we apply some field theory that can be found in [Bo, Chapter V]. We write K = (x 1 , . . . , x N ) for the fraction field of R, and note that the fraction field of R p is K p . Next we observe that {e 1 , . . . , e N } form a separating transcendence basis of K over in the sense of [Bo,Definition V.16.7.1]. Therefore, by [Bo,Theorem V.16.7.5], we have that {de 1 , . . . , de N } form a K-basis of the space Ω (K) of -derivations of K. Since any D ∈ Ω (K) annihilates K p , we have that Ω K p (K) = Ω (K), so that {de 1 , . . . , de N } is a Kbasis of Ω K p (K). Then we can apply [Bo,Theorem V.13.2.1] to deduce that {e 1 , . . . , e N } is a p-basis of K over K p , in the sense of [Bo,Definition V.13.1.1]. By definition of a p-basis we have that the p-restricted monomials in {e 1 , . . . , e N } are a basis of K over K p , and thus in particular are linear independent as required.
5.3. Highest weight modules for U(g, e). Through the isomorphism φ : Y n,l (σ) → U(g, e), which we know is T n -equivariant, we have a notion of highest weight modules for U(g, e). We use the notation and terminology introduced in §5.2 also for U(g, e). We are mainly interested in considering the restriction of highest weight U(h)-modules to U(g, e), and our main result is Lemma 5.6. We move on to show that elements of Z(g, e) 0 are determined by their action on highest weight vectors in Corollary 5.7.
We recall the good grading g = i∈Z g(i) from (3.5) and the notation h := g(0) and p = i≥0 g(i) from (3.6). We recall that the heights of the columns in π are q 1 , . . . , q l , and so h ∼ = gl q 1 ( ) ⊕ · · · ⊕ gl q l ( ). We let b h be the Borel subalgebra of h with basis {e i,j | col(i) = col(j), row(i) ≤ row(j)}, which is the direct sum of the Borel subalgebras of upper triangular matrices in each of the gl q i ( ).
For A ∈ Tab (π) we define the weight λ A ∈ t * by We let which is a "shifted choice of ρ for the Borel subalgebra b h of h". Then we define where we recall that η is defined in (3.10).
We define A to be the 1-dimensional t-module on which t acts via λ A − ρ, and view it also as a module for b h on which the nilradical acts trivially. Then we define the Verma module M h (A) = U(h) ⊗ U (b h ) A for U(h), and we write m A := 1 ⊗ 1 A for the highest weight vector. We may view M h (A) as a U(p)-module on which the nilradical i>0 g(i) of p acts trivially, and then restrict it to U(g, e) ⊆ U(p). We write M h (A) for the restriction of M h (A) to U(g, e), and write m A for m A viewed as an element of M h (A).
The following lemma shows that m A is a highest weight vector in M h (A) with weight A, and further gives the action of ξ p (gr ′ D (r) i ) on m A . We note that a proof of (b) could be given based on the last paragraph of the proof of [BK2,Theorem 7.9]; however we give a more direct approach here, which can also be used to prove (c).
Lemma 5.6. Let A ∈ Tab (π) and let M h (A) and m A be as defined above. Then i m A = e r (a i,1 + (i − 1), . . . , a i,p i + (i − 1)) for all (i, r) ∈ J D ; and (c) ξ p (gr ′ D (r) i )m A = e r (a p i,1 − a i,1 , . . . , a p i,p i − a i,p i ) for all (i, r) ∈ J D . Proof. First we note that M h (A) is isomorphic as a U(p)-module to U(p)/I p (A), where I p (A) is the left ideal of U(p) generated by {e i,j − δ i,j (λ A − ρ)(e i,i ) | col(i) = col(j), row(i) ≤ row(j)} ∪ {e i,j | col(i) > col(j)}. Next we observe that T e ∼ = T n acts on p by the adjoint action, and this induces an action of T n on I p (A). Using the same proof as Lemma 5.1(a) we see that any element of U(p) with a positive T n weight annihilates m A . Now part (a) follows as E (r) i ∈ U(g, e) ⊆ U(p) has positive T n -weight. We move on to prove (b), where we use the explicit formula for D (r) i given in (3.11). We set up some notation to simplify the proof. The formula (3.11) is given as a sum of terms indexed by integers 1 ≤ i 1 , ..., i s , j 1 , ..., j s ≤ N subject to conditions (a)-(f). We write i = (i 1 , ..., i s ), j = (j 1 , ..., j s ) andẽ i,j for the summand corresponding to i, j.
First we observe that if s < r, then condition (a) ensures that col(j k ) > col(i k ) for some k, which implies thatẽ i,j kills m A . Now we consider sequences i, j with s = r. Then we have col(i k ) = col(j k ) for all k, so thatẽ i,j ∈ U(h). Suppose that i k = j k for all k. Using conditions (d), (e) and (f) we see that there is some k such that i k < j k , and we choose the maximal such k. We certainly have thatẽ i k ,j k = e i k ,j k kills m A . Further by condition (c) and (e), we have col(i m ) > col(i k ) for all m > k, so that e i k ,j k commutes withẽ im,jm . We deduceẽ i,j kills m A .
To prove (c), we can argue exactly as above and use the formula for ξ p (D We have that λ A (e p i k ,i k − e i k ,i k ) = a p i,k − a i,k whilstρ(e p i k ,i k − e i k ,i k ) =ρ(e i k ,i k ) p −ρ(e i k ,i k ) = 0. Hence, ξ p (gr ′ D (r) i ) acts on m A via e r (a p i,1 − a i,1 , . . . , a p i,p i − a i,p i ) as required. To end the subsection, we record a version of Corollary 5.5(b) for the algebra U(g, e). We define Z(g, e) 0 to be the subspace of Z(g, e) which is spanned by the PBW monomials appearing in (4.17) such that u = v = 0.
Proof. It follows from Lemma 4.4, along with (5.4), that Z r ∈ Z HC (g, e) acts on m A via the rth elementary symmetric function in {a i,j | 1 ≤ i ≤ n, 1 ≤ j ≤ p i }. Also by Lemma 5.6(c), we know that ξ p (gr ′ D (r) i ) ∈ Z p (g, e) acts on m A by the rth elementary symmetric polynomial in {a p i,1 − a i,1 , . . . , a p i,p i − a i,p i }. Therefore, we may apply precisely the same argument as for Corollary 5.5(b) to complete the proof.

The isomorphism of restricted versions
The main goal of this section is to prove Theorem 1.1. We continue to use that notation introduced in Sections 3-5.
(a) φ((Ė (r) i,j ) p ) ∈ Z p (g, e) + Z(g, e). (b) φ((Ḟ (s) i,j ) p ) ∈ Z p (g, e) + Z(g, e). i ) ∈ Z p (g, e) + Z(g, e). Proof. Recall from §4.5 that ξ p (gr ′ E (r) i,j p ), ξ p (gr ′ F (r) i,j p ) ∈ Z p (g, e) + . The basis elements of Z(g, e) in (4.17) with nonzero weight have u = 0 or v = 0, so that these elements lie in Z p (g, e) + Z(g, e). Now (a) and (b) follow from the facts thatĖ (r) i,j has T n -weight p(ε i − ε j ) andḞ (r) i,j has T n -weight p(ε j − ε i ) along with the fact that φ is T n -equivariant. By Lemmas 5.4 and 5.6 we know that φ(B i ) is a span of elements with u = v = 0 modulo terms lying in Z p (g, e) + Z(g, e). We may now apply Corollary 5.7 to deduce that φ( B (rp) i ) − ξ p (gr ′ D (r) i ) ∈ Z p (g, e) + Z(g, e) as required.
We are now ready to deduce our main theorem.