# Restricted shifted Yangians and restricted finite -algebras

## Abstract

We study the truncated shifted Yangian over an algebraically closed field of characteristic which is known to be isomorphic to the finite , -algebra associated to a corresponding nilpotent element We obtain an explicit description of the centre of . showing that it is generated by its Harish-Chandra centre and its , We define -centre. to be the quotient of by the ideal generated by the kernel of trivial character of its Our main theorem states that -centre. is isomorphic to the restricted finite -algebra As a consequence we obtain an explicit presentation of this restricted . -algebra.

## 1. Introduction

Let be a reductive algebraic group over an algebraically closed field of characteristic with Lie algebra , The centre of . admits a large * -centre* which is isomorphic to the coordinate ring of (the Frobenius twist of) -equivariantly For . the reduced enveloping algebra is defined to be the quotient of , by the ideal generated by the maximal ideal of corresponding to The most important aspects of the representation theory of . are understood by studying and the early work of KacтАУWeisfeiler, in -modules,Reference KW, shows that it suffices to consider the case nilpotent, meaning identifies with a nilpotent element under some choice of isomorphism -equivariant (we assume the standard hypotheses). We refer to Reference Ja for a survey of this theory up to 2004, and also to Reference BM for major developments based on deep connections with the geometry of Springer fibres. In Reference Pr1 Premet made a significant breakthrough: he showed that any such is Morita equivalent to a certain algebra now known as the ,*restricted finite -algebra*.

In this paper, we consider the case so that , Our main theorem provides an explicit presentation for the restricted finite . -algebra This is achieved by exhibiting an isomorphism with a restricted version of a truncated shifted Yangian, as stated in Theorem .1.1 below. In future work we will employ this presentation in studying the representation theory of The fundamental advantage of studying . via these Yangians is that the rank of the Yangian associated to -modules corresponds to the number of Jordan blocks of the nilpotent For example, the -character. with a two-block nilpotent -modules are described via a Yangian that is computationally accessible. -character

Before we proceed, we recall some relevant history. In Reference Pr1, Section 4 Premet constructed finite 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 -algebrasReference Pr3 and Reference Lo for surveys on this theory.

In Reference BK1, BrundanтАУKleshchev made a breakthrough by providing a presentation of the complex finite for the case -algebra 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 in -algebrasReference BK2.

Building on PremetтАЩs seminal work using the method of modular reduction of finite first considered in -algebras,Reference Pr2 and exploited further in Reference Pr4, the authors developed a direct approach to theory of finite -algebras over in Reference GT1. Very briefly, for a choice of nilpotent corresponding to the algebra , is a filtered deformation of a good transverse slice to the coadjoint orbit Further, . admits a -centre isomorphic to the coordinate algebra of (the Frobenius twist of) Then the restricted . -algebra is the quotient of by the ideal generated by the ideal of corresponding to .

In joint work with Brundan Reference BT the second author developed the theory of shifted Yangians over One of the key features which differs from characteristic zero is the existence of a large central subalgebra . called the ,* -centre*, which is constructed using some very natural power series formulas.

In subsequent work Reference GT2, the authors showed that BrundanтАУKleshchevтАЩs isomorphism descends to positive characteristic. To explain this, we require a little notation, and from now on we take To each nilpotent element . with Jordan type we may associate a choice of shift matrix , and thus a shifted Yangian , which is a subalgebra of the Yangian , The beautiful formulas introduced in .Reference BK1 lead to a surjective algebra homomorphism Unsurprisingly the kernel of . has the same description as in characteristic zero, and so there is an isomorphism

where is the truncated shifted Yangian of level first defined in characteristic zero in ,Reference BK1, Section 6.

Making use of the explicit presentation of obtained through the isomorphism it was proved in ,Reference GT2 that every of minimal dimension is parabolically induced. This result is a modular analogue of M┼УglinтАЩs famous theorem on completely prime primitive ideals, see -moduleReference M┼У, and some of our methods adapt those in the proof given by Brundan in Reference Br.

In this paper we define the -centre of to be the image of under the quotient map This leads to a .*restricted truncated shifted Yangian* by taking the quotient of by the ideal generated by the generators of .

We emphasise here that the origin of is totally distinct from the construction of Nevertheless, our main theorem states that the isomorphism . factors through the restricted quotients.

Since is defined by generators and relations, the above theorem provides an explicit presentation for .

The main ingredients of the proof are a detailed study of the centres of and together with an analysis of highest weight modules for both algebras. We emphasise that Theorems 4.2 and 4.7 are significant results in their own right, describing the structures of the centres of and 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 ┬з2.6, where we consider the centre of the universal enveloping algebra of the centralizer of In particular, we use .Reference BB to give precise formulas for the generators of sharpening the main results of ,Reference To. Also in ┬з2.7, we observe that is isomorphic to a truncated shifted current Lie algebra, which is helpful later in the sequel.

In Section 3, we recall the structural features of the shifted Yangian and the finite -algebra drawing on ,Reference BT, Reference GT1 and Reference 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 map lying at the core of our main theorem. In ┬з3.3 we introduce the truncation at level and use the shifted current algebra to simplify the proof of the PBW theorem for , 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 and These integral forms, introduced in ┬з .3.4, are useful tools in some of our later proofs as they allow us to reduce modulo 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 and Our results are perfect analogues of VeldkampтАЩs classical description of the centre . of see for example ;Reference BG, Theorem 3.5 and the references there. We give definitions of the Harish-Chandra centres of and these are denoted by ; and and they are defined so that they тАЬliftтАЭ the centre in characteristic zero. The , -centres and of and are also introduced here. In Theorem 4.2 we give a detailed description of the centre of in particular showing that is generated by , and The next significant result is Theorem .4.7 in which we deduce an analogous result for the centre of We mention that in recent work, ShuтАУZeng have stated a more general result about the centre of modular finite . associated to arbitrary connected reductive groups, under certain hypotheses, see -algebrasReference SZ, Theorem 1. The more detailed description we give here is a necessary step in the proof of our main theorem, and will play a role in future work. A precise description of a set of generators for is given in ┬з4.4, and this is important in the sequel. We also draw attention to Corollary 4.5 which shows that preserves the Harish-Chandra centres. In ┬з4.3 and ┬з4.5 we discuss the restricted quotients and and their PBW bases.

In Section 5 we develop some highest weight theory for and study the action of 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 ┬з5.1. The key results after that are Lemmas 5.4 and 5.6(c) which describe how the generators of the -centres and 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 and .

Finally, in Section 6, we combine our results to observe that the generators of act on highest weight vectors in precisely the same manner as the generators for We use results from Section .5 to show that the ideal of generated by the kernel of the trivial character of is mapped to the ideal of generated by the kernel of the trivial character of and the main theorem follows quickly. We remark that our proof does not show that , and so it remains an interesting open problem to decide if these centres really do line up. ,

## 2. Preliminaries and recollection

Throughout this paper, let be a prime number, let be the field of elements and let be an algebraically closed field of characteristic .

### 2.1. A useful identity

We require a standard identity in the polynomial ring for the proof of Lemma 5.4, and we recall it here. Each satisfies so for an indeterminate , we deduce that ,

in More generally, for any . we have the following equality in ,

Observe that for the coefficient of in the left hand side of Equation 2.1 is where , denotes the elementary symmetric polynomial in indeterminates th It follows that . in for this gives a short alternative proof of ;Reference 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 readerтАЩs convenience. The first lemma is well-known. Let be a commutative and -algebra subalgebras. If is generated by then it follows that there is a surjective homomorphism .

The next result concerns free modules for a commutative -algebra It is well-known that a surjective endomorphism of a finitely generated . is an isomorphism; this can be proved using NakayamaтАЩs lemma, see for example -moduleReference Ma, Theorem 2.4.

The final result in this subsection is required several times in the sequel, and included for convenience of reference. Let be a non-negatively filtered (not necessarily commutative) with filtered pieces -algebra for Also let . be a non-negatively filtered with filtered pieces -module for We write . for the associated graded algebra of and for the associated graded module of If . then the notation is used throughout the paper. The following lemma can be proved with a standard filtration argument.

### 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 be a linear algebraic group over and let , be the Lie algebra of We write . for the universal enveloping algebra of and , for the centre of We denote the . filtered piece of th in the standard PBW filtration by The associated graded algebra . is identified with the symmetric algebra of , .

The adjoint action of on extends to an action on Also . has adjoint actions of and We use the standard notation . and for these actions, where , and , or For a closed subgroup . of and subspace -stable of or of we write , for the invariants of in and for the invariants of in Given . we write , for the centralizer of in and we write , for the centralizer of in .

We have that is a restricted Lie algebra and we write for the map. The -power of -centre is the subalgebra of generated by There is a . isomorphism -equivariant determined by , for here ; denotes the Frobenius twist of .

### 2.4. Combinatorial notation

We require various pieces of combinatorial notation, which we set out below.

By a *composition* we simply mean a sequence where , and only finitely many are nonzero. When is a composition, and for all we write , Given a composition . we define , and say that is a composition of Also we define . In this paper a composition . is called a *partition* if for all Given two compositions . and we say that , is a *subcomposition* of if for all and in this case we write , .

Let By a .*shift matrix* of size we mean a matrix with entries in such that whenever or , We note that this implies that . for all and that , is completely determined by the entries and for .

Let and let be a composition of such that for some we have and let , We define the pyramid . to be the diagram made up of boxes stacked in columns of heights We let . be the partition of giving the row lengths of from top to bottom; note that the number is often referred to as *the level*. The boxes in are labelled with along rows from left to right and from top to bottom. The columns of are labelled from left to right and the rows are labelled from top to bottom. The box in containing is referred to as the box, and we write th and for the row and column of the box respectively. We define the shift matrix th from by setting to be the left indentation of the row of th relative to the row, and th to be the right indentation of the row of th relative to the row, for th .

As an example we consider The pyramid is .

Then we obtain the partition and the shift matrix ,

Evidently the data encoded in the composition is equivalent to the data given by the pyramid We have explained how to construct a shift matrix and a level . from a pyramid. To complete the picture we observe that we can build the pyramid from knowledge of by starting with a bottom row of length , and indenting the higher rows according to ,