One dimensional representations of finite W-algebras, Dirac reduction and the orbit method
OONE DIMENSIONAL REPRESENTATIONS OF FINITE W -ALGEBRAS,DIRAC REDUCTION AND THE ORBIT METHOD LEWIS TOPLEY
Dedicated to my teacher Sasha Premet, with admiration and gratitude.
Abstract.
In this paper we study the variety of one dimensional representations of the finite W -algebra for classical Lie algebras, giving a precise description of the dimensions of the irreduciblecomponents. We apply this to prove a conjecture of Losev describing the image of his orbit methodmap. In order to do so we first establish new Yangian-type presentations of semiclassical limits ofthe W -algebras attached to distinguished nilpotent elements in classical Lie algebras, using Diracreduction. Introduction
Let G be a complex connected reductive algebraic group with Lie algebra g , nilpotentcone N p g q . Identify g with g ˚ by a choice of non-degenerate G -invariant trace form on g . Theprimitive spectrum Prim U p g q is the set of all primitive ideals of the enveloping algebra, equippedwith the Jacobson topology. These ideals are classically studied via their invariants, and the mostimportant of these are the associated variety and Goldie rank . The associated variety VA p I q isdefined to be the vanishing locus in g of the associated graded ideal gr I Ď S p g q “ C r g s withrespect to the PBW filtration. A celebrated theorem of Kostant states that N p g q is the vanishinglocus of the positive degree invariant polynomials S p g q G ` whilst Joseph’s irreducibility theoremstates that VA p I q is irreducible (see [Ja04] for a detailed survey). Together with Dixmier’s lemma,these results show that the associated variety is the closure of a nilpotent orbit. The Goldie rankis defined to be the uniform dimension of the primitive quotient U p g q{ I .It is natural to consider the decomposition Prim U p g q “ Ť O Prim O U p g q , were the union is takenover all nilpotent G -orbits O and Prim O U p g q “ t I P Prim U p g q | VA p I q “ O u . Now fix an orbit O Ď N p g q and e P O , and let U p g , e q denote the finite W -algebra, first associated to p g , e q byPremet [Pr02]. The reductive part of the centraliser G e p q acts naturally on U p g , e q by algebra au-tomorphisms, and this induces an action of the component group Γ “ G e p q{ G e p q ˝ on the categoryof finite dimensional modules. Losev famously gave a new construction of U p g , e q via deformationquantization [Lo10a] and used this to show that Prim O U p g q is in bijection with U p g , e q -mod f . d . { Γ[Lo11]. The one dimensional representations of U p g , e q play an especially important role here fortwo reasons: on one hand the images under Skryabin’s equivalence are all completely prime, andtherefore play a key role in Joseph’s theory of Goldie rank polynomials [Lo15], and on the otherhand they classify quantizations of G -equivariant coverings of O [Lo10b]. The above narrative leads us to consider the affine scheme E p g , e q : “ Spec U p g , e q ab associatedto the maximal abelian quotient. By Hilbert’s nullstellensatz the closed points classify the one a r X i v : . [ m a t h . R T ] F e b dimensional representations of U p g , e q . The work of Losev and Premet [Lo10a, Pr14] shows that E p g , e q is nonempty and in [Pr10, PT14] the first steps were made towards a full description of E p g , e q .Recall that the sheets of g are the maximal irreducible subsets consisting of orbits of constantdimension. They are classified via the theory of decomposition classes which, in turn, are classifiedby the Lusztig–Spaltenstein induction data. One of the main themes of [PT14], which we buildupon in this paper, is the interplay between the sheets of g and the structure of E p g , e q . In the casewhere g is classical (i.e. a simple Lie algebra of type A, B, C or D ) we described a combinatorialprocedure for describing the sheets of g containing a given orbit O , and we named it the Kempken–Spaltenstein (KS) algorithm . This procedure was one of the key tools for [PT14, Theorem 1], whichstates that for g classical the variety E p g , e q is an affine space if and only if e lies in a unique sheetof g . The first goal of this paper is to elucidate the structure of E p g , e q when g is classical and e is singular , i.e. lies in multiple sheets.Once again let G be connected and reductive and let S , ..., S l be the set of all sheets containing O P N p g q{ G . If e ` g f denotes the Slodowy slice to O at e then we define the Katsylo variety e ` X : “ p e ` g f q X l ď i “ S i . (1.1)In [Ka82] Katsylo used this variety to construct a geometric quotient of the variety Ť li “ S i . Perhapsthe first indication that e ` X should influence the representation theory of U p g , e q appeared in[Pr10]. Premet used reduction modulo p to show that there is a surjective map on the sets ofirreducible components Comp E p g , e q Ý (cid:16) Comp p e ` X q (1.2)which restricts to a dimension preserving bijection on some subset of Comp E p g , e q . This is ourfirst main result. Theorem 1.1.
When g is a simple Lie algebra of classical type, the map (1.2) is a dimensionpreserving bijection. The dimensions of the irreducible components of e ` X can be calculated from the KS algorithm,which depends only on the partition associated to e ; see Proposition 6.2(2) and Proposition 8.2.Thus Theorem 1.1 provides an effective method for computing dimensions of all components of E p g , e q . We note that these dimensions were calculated in low ranks in [BG18]. In [Lo16] Losev demonstrated that for every conic symplectic singularity there is an initialobject in the category of filtered quantizations of Poisson deformations (see also [ACET20]). Wecall this the universal quantization . Using this result, he then showed that every coadjoint orbituniquely gives rise to a quantization of the affinization of a certain cover of a nilpotent orbit,and that each such quantization give rise to a completely prime primitive ideal. Thus we have amap J : g ˚ { G Ñ Prim U p g q , which is known to be an embedding whenever g is classical [Lo16,Theorem 5.3]. The search for such a map is motivated by the orbit method of Kostant and Kirillov,and we will refer to the map as Losev’s orbit method map . An introduction to the classical theoryof the orbit method can be found in [Vo94].
It is important to understand and characterise the primitive ideals appearing in the imageof the orbit method map for g . Losev has conjectured that they are precisely the ideals oneobtains by applying Skryabin’s equivalence to the one dimensional representations of the W -algebras associated to g , and taking their annihilators. In the final Section of this paper we deducehis conjecture from Theorem 1.1. Theorem 1.2.
For g classical, the image of J consists of primitive ideals obtained from onedimensional representations of W -algebras under Skryabin’s equivalence. The main technique appearing in the proof is the application of quantum Hamiltonian reductionto the universal quantizations of affinizations of nilpotent orbit covers.
There are several new tools involved in the proof of Theorem 1.1, and we now brieflydescribe the most important ones. One of the basic ideas comes from deformation theory. Thefinite W -algebra is a filtered quantization of the transverse Poisson structure on the Slodowy slice C r e ` g f s and so there are two natural degenerations associated to E p g , e q . On the one hand, wemay degenerate U p g , e q to the classical finite W -algebra S p g , e q – C r e ` g f s and then abelianise,which leads to the spectrum of the maximal Poisson abelian quotient F p g , e q : “ Spec S p g , e q ab .On the other hand we may abelianise and then degenerate, which leads us to the asymptotic coneof E p g , e q , denoted C E p g , e q : “ Spec p gr U p g , e q ab q .One of the general results of this paper states that there is a closed immersion of schemesinducing a bijection on closed points, which we prove using reduction to prime characteristic C E p g , e q ã ÝÑ F p g , e q . (1.3)Furthermore by considering the rank strata and symplectic leaves of the Poisson structure of e ` g f we see that the reduced subscheme associated to F p g , e q is e ` X . Therefore the main theoremwill follow if we can show that Comp E p g , e q is no larger than Comp C E p g , e q . It is an elementaryfact from commutative algebra that Comp E p g , e q ď Comp C E p g , e q provided C E p g , e q is reduced,and so our approach is to show that gr U p g , e q ab has no nilpotent elements. By (1.3) it suffices toshow that S p g , e q ab is reduced.In this paragraph we take g classical. By passing to the completion of S p g , e q ab at the maximalgraded ideal and using the fact that the Slodowy slice is transverse to every point of e ` X we areable to reduce the problem of showing that S p g , e q ab is reduced to the case where e is distinguished.To be more precise, S p g , e q ab is reduced if and only if the completion at the maximal graded idealis so, and we show that this completion is reduced if and only if p S p ˜ g , ˜ e q ^ x q ab is reduced, where x isa point on the Slodowy slice attached to a distinguished element in a larger classical Lie algebra.For the proof we make use of the fact that transverse Poisson manifolds are locally diffeomorphic,which follows from Weinstein’s splitting theorem [We83, Theorem 2.1]. Now Theorem 1.1 will follow if we can show that S p g , e q ab is reduced for distinguishedelements e . Again we introduce some new ideas to attack this problem. If X is a complex Poissonscheme of finite type and H is a reductive group acting rationally by Poisson automorphisms thenthe invariant subscheme X H can be equipped with a Poisson structure via Dirac reduction. Thereduced Poisson algebra will be denoted R p k r X s , H q . Now if g “ Lie p G q for G reductive and H Ď Aut p g q is a reductive group fixing an sl -triple t e, h, f u Ď g then we prove the following isomorphism of Poisson algebras R p S p g , e q , H q „ ÝÑ S p g H , e q . (1.4)Some special cases of (1.4) were discovered by Ragoucy [Ra01].We apply this isomorphism in the case where g “ gl n and H “ Z { Z is generated by someinvolution τ . Then g τ “ so n or g “ sp n . It follows from the work of Brundan and Kleshchev that S p g , e q is a quotient of a (semiclassical) shifted Yangian y n p σ q depending on e . If e is distinguishedthen we can identify an involution τ on y n p σ q defined so that the Poisson homomorphism y n p σ q (cid:16) S p g , e q is τ -equivariant. As a consequence we can apply the Dirac reduction procedure to theshifted Yangian y n p σ q and thus obtain a presentation of the Poisson structure on S p g , e q . Wemention that obtaining presentations of finite W -algebras outside of type A is one of the key openproblems in the field, and so we hope this result will present fertile ground for further discoveries.Let m be the maximal graded ideal. Finally we use the presentation of S p g , e q ab to calculategenerators and certain relations of gr m S p g , e q ab . Considering the relationship between F p g , e q and e ` X mentioned earlier we see that the reduced algebra of gr m S p g , e q ab is naturally identified withthe coordinate ring on the tangent cone C r TC e p e ` X qs . Micha¨el Bulois has recently demonstratedthat the sheets of g containing e are transversal at e (work in preparation [Bu]), which allowsus to calculate the dimensions of the irreducible components of TC e p e ` X q , once again theyare determined by the KS algorithm. Using a combinatorial argument we then show that therelations mentioned above give a full presentation for gr m S p g , e q ab . It follows quickly that bothgr m S p g , e q ab and S p g , e q ab are reduced, which allows us to conclude the proof of Theorem 1.1. Since the presentation of the distinguished semiclassical finite W -algebras in types B, C,D is an important result in its own right we formulate it straight away. For the proof combineTheorem 3.7 and Theorem 4.9.
Theorem 1.3.
Let g “ so n or sp n and let e be a distinguished nilpotent element with partition λ “ p λ , ..., λ n q . Then S p g , e q is generated as a Poisson algebra by elements t η p r q i | ď i ď n, r P Z ą u Y t θ p r q i | ď i ă n, r ´ λ i ` ´ λ i P Z ą u (1.5) together with the following relations t η p r q i , η p s q j u “ (cid:32) η p r q i , θ p s q j ( “ p δ i,j ´ δ i,j ` q r ´ ÿ t “ η p t q i θ p r ` s ´ ´ t q j (1.7) (cid:32) θ p r q i , θ p s q i ( “ for r ` s ´ odd (1.8) (cid:32) θ p r q i , θ p s q i ( “ s ´ ÿ t “ r θ p t q i θ p m ´ t q i ` p´ q s ` s i,i ` m ÿ t “ η p m ´ t q i ` r η p t q i for r ă s, r ` s ´ “ m (1.9) (cid:32) θ p r ` q i , θ p s q i ` ( ´ (cid:32) θ p r q i , θ p s ` q i ` ( “ θ p r q i θ p s q i ` (1.10) (cid:32) θ p r q i , θ p s q j ( “ for | i ´ j | ą ! θ p r q i , (cid:32) θ p s q i , θ p t q j () ` ! θ p s q i , (cid:32) θ p r q i , θ p t q j () “ for | i ´ j | “ and r ` s odd (1.12) ! θ p r q i , (cid:32) θ p s q i , θ p t q j () ` ! θ p s q i , (cid:32) θ p r q i , θ p t q j () “ (1.13) 2 p´ q s ` s i,i ` ´ δ i,j ` m ´ ÿ m “ m ÿ m “ η p p m ´ m ´ qq i ` r η p m q i θ p p m ´ m q` t q j ` p´ q s ` s i,i ` ´ δ i ` ,j m ´ ÿ m “ m ´ m ´ ÿ m “ r η p m q i η p m q i ` θ p p m ´ m ´ m ´ q` t q j for | i ´ j | “ and r ` s “ m even η p r q i “ for r ą λ (1.14) λ i { ÿ t “ η p t q i θ p λ i ´ t ` s , ` q i “ for i “ , ..., n ´ when g “ sp n . (1.15) where we adopt the convention η p q i “ r η p q i “ and the elements t r η p r q i | r P Z ě u are defined viathe recursion r η p r q i : “ ´ r ÿ t “ η p t q i r η p r ´ t q i . (1.16) To conclude the introduction we describe the structure of the paper, which is divided intotwo parts. The first part is very algebraic, dealing with Dirac reduction and the presentation ofsemiclassical W -algebras in the distinguished case. The second part is more geometric, studyingthe degenerations of E p g , e q using Lusztig–Spaltenstein induction and the tangent cone of theKatsylo variety. Part I:
We begin Section 2 by giving an elementary introduction to the version of Dirac reductionused in this paper. In Subsection 2.3 we prove the isomorphism (1.4), which should have indepen-dent interest. In Section 3 we describe the semiclassical shifted Yangian y n p σ q by generators andrelations. The main results on the Dirac reduction of y n p σ q are presented in Subsection 3.4, includ-ing the canonical grading, the loop filtration, the PBW theorem and the presentation by generatorsand relations. All of the results about R p y n p σ q , τ q are ultimately deduced from similar results on y n p σ q . In Section 4 we recall the definition of Brundan–Kleshchev’s isomorphism y n p σ q (cid:16) S p g , e q and show by an explicit calculation that this is τ -equivariant for a suitable choice of involution on y n p σ q . By (1.4) this leads to a surjection R p y n p σ q , τ q (cid:16) S p g τ , e q and in Subsection 4.5 we completethe picture by describing a full set of Poisson generators for the kernel. Part II:
In Section 5 we gather together some important general facts about degenerationsand completions of schemes, as well as reviewing the theory of rank stratification and symplecticleaves of a Poisson scheme. In Section 6 we explain how S p g , e q ab is related to Katsylo variety e ` X , and recall the aforementioned results of Bulois allowing us to enumerate the irreduciblecomponents of TC e p e ` X q and calculate their dimensions. In Section 7 we introduce quantumfinite W -algebras and prove the existence of the closed immersion (1.3). The proof of the latteruses a reduction modulo p argument similar to Premet’s construction of the component map (1.2) in [Pr10, Theorem 1.2], along with the identification of reduced schemes F p g , e q red “ C r e ` X s from Section 6. Finally in Section 8 we describe the Kempken–Spaltenstein algorithm, as well asits relationship with sheets, and then use this to construct an algebraic variety X λ associated to adistinguished nilpotent orbit O with partition λ , which we call the combinatorial Katsylo variety .In Theorem 8.7 we use the presentation of S p g , e q ab obtained in part I of the paper to demonstratethat C r X λ s (cid:16) S p g , e q ab (cid:16) C r TC e p e ` X qs , and we show that these are isomorphisms by comparingthe dimensions of the irreducible components. In particular this implies that S p g , e q ab is reducedfor e distinguished. Theorem 8.8 reduces the general case to the distinguished case. Finally weconclude the proof of the main theorem in Subsection 8.7, making use of deformation techniquesgathered in Section 5.In Subsection 9.1 we recall the definition of Losev’s orbit method map and we prove a slightrefinement of Theorem 1.2. In Subsection 9.2, 9.3 we recall the role of the Namikawa–Weyl groupin the classification of quantizations of conic symplectic singularities, and study the relationshipwith primitive ideals. Finally in Subsection 9.4 we prove the main theorem by applying quantumHamiltonian reduction to the quantizations of orbit covers arising from generalised Springer maps. Contents
1. Introduction 1Notation and conventions 7Acknowledgements 7
Part 1. Presentations of classical W -algebras
82. Dirac reduction for classical finite W -algebras 82.1. Invariants via Dirac reduction 82.2. Classical finite W -algebras 92.3. Dirac reduction applied to W -algebras 103. Dirac reduction for shifted Yangians 133.1. Poisson algebras by generators and relations 133.2. Chevalley–Serre presentations for shifted current Lie–Poisson algebras 133.3. The semiclassical shifted Yangian 163.4. The Dirac reduction of the shifted Yangian 194. Finite W -algebras for classical Lie algebras 234.1. Partitions for nilpotent orbits in classical Lie algebras 234.2. The Dynkin pyramid and the centraliser 234.3. Symplectic and orthogonal subalgebras 244.4. Generators of the W -algebra 264.5. The semiclassical Brundan–Kleshchev homomorphism 27 Part 2. One dimensional representations of W -algebras W -algebras and Slodowy slices 326.3. The Katsylo variety and the tangent cone 337. Abelian quotients of finite W -algebras 347.1. The finite W -algebra 347.2. Bounding the asymptotic cone 358. The abelian quotient of the classical W -algebra 378.1. Rigid, singular and distinguished partitions 378.2. The Kempken–Spaltenstein algorithm 388.3. Distinguished elements and induction 388.4. The combinatorial Katsylo variety 398.5. The semiclassical abelianisation I: the distinguished case 418.6. The semiclassical abelianisation II: the general case 438.7. The abelianisation of the finite W -algebra via deformation theory 449. The orbit method 449.1. Losev’s orbit method map 449.2. The Namikawa–Weyl group and the Poisson automorphism group 469.3. Primitive ideals arising from generalised Springer maps 469.4. Quantum Hamiltonian reduction of universal quantizations 48References 50 Notation and conventions.
The following notation will be used throughout the paper. For i P Z we write Z ą i “ t i ` , i ` , ... u . All algebras and vector spaces defined over C , except in Section 7where we use reduction modulo a large prime. We use capital letters G, H, ... for algebraic groupsand gothic script g , h , ... for their Lie algebras. All schemes appearing in this paper will be affineschemes of finite type over C , thus the reader may almost always think of complex affine varieties,except that the coordinate rings will often be non-reduced (the consideration of nilpotent elementswill be vital to our main results). Occasionally we write m-Spec p A q for the variety of closed pointsin the prime spectrum of a commutative algebra A . If X is a noetherian scheme then we writeComp p X q for the set of irreducible components of X .The associated graded algebra of an almost commutative, filtered associative algebra is equippedwith a Poisson structure in the usual manner. If A is a commutative algebra and I Ď A an ideal,then we write gr I A for the graded algebra with respect to the I -adic filtration. Acknowledgements.
I would like to offer thanks to Simon Goodwin, Ivan Losev, Sasha Premetand Dmytro Matvieievskyi for useful comments on an early draft of this paper. I’m especiallygrateful to Ivan for suggesting some of the constructions used in the proof of Theorem 1.2, and toSasha to whom this paper is dedicated - his insightful teaching first introduced to these fascinatingproblems. I have also benefited from many interesting conversations and email correspondence with Jon Brundan, Micha¨el Bulois, Paul Levy, Anne Moreau, and Daniele Valeri, and am grate-ful for all their help. Some of these results were announced at the conference “Geometric andautomorphic aspects of W -algebras”, Lille 2019. This research is supported by the UKRI Fu-ture Leaders Fellowship project “Geometric representation theory and W -algebras”, grant numberMR/S032657/1. Part Presentations of classical W -algebras Dirac reduction for classical finite W -algebras Invariants via Dirac reduction.
Let X be an complex affine Poisson variety and suppose z , ..., z n P C r X s such that the determinant of the matrix pt z i , z j uq ď i,j ď n is a unit. In his seminalpaper [Dir50] Dirac defined a new Poisson bracket on C r X s , such that the z i are Casimirs, thusequipping C r X s{p z , ..., z n q with a Poisson structure. In fact this is a special case of the followingprocedure: say that I Ď C r X s is a Dirac ideal if N C r X s p I q (cid:16) C r X s{ I surjects where N C r X s p I q : “t f P C r X s | t f, I u Ď I u denotes the Poisson idealiser. Then the subscheme associated to I inheritsa Poisson structure from X , and we call this induced structure the Dirac reduction [LPV13, 5.4.3].Now let H be a linearly reductive group acting on X by Poisson automorphisms. We regard theset of invariants X H as a (not necessarily reduced) affine scheme with global sections C r X H s : “ C r X s H , where C r X s H “ C r X s{ I H and I H : “ p h ¨ f ´ f | h P H, f P C r X sq . Although I H is usuallynot a Poisson ideal, it is always a Dirac ideal (this is a corollary of Lemma 2.1) so that X H acquiresthe structure of a Poisson scheme.The following suggests an alternative approach of the Poisson structure on X H , better-suited tocalculations. Lemma 2.1.
The map C r X s H Ñ C r X H s is surjective and C r X s H { C r X s H X I H „ ÝÑ C r X H s (2.1) is a Poisson isomorphism.Proof. If f P C r X s H , g P C r X s and h P H then t f, h ¨ g ´ g u “ h ¨ t f, g u ´ t f, g u and so C r X s H Ď N C r X s p I H q is a Poisson subalgebra. Since H is linearly reductive and acts locally finitely we candecompose C r X s “ C r X s H ‘ C r X s where C r X s is the kernel of the H -equivariant projection C r X s Ñ C r X s H . If V Ď C r X s is an irreducible H -submodule then t h ¨ v ´ v | h P H, v P V u Ď V is H -stable and nonzero hence equal to V , and it follows that I H “ p C r X s q . Therefore thecomposition C r X s H ã Ñ N C r X s p I H q Ñ C r X H s “ C r X s H is surjective. (cid:3) Now let A be any Poisson algebra with a locally finite action of H by Poisson automorphisms.Write I H : “ p h ¨ a ´ a | h P H, a P A q and define the Dirac reduction of A by H by R p A, H q : “ A H { A H X I H (2.2)If τ P Aut p A q is a semisimple Poisson automorphism of finite order, then we often abuse notationwriting R p A, τ q for R p A, H q where H is the group generated by τ .Since H is linearly reductive the functor of H -invariants is exact on the category of locally finite H -modules. This implies that R p´ , H q is a right exact functor from the category of H -locallyfinite Poisson algebras, to the category of Poisson algebras. Lemma 2.2.
Suppose the following: (1) V is a direct sum of locally finite H -modules with H -stable decomposition V “ V H ‘ V . (2) S p V q is a Poisson algebra with H acting by Poisson automorphisms.Then the natural map S p V H q Ñ R p S p V q , H q is an isomorphism of commutative algebras.Proof. Since S p V q “ S p V H q ‘ p V q and S p V H q is H -fixed we must have S p V q Ď p V q . Combiningwith V Ď S p V q we deduce that I H “ p V q which proves the map S p V H q Ñ R p S p V q , H q is anisomorphism of commutative algebras. (cid:3) Remark . Let g be a Lie algebra and H a linearly reductive group of automorphisms of g acting locally finitely. In this case the composition S p g H q Ñ S p g q H Ñ R p S p g q , H q is a Poissonhomomorphism and so Lemma 2.2 shows that S p g H q „ ÝÑ R p S p g q , H q as Poisson algebras.2.2. Classical finite W -algebras. For the rest of the Section we fix a connected reductive al-gebraic group G such that the derived subgroup is simply connected, and write g “ Lie p G q . Let κ be a choice of non-degenerate trace form on g which is preserved by Aut p g q . Pick a nilpotentelement e P g and write χ : “ κ p e, ¨q P g ˚ . Pick an sl -triple t e, h, f u and write g “ À i P Z g p i q forthe grading by ad p h q -eigenspaces. Throughout the paper we use the notation g pď i q “ À j ď i g p j q and similar for g pă i q . Since e P g p q we see that χ restricts to a character on g pă –1 q . Make thefollowing notation g pă –1 q : “ t x ´ χ p x q | x P g pă –1 qu Ď S p g q . The nilpotent Lie algebra g pă q isalgebraic and we write g pă q “ Lie G pă q .The (classical) finite W -algebra associated to p g , e q is a Poisson reduction of S p g q S p g , e q : “ p S p g q{ S p g q g pă –1 q χ q G pă q . (2.3)In more detail, the Poisson normaliser N “ t f P S p g q | t f, g pă –1 q χ u Ď S p g q g pă –1 q χ q is a Poissonsubalgebra of S p g q with N X S p g q g pă –1 q χ embedded as a Poisson ideal, and S p g , e q is equippedwith a Poisson structure via the isomorphism N { N X S p g q g pă –1 q χ „ ÝÑ S p g , e q .The Kazhdan grading is defined on S p g q by placing g p i q in Kazhdan degree i `
2. Notice that g pă –1 q χ generates a homogeneous ideal and that S p g q{ S p g q g pă –1 q χ inherits a connected gradingin non-negative degrees, with S p g , e q embedded as a graded subalgebra, with Poisson bracket indegree ´ e the map g e Ñ S p g q{ S p g q g pă –1 q χ is injective. Theorem 2.4.
Let m Ď S p g q{ S p g q g pă –1 q χ denote the unique maximal graded ideal. (1) There exists a Kazhdan graded map θ : g e Ñ S p g , e q such that θ p x q ´ x P m . (2.4) Furthermore θ can be chosen to be equivariant with respect to any reductive group of Poissonautomorphisms acting rationally on S p g , e q by graded automorphisms. (2) If θ is any map satisfying (2.4) then S p g e q Ñ S p g , e q is an isomorphism of commutativealgebras.Proof. It follows from [GG02, Lemma 2.1] that the restriction homomorphism S p g q{ S p g q g pă –1 q χ – C r χ ` g pă –1 q K s Ñ C r χ ` g f s – S p g e q gives a G e p q -equivariant isomorphism S p g , e q Ñ S p g e q of commutative algebras, where g pă –1 q K : “ t η P g ˚ | η p g pă –1 qq “ u . Taking the inverse isomor-phism restricted to g e Ď S p g e q gives the desired map θ . If H is any reductive group of Poissonisomorphisms then θ can be replaced with an H -equivariant map using the standard trick ofprojecting onto isotypic components of S p g , e q for the H -action.Now if θ is any map satisfying (2.4) then S p g e q Ñ S p g , e q and it suffices to show that thismap is surjective. If x , ..., x r P g e is a homogenous basis then [Ja04, Lemma 7.1] shows that θ p x q , ..., θ p x r q generate a graded radical ideal of finite codimension. The only such ideal is themaximal graded ideal of S p g , e q and this implies that θ p x q , ..., θ p x r q generate S p g , e q . (cid:3) Dirac reduction applied to W -algebras. Now fix a reductive subgroup H Ď Aut p g q fixingour choice of sl -triple. Make the notation g : “ g H and let g denote an H -invariant complementto g in g . Also write G Ď G for the connected component of the subgroup consisting of g P G such that r Ad p g q , H s “ Lemma 2.5. (1)
The restriction of κ to g is non-degenerate. (2) G is reductive and g “ Lie p G q .Proof. Since g is spanned by elements t h ¨ x ´ x | h P H, x P g u it follows that κ p g , g q “
0. Since κ is non-degenerate we deduce part (1).Now suppose that κ is defined from a representation ρ : g Ñ gl p V q . If n Ď g is a nilpotent idealthen by Engel there exists k ą ρ p n q k “
0. Hence p ρ p x q ρ p n qq k “ x P g and n P n , so κ p x, n q “
0. Since κ is non-degenerate the nilradical of g is trivial.If Ad G : G Ñ GL p g q is the adjoint representation then we consider ρ : “ Ad GL p g q ˝ Ad G : G Ñ GL p End p g qq and let ρ : G Ñ GL p W q be any faithful representation admitting ρ as a directsummand. If we identify G (resp. g ) with its image in GL p W q (resp. gl p W q ) via ρ (resp. d ρ ),and identify H with a subset of End p g q Ď W , then G is precisely the subgroup of GL p W q fixing H , and similar for g . Now apply [Hum75, Theorem 13.2] to see that g “ Lie p G q . Since thenilradical of g is trivial, G is reductive thanks to [Hum75, Theorem 13.5]. (cid:3) Since g is a graded subalgebra of g containing e we can consider S p g , e q . Furthermore H preserves g pă –1 q χ and the induced action on S p g q{ S p g q g pă –1 q χ stabilises the G pă q -invariants, sothat H acts by Poisson automorphisms on S p g , e q .We are now ready to formulate one of our first main theorems, stated in (1.4). Theorem 2.6.
There is a natural Poisson isomorphism R p S p g , e q , H q „ ÝÑ S p g , e q . (2.5)Since κ is non-degenerate on g and g and Aut p g q -invariant, it follows that g K “ t x P g | κ p x, g q “ u is an H -stable complement to g in g . Hence g “ g K . This implies(2.6) χ p g q “ . Lemma 2.7. (1) S p g q X S p g q g pă –1 q χ “ S p g q g pă –1 q χ ; (2) S p g q g X S p g q g pă –1 q χ “ S p g qp g pă –1 q ` g g pă –1 q χ q .Proof. Observe that g and g are graded subspaces of g . It suffices to show that the left handside is contained in the right, for both (1) and (2). Let x P g pă –1 q and f P S p g q , and let f “ f ` f and x “ x ` x be the decompositions over S p g q “ S p g q ‘ S p g q g and g “ g ‘ g respectively. By (2.6) the projections of f p x ´ χ p x qq to S p g q and S p g q g are f p x ´ χ p x qq and f x ` f p x ´ χ p x qq respectively. Therefore if f p x ´ χ p x qq lies in S p g q we must have f x ` f p x ´ χ p x qq “ f p x ´ χ p x qq “ f p x ´ χ p x qq , proving (1).Now suppose that f p x ´ χ p x qq P S p g q g X S p g q g pă –1 q χ and that x “ x ` x where x i P g i pă –1 q .Then f p x ´ χ p x qq P S p g q g pă –1 q by (2.6), whilst f p x ´ χ p x qq P S p g q g implies that f P S p g q g so that f p x ´ χ p x qq P S p g qp g g pă –1 q χ q . This proves (2). (cid:3) Lemma 2.8.
The ideal S p g q g pă –1 q χ is the direct sum of its intersections with S p g q and S p g q g .Therefore S p g q “ S p g q ‘ S p g q g gives a G pă q -module decomposition S p g q{ S p g q g pă –1 q χ “ S p g q{ S p g q g pă –1 q χ ‘ S p g q g { S p g qp g pă –1 q ` g g pă –1 q χ q . (2.7) Proof.
We begin by proving the claim S p g q g pă –1 q χ “ S p g q X S p g q g pă –1 q χ ‘ S p g q g X S p g q g pă –1 q χ . (2.8)It suffices to show that the right hand side contains the left. Let f P S p g q and x ´ χ p x q P g pă –1 q χ .Since g pă –1 q “ g pă –1 q ‘ g pă –1 q we can consider two cases: (i) if x P g then by (2.6) we have f p x ´ χ p x qq P S p g q g X S p g q g pă –1 q χ ; (ii) if x P g then we can write f “ f ` f P S p g q ‘ S p g q g ,in which case f p x ´ χ p x qq P S p g q g pă –1 q χ Ď S p g q X S p g q g pă –1 q χ and f p x ´ χ p x qq P S p g q g X S p g q g pă –1 q χ . Now the Lemma follows from Lemma 2.7. (cid:3) We write S p g q for the H -invariants and S p g q for the unique H -stable complement. Use similarnotation for S p g , e q . Consider the two sets N : “ t f P S p g q | g ¨ f ´ f P S p g q g pă –1 q χ for all g P G pă qu ;(2.9) N : “ t f P S p g q | g ¨ f ´ f P S p g q g pă –1 q χ for all g P G pă qu . (2.10)By differentiating the locally finite actions on G pă q on S p g q{ S p g q g pă –1 q χ and of G pă q on S p g q{ S p g q g pă –1 q χ we see that t g pă –1 q , N X S p g q u Ď N X S p g q g pă –1 q χ ; t g pă –1 q , N u Ď N X S p g q g pă –1 q χ . (2.11)We consider the projection ρ : S p g q Ñ S p g q (2.12)across the decomposition S p g q “ S p g q ‘ S p g q g . Proposition 2.9. (i) N X S p g q Ď S p g q and N Ď S p g q are Poisson subalgebras; (ii) ρ p N X S p g q q Ď N and the map ρ : N X S p g q Ñ N is a Poisson homomorphism; (iii) The map π : S p g q Ñ S p g q{ S p g q g pă –1 q χ restricts to a surjective Poisson homomorphism N X S p g q (cid:16) S p g , e q ;(iv) The map π : S p g q Ñ S p g q{ S p g q g pă –1 q χ restricts to a surjective Poisson homomor-phism N (cid:16) S p g , e q ; (v) The kernel of the map N X S p g q (cid:16) S p g , e q is contained in the kernel of the map π ˝ ρ .Thus ρ induces a Poisson homomorphism S p g , e q (cid:16) S p g , e q ; (vi) S p g , e q X S p g , e q S p g , e q is contained in the kernel of S p g , e q (cid:16) S p g , e q , inducing a sur-jective Poisson homomorphism R p S p g , e q , H q (cid:16) S p g , e q . (2.13) Proof.
Since H acts on S p g q by Poisson automorphisms the invariant subspace S p g q is a Poissonsubalgebra. Therefore the proofs of the two claims in (i) are identical, and we will only prove that N Ď S p g q is a Poisson subalgebra. It is evidently closed under multiplication so we only needto show that it is closed under the bracket. Suppose that f , f P N , that g P G pă q and that g ¨ f i ´ f i “ h i P S p g q g pă –1 q χ . We have g ¨ t f , f u “ t g ¨ f , g ¨ f u “ t f ` h , f ` h u andso (i) will follow if we can show that t f , h u , t h , f u , t h , h u P S p g q g pă –1 q χ . Since χ vanisheson g pă´ q we have r g pă –1 q χ , g pă –1 q χ s Ď g pă –1 q χ , therefore t h , h u P S p g q g pă –1 q χ by theLeibniz rule. To complete the proof of (i) we observe that N X S p g q g pă –1 q χ is a Poisson idealof N , which follows quickly from (2.11).We now address (ii). Let f P N X S p g q and write f “ f ` f according to the decomposition S p g q “ S p g q ‘ S p g q g . If g P G pă q then g ¨ f ´ f P S p g q X S p g q g pă –1 q χ “ S p g q g pă –1 q χ byLemma 2.7 and Lemma 2.8. This shows that ρ p f q “ f P N .Finally, to see that ρ : N X S p g q Ñ N is a Poisson homomorphism it suffices to show that N X S p g q X S p g q g is a Poisson ideal of N X S p g q . Recall that g “ t h ¨ x ´ x | h P H, x P g u and S p g q “ t h ¨ x ´ x | h P H, x P S p g qu . If f P S p g q then t f, h ¨ x ´ x u “ h ¨ t f, x u ´ t f, x u P S p g q for any x P S p g q . Lemma 2.2 shows that the ideals generated by g and S p g q coincide, hence t S p g q , g u Ď S p g q g , which shows that S p g q X S p g q g is a Poisson ideal of S p g q . This completesthe proof of (ii).The map π restricts to a surjection N (cid:16) S p g , e q by definition. Since the latter map is H -equivariant we get N X S p g q Ñ S p g , e q , which proves (iii), whilst (iv) is proven similarly.We move on to (v). The kernel of N X S p g q Ñ S p g , e q is N X S p g q X S p g q g pă –1 q χ . This ismapped to S p g q X S p g q g pă –1 q χ “ S p g q g pă –1 q χ by ρ , thanks to Lemma 2.7 and Lemma 2.8.Finally S p g q g pă –1 q χ lies in the kernel of π , which proves (v). Now we take f P N X S p g q suchthat π p f q P S p g , e q S p g , e q . Using Lemma 2.8 again we see that S p g q{ S p g q g pă –1 q χ decomposes asthe direct sum of the image of S p g q and the image of S p g q g . Therefore S p g , e q , and the idealwhich it generates, are contained in the image of S p g q g . It follows immediately that f P S p g q S p g q ` S p g q g pă –1 q χ ρ p f q P S p g q g pă –1 q χ X S p g q “ S p g q g pă –1 q χ Hence ρ p f q is in the kernel of π , completing (vi). This concludes the proof. (cid:3) Proof of Theorem 2.6.
Thanks to Proposition 2.9 we have φ : R p S p g , e q , H q (cid:16) S p g , e q .Pick an H -equivariant map θ : g e Ñ S p g , e q satisfying the properties of Theorem 2.4, and define θ : g e Ñ S p g , e q via θ p x q : “ φ p θ p x q ` S p g , e q X S p g , e q S p g , e q q Property (2.4) for θ implies (2.4) for θ . By Theorem 2.4(2) this implies that φ is surjective.Applying Proposition 2.2 we see that R p S p g , e q , H q is a polynomial algebra generated by the image of θ p g e q under the map S p g , e q Ñ R p S p g , e q , H q . It follows that φ maps a basis of R p S p g , e q , H q to a basis of S p g , e q , hence it is an isomorphism. (cid:3) Dirac reduction for shifted Yangians
Poisson algebras by generators and relations.
Let X be a set. The free Lie algebra L X on X is the initial object in the category of (complex) Lie algebras generated by X and can beconstructed as the Lie subalgebra of the free algebra C x X y generated by the vector space spannedby X . When L is a Lie algebra generated by X we say that L has relations Y Ď L X if Y generatesthe kernel of L X (cid:16) L .The free Poisson algebra generated by X is the initial object in the category of (complex) Poissonalgebras generated by X . It can be constructed as the symmetric algebra S p L X q together withits Poisson structure. If there is a Poisson surjection S p L X q (cid:16) A then we say that A is Poissongenerated by X . It is important to distinguish this from A being generated by X as a commutativealgebra, as both notions will occur frequently.We say that a (complex) Poisson algebra A has Poisson generators X and relations Y Ď S p L X q if there is a surjective Poisson homomorphism S p L X q (cid:16) A and the kernel is the Poisson idealgenerated by Y .Let X be a set and Y Ď L X Ď S p L X q . Write I (resp. J ) for the ideal of S p L X q (resp. L X )generated by Y . It is easy to see that the natural map S p L X q Ñ S p L X q{ I induces an isomorphism S p L X { J q „ ÝÑ S p L X q{ I. (3.1)3.2. Chevalley–Serre presentations for shifted current Lie–Poisson algebras.
Through-out this Section we fix n P Z ą . Following [BK06] a shift matrix is an n ˆ n array σ “ p s i,j q ď i,j ď n of non-negative integers with zero on the diagonal, satisfying s i,k “ s i,j ` s j,k (3.2)whenever i ď j ď k or k ď j ď i . A shift matrix is said to be symmetric if it is equal to itstranspose and is said to be even if the entries are integers. This nomenclature arises from the factthat shift matrices classify certain gradings [BK06, (7.6)] and an even shift matrix corresponds to agrading supported on the even integers. Symmetric shift matrices correspond to Dynkin gradings,i.e. those which arise from sl -triples.The current algebra is the Lie algebra c n : “ gl n b C r t s . It has a basis consisting of elements t e i,j t r | ď i, j ď n, r P Z ě u where we write x b t r “ xt r for x P gl n and r ě
0. For any shiftmatrix σ “ p s i,j q ď i,j ď n we define the shifted current algebra c n p σ q to be the subalgebra spannedby t e i,j t r | ď i, j ď n, r ´ s i,j P Z ě u . (3.3) Lemma 3.1.
The Lie subalgebra u n p σ q Ď c n p σ q spanned by elements (3.3) with i ă j is generatedas a complex Lie algebra by t e i ; r | ď i ă n, r ´ s i,i ` P Z ě u (3.4) subject to the relations “ e i ; r , e j ; s ‰ “ for | i ´ j | ‰ , (3.5) “ e i ; r ` , e i ` s ‰ ´ “ e i ; r , e i ` s ` ‰ “ , (3.6) ” e i ; r “ e i ; r , e j ; r ‰ı ` ” e i ; r “ e i ; r , e j ; r ‰ı “ for all | i ´ j | “ . (3.7) Proof.
Write 0 for the n ˆ n zero matrix. It follows from (3.2) that the linear map u n p σ q Ñ u n p q defined by e i,j t r ÞÑ e i,j t r ´ s i,j is a Lie algebra isomorphism and so it suffices to prove the Lemmawhen σ “ p u n be the Lie algebra with generators (3.4) and relations (3.5)–(3.7), with σ “
0. Weinductively define elements e i,j ; r P p u n by setting e i,i ` r : “ e i ; r and e i,j ; r : “ r e i,j ´ r , e j ´ ,j ;0 s for 1 ď i ă j ď n . There is a homomorphism p u n (cid:16) u n p q given by e i ; r ÞÑ e i,i ` t r and, in order to show thatit is an isomorphism, we show that p u n is spanned by the elements t e i,j ; r | ď i ă j ď n, r P Z ě u .Following (1)–(7) in the proof of [BK05, Lemma 5.8] verbatim we have for all i, j, k, l, r, s r e i,j ; r , e k,l ; s s “ δ j,k e i,l ; r ` s ´ δ i,l e k,j ; r ` s . (3.8)Define an ascending filtration on p u n “ Ť d ą F d p u n satisfying F d p u n “ ř d ` d “ d r F d p u n , F d p u n s by placing e i ; r in degree 1. We prove by induction that F d p u n is spanned by elements e i,j ; r with j ´ i ď d . The base case d “ d ` d “ d ą d p u n and F d p u n are spanned by elements e i,j ; r . Using (3.8) we complete theinduction, which finishes the proof. (cid:3) Theorem 3.2. S p c n p σ qq is Poisson generated by t d i ; r | ď i ď n, r P Z ě u Y t e i ; r | ď i ă n, r ´ s i,i ` P Z ě uYt f i ; r | ď i ă n, r ´ s i ` ,i P Z ě u (3.9) subject to relations (cid:32) d i ; r , d j ; s ( “ , (3.10) (cid:32) d i ; r , e j ; s ( “ p δ i,j ´ δ i,j ` q e j ; r ` s , (3.11) (cid:32) d i ; r , f j ; s ( “ p δ i,j ` ´ δ i,j q f j ; r ` s , (3.12) (cid:32) e i ; r , e i ` r ` ( ´ (cid:32) e i ; r ` , e i ` r ( “ , (3.13) (cid:32) f i ; r , f i ` r ` ( ´ (cid:32) f i ; r ` , e i ` r ( “ , (3.14) (cid:32) e i ; r , e j ; s ( “ for | i ´ j | ‰ , (3.15) (cid:32) f i ; r , f j ; s ( “ for | i ´ j | ‰ , (3.16) ! e i ; r , (cid:32) e i ; r , e j ; r () ` ! e i ; r , (cid:32) e i ; r , e j ; r () “ for | i ´ j | “ , (3.17) ! f i ; r , (cid:32) f i ; r , f j ; r () ` ! e i ; r , (cid:32) e i ; r , e j ; r () “ for | i ´ j | “ . (3.18) Proof.
By (3.1) it suffices to show that c n p σ q is generated as a Lie algebra by (3.9) subject torelations (3.10)–(3.14). Let p p c n p σ q , t¨ , ¨uq denote the Lie algebra with these generators and relations.Define a map from the set (3.9) to c n p σ q by d i ; r ÞÑ e i,i t r , e i ; r ÞÑ e i,i ` t r , f i ; r ÞÑ e i ` ,i t r . One caneasily verify using (3.8) that this extends to a surjective Lie algebra homomorphism p c n p σ q (cid:16) c n p σ q . To show that this is an isomorphism it suffices to show that the elements t e i,j ; r | ď i, j ď n, r ´ s i,j P Z ě u Ď p c n p σ q (3.19)defined inductively by setting e i,i ` r : “ e i,r , e i ` ,i ; r : “ f i ; r and e i,j ; r : “ t e i ; s i,i ` , e i ` ,j ; r ´ s i,i ` u for i ă j ;(3.20) e i,j ; r : “ t f i ´ s i,i ´ , e i ´ ,j ; r ´ s i,i ´ u for i ą j. (3.21)form a spanning set. Using (3.10),(3.11), (3.12) and a simple inductive argument one can see that p c n p σ q is a direct sum of three subalgebras: the diagonal subalgebra, spanned by the elements d i ; r ,and the upper and lower triangular subalgebras u ` n p σ q and u ´ n p σ q generated by the elements e i ; r ,respectively by the elements f i ; r .In order to complete the proof it suffices to show that u ` n p σ q and u ´ n p σ q are spanned by theelements defined in (3.20) and (3.21) respectively. Since the argument is identical for u ` n p σ q and u ´ n p σ q we only need to consider the former, where the claim follows from Lemma 3.1 (cid:3) The following theorem is one of the key stepping stones for understanding the Dirac reduction ofthe shifted Yangian. The algebra described here is the twisted shifted current Lie–Poisson algebra . Theorem 3.3. If σ is even and symmetric then S p c n p σ qq admits a Poisson automorphism τ : e i,j t r ÞÑ p´ q r ´ ` s i,j e j,i t r . (3.22) The Dirac reduction R p S p c n p σ q , τ q “ S p c n p σ q τ q is Poisson generated by t η i ;2 r ´ | ď i ď n, r P Z ą u Y t θ i ; r | ď i ď n, r P Z ě s i,i ` u (3.23) subject to the relations (cid:32) η i ;2 r ´ , η j ;2 s ´ ( “ , (3.24) (cid:32) η i ;2 r ´ , θ j ; s ( “ p δ i,j ´ δ i,j ` q θ j ;2 r ´ ` s , (3.25) (cid:32) θ i ; r , θ j ; s ( “ if | i ´ j | ą , (3.26) (cid:32) θ i ; r , θ i ` s ` ( “ (cid:32) θ i ; r ` , θ i ` s ( , (3.27) (cid:32) θ i ; r , θ i ; s ( “ p´ q s ´ ` s i,i ` p η i ; r ` s ´ η i ` r ` s q if r ` s is odd if r ` s is even , (3.28) ! θ i ; r , (cid:32) θ i ; r , θ j ; r () ` ! θ i ; r , (cid:32) θ i ; r , θ j ; r () “ for | i ´ j | “ , r ` r odd . (3.29) ! θ i ; r , (cid:32) θ i ; r , θ j ; r () ` ! θ i ; r , (cid:32) θ i ; r , θ j ; r () “ p´ q r ´ ` s i,i ` p δ i ` ,j ` δ i,j ` q θ j ; r ` r ` r (3.30) for | i ´ j | “ , r ` r even . Proof.
Relation (3.8) implies that τ gives a Poisson automorphism. By Remark 2.3 we can iden-tify R p S p c n p σ q , τ q with S p c n p σ q τ q , so it suffices to check that S p c n p σ q τ q has the stated Poissonpresentation. Use (3.1) to reduce the claim to a statement about Lie algebras.Consider the Lie algebra p c n p σ q τ which is generated by the set (3.23) subject to relations (3.24)–(3.30). We define a map from the set (3.23) to c n p σ q τ by sending η i ;2 r ´ ÞÑ e i,i t r ´ P c n p σ q τ , and sending θ i ; r ÞÑ θ i,i ` t r P c n p σ q τ . One can check that this determines a surjective Lie algebrahomomorphism p c n p σ q τ (cid:16) c n p σ q τ , indeed, checking that relations (3.24)–(3.30) hold amongst thecorresponding elements of c n p σ q τ is a routine calculation using (3.8).In order to complete the proof of (2) it is sufficient to show that this map is an isomorphism.For 1 ď i ă j ď n and r P Z ě s i,j we inductively define elements θ i,j ; r : “ t θ i,i ` s i,i ` , θ i ` ,j ; r ´ s i,i ` u P p c n p σ q τ . (3.31)where θ i,i ` r : “ θ i ; r . It remains to check that p c n p σ q τ is spanned by the elements t η i ;2 r ´ | ď i ď n, r P Z ą u Y t θ i,j ; r | ď i ă j ď n, r ´ s i,j P Z ě u (3.32)We define a filtration p c n p σ q τ “ Ť i ą F i p c n p σ q τ by placing the generators (3.23) in degree 1and satisfying F d p c n p σ q τ “ ř d ` d “ d t F d p c n p σ q τ , F d p c n p σ q τ u . By convention F p c n p σ q τ “
0. Theassociated graded Lie algebra gr p c n p σ q τ “ À i ą F i p c n p σ q τ { F i ´ p c n p σ q τ is generated by elements η i ;2 r ´ : “ η i t r ´ ` F p c n p σ q τ for 1 ď i ď n, r P Z ě (3.33) θ i ; r : “ θ i t r ` F p c n p σ q τ for 1 ď i ă n, r P Z ě s i,i ` . (3.34)These generators of gr p c n p σ q τ satisfy the top graded components of the relations (3.24)–(3.30).Let a be an abelian Lie algebra with basis t d i t r | ď i ď n, r P Z ą u . Let u n p σ q be the Liesubalgebra of c n p σ q described in Lemma 3.1. Then a ‘ u n p σ q is a Lie algebra with a an abelianideal. Comparing the top components (3.24)–(3.30) to (3.5)–(3.7) we see that there is surjectiveLie algebra homomorphism a ‘ u n p σ q (cid:16) gr p c n p σ q τ defined by d i t r ÞÑ η i ;2 r ´ and e i,i ` t r ÞÑ θ i,r . Thealgebra a ‘ u n p σ q has basis consisting of element d i t r , e j,k t s where i “ , ..., n, r P Z ą , ď j ă k ď n, s P Z ě s j,k . It follows that gr p c n p σ q τ is spanned by elements η i ;2 r ´ , θ j,k ; s where the indexes varyin the ranges specified in (3.32), and θ j,k ; s defined inductively from (3.34), analogously to (3.31).We deduce that p c n p σ q τ is spanned by the required elements, which completes the proof. (cid:3) The semiclassical shifted Yangian.
In this Section we fix n ą
0, an even shift matrix σ of size n and ε P t˘ u . The (semiclassical) shifted Yangian y n p σ q is the Poisson algebra generatedby the set(3.35) t d p r q i | ď i ď n, r ą u Y t e p r q i | ď i ă n, r ą s i,i ` uY t f p r q i | ď i ă n, r ą s i ` ,i u subject to the following relations (cid:32) d p r q i , d p s q j ( “ , (3.36) (cid:32) e p r q i , f p s q j ( “ ´ δ i,j r ` s ´ ÿ t “ d p r ` s ´ ´ t q i ` r d p t q i , (3.37) (cid:32) d p r q i , e p s q j ( “ p δ i,j ´ δ i,j ` q r ´ ÿ t “ d p t q i e p r ` s ´ ´ t q j , (3.38) (cid:32) d p r q i , f p s q j ( “ p δ i,j ` ´ δ i,j q r ´ ÿ t “ f p r ` s ´ ´ t q j d p t q i , (3.39) (cid:32) e p r q i , e p s q i ( “ s ´ ÿ t “ r e p t q i e p r ` s ´ ´ t q i if r ă s, (3.40) (cid:32) f p r q i , f p s q i ( “ r ´ ÿ t “ s f p r ` s ´ ´ t q i f p t q i if r ą s, (3.41) (cid:32) e p r ` q i , e p s q i ` ( ´ (cid:32) e p r q i , e p s ` q i ` ( “ e p r q i e p s q i ` , (3.42) (cid:32) f p r q i , f p s ` q i ` ( ´ (cid:32) f p r ` q i , f p s q i ` ( “ f p s q i ` f p r q i , (3.43) (cid:32) e p r q i , e p s q j ( “ | i ´ j | ą , (3.44) (cid:32) f p r q i , f p s q j ( “ | i ´ j | ą , (3.45) ! e p r q i , (cid:32) e p s q i , e p t q j () ` ! e p s q i , (cid:32) e p r q i , e p t q j () “ | i ´ j | “ , r ‰ s, (3.46) ! f p r q i , (cid:32) f p s q i , f p t q j () ` ! f p s q i , (cid:32) f p r q i , f p t q j () “ | i ´ j | “ , r ‰ s. (3.47)for all admissible i, j, r, s, t . In these relations, the notation d p q i “ r d p q i : “ r d p r q i for r ą r d p r q i : “ ´ r ÿ t “ d p t q i r d p r ´ t q i (3.48)In order to describe the structure of y n p σ q as a commutative algebra we make the notation e p r q i,i ` : “ e p r q i for 1 ď i ă n, s i,i ` ă r ;(3.49) f p r q i,i ` : “ f p r q i for 1 ď i ď n, s i ` ,i ă r, (3.50)and inductively define e p r q i,j : “ t e p r ´ s j ´ ,j q i,j ´ , e p s j ´ ,j ` q j ´ u for 1 ď i ă j ď n, s i,j ă r ;(3.51) f p r q i,j : “ t f p s j,j ´ ` q j ´ , f p r ´ s j,j ´ q i,j ´ u for 1 ď i ă j ď n, s j,i ă r. (3.52)The shifted Yangian admits a Poisson grading y n p σ q “ À r ě y n p σ q r , which we call the canonicalgrading . It places d p r q i , e p r q i , f p r q i in degree r and the bracket lies in degree ´
1, meaning t¨ , ¨u : y n p σ q r ˆ y n p σ q s Ñ y n p σ q r ` s ´ . There is also an important Poisson filtration y n p σ q “ Ť i ě F i y n p σ q ,called the loop filtration , which places d p r q i , e p r q i , f p r q i in degree r ´
1. Again the bracket is in degree ´ r y n p σ q ˆ F s y n p σ q Ñ F r ` s ´ y n p σ q . The associated graded Poisson algebra is denotedgr y n p σ q .The following theorem is a semiclassical analogue of [BK06, Theorem 2.1], which we ultimatelydeduce from the noncommutative setting. Theorem 3.4.
There is a Poisson isomorphism S p c n p σ qq „ ÝÑ gr y n p σ q defined by e i ; r ´ ÞÝÑ e p r q i ` F r ´ y n p σ q for ď i ă n, r P Z ą s i,i ` ; f i ; r ´ ÞÝÑ f p r q i ` F r ´ y n p σ q for ď i ă n, r P Z ą s i ` ,i ; d i ; r ´ ÞÝÑ d p r q i ` F r ´ y n p σ q for ď i ď n, r P Z ą . (3.53) As a consequence y n p σ q is isomorphic to the polynomial algebra on infinitely many variables (3.54) t d p r q i | ď i ď n, r ą u Y t e p r q i,j | ď i ă j ď n, r ą s i,j uY t f p r q i,j | ď i ă j ď n, r ą s j,i u . Proof.
Comparing the top graded components of the relations (3.36)–(3.47) with respect to theloop filtration, with relations (3.10)–(3.18), it is straightforward to see that (3.53) gives a surjectivePoisson homomorphism. To prove the theorem we demonstrate that the ordered monomials in theelements (3.54) are linearly independent.Consider the set X : “ t E p r q i , F p s q i , D p t q j | ď i ă n, ď j ď n, s i,i ` ă r, s i ` ,i ă s, ă t u . In [BK06, §
2] the shifted Yangian Y n p σ q is defined as a quotient of the free algebra C x X y by theideal generated by the relations [BK06, (2.4)–(2.15)]. Let L : “ L X be the free Lie algebra on X and define a grading L “ À i ě L i by placing E p r q i , F p r q i , D p r q i in degree r ´
1. Then we place afiltration on the enveloping algebra U p L q so that L i lies in degree i `
1. By the PBW theorem for U p L X q we see that gr U p L q – S p L q .The universal property of U p L q ensures that there is a surjective algebra homomorphism U p L q (cid:16) C x X y and by [Ser06, I, Ch. IV, Theorem 4.2] this is an isomorphism. Identifying these algebras,the filtration on U p L q descends to Y n p σ q “ Ť i ě F i Y n p σ q , and this resulting filtration is com-monly referred to as the canonical filtration [BK06, § Y n p σ q is equipped with a Poisson structure in the usual manner. Comparing relations (3.36)–(3.47) withthe top graded components of relations [BK06, (2.4)–(2.15)] we see that the Poisson surjection S p L q (cid:16) gr Y n p σ q factors through S p L q Ñ y n p σ q . As a result there is a surjective Poisson homo-morphism π : y n p σ q (cid:16) gr Y n p σ q given by e p r q i ÞÑ E p r q i ` F r ´ Y n p σ q , f p r q i ÞÑ F p r q i ` F r ´ Y n p σ q , d p r q i ÞÑ D p r q i ` F r ´ Y n p σ q . Following [BK06, (2.18), (2.19)] we introduce elements E p r q i,j , F p r q i,j of Y n p σ q lyingin filtered degree r . By the definition of the filtration and the elements (3.51), (3.52) we have π p e p r q i,j q “ E p r q i,j ` F r ´ Y n p σ q ; π p f p r q i,j q “ F p r q i,j ` F r ´ Y n p σ q . By [BK06, Theorem 2.3(iv)] the ordered monomials in E p r q i,j , F p r q i,j , D p r q i are linearly independent in Y n p σ q and so we deduce that the images of these monomials in gr Y n p σ q are linearly independent.This completes the proof. (cid:3) We record two formulas for future use, which are semiclassical analogues of [BT18, (4.32)]
Lemma 3.5.
The following hold for i “ , ..., n , j “ , ..., n ´ , r ą and s ą s j,j ` : t r d p r q i , e p s q j u “ p δ i,j ` ´ δ i,j q r ´ ÿ t “ r d p t q i e p r ` s ´ ´ t q j , (3.55) t r d p r q i , f p s q j ( “ p δ i,j ´ δ i,j ` q r ´ ÿ t “ f p r ` s ´ ´ t q j r d p t q i . (3.56) Proof.
We only sketch (3.55), as the proof of (3.56) is almost identical. The argument is byinduction based on (3.48). Note that r d p q i “ ´ d p q i and so (3.55) is equivalent to (3.38) in this case. We have t r d p r q i , e p s q j u “ t ř r ´ t “ d p r ´ t q i r d p t q i , e p s q j u and relation (3.38) together with the inductivehypothesis imply that the coefficient of e p s ` t q j is p δ i,j ` ´ δ i,j qp d p r ´ ´ t q i ` ř r ´ m “ t ` p r d p r ´ m q i d p m ´ t ´ q i ´ d p r ´ m q i r d p m ´ t ´ q i qq . Using d p q i “ r d p q i “ p δ i,j ` ´ δ i,j q r d p r ´ ´ t q i , which concludes the induction. (cid:3) The Dirac reduction of the shifted Yangian.
In this Section we suppose that σ issymmetric and even. Examining the relations (3.36)–(3.47) we see that there is unique Poissonautomorphism τ of y n p σ q determined by τ p d p r q i q : “ p´ q r d p r q i ; τ p e p r q i q : “ p´ q r ` s i,i ` f p r q i ; τ p f p r q i q : “ p´ q r ` s i ` ,i e p r q i . (3.57)Examining the relations (3.36)–(3.47) we see that τ extends to an involutive Poisson automorphismof y n p σ q . Our present goal is to give a complete description of the Dirac reduction R p y n p σ q , τ q .For i “ , ..., n and r P Z ą s i,i ` we write p e p r q i : “ e p r q i ` p´ q r ` s i,i ` f p r q i P y n p σ q ; q e p r q i : “ e p r q i ´ p´ q r ` s i,i ` f p r q i P y n p σ q , (3.58)so that p e p r q i and d p s q j are τ -invariants, whilst q e p r q i and d p s ´ q j each span a non-trivial representationof the cyclic group of order 2 generated by τ . We can recover e p r q i and f p r q i via e p r q i “ p p e p r q i ` q e p r q i q ; f p r q i “ p´ q r ` s i,i ` p p e p r q i ´ q e p r q i q . (3.59)Now let q y n p σ q be the ideal of y n p σ q generated by t q e p r q i , d p s ´ q j | i “ , ..., n ´ , j “ , ..., n, r P Z ą s i,i ` , s P Z ą u . Also write q y n p σ q τ : “ q y n p σ q X y n p σ q τ . By Lemma 2.2 we see that R p y n p σ q , τ q “ y n p σ q τ { q y n p σ q τ isPoisson generated by elements θ p r q i : “ p e p r q i ` q y n p σ q τ for i “ , ..., n ´ , r ą s i,i ` ,η p r q j : “ d p s q j ` q y n p σ q τ for i “ , ..., n, r ą , (3.60)with Poisson brackets induced by the bracket on y n p σ q . Furthermore R p y n p σ q , τ q is generated as acommutative algebra by elements t η p r q i | ď i ď n, r P Z ą u Y t θ p r q i,j | ď i ă j ď n, r ´ s i,j P Z ą u (3.61)where the θ p r q i,j : “ e p r q i,j ` p´ q r ` s i,j f p r q i,j ` q y n p σ q τ P R p y n p σ q , τ q . Using an inductive argument and(3.37) we see the elements θ p r q i,j can also be defined via the following recursion θ p r q i,i ` : “ θ p r q i for 1 ď i ă n, r ´ s i,j P Z ą θ p r q i,j : “ t θ p r ´ s j ´ ,j q i,j ´ , θ p s j ´ ,j ` q j ´ u for 1 ď i ă j ď n, r ´ s i,j P Z ą . (3.62)The next lemma can be deduced from (3.38), (3.39), (3.55), (3.56). Lemma 3.6.
The following equalities hold in R p y n p σ q , τ q : t d p r ´ q i , q e p s q j u ` q y n p σ q τ “ p δ i,j ´ δ i,j ` q r ´ ÿ t “ η p t q i θ p p r ´ t ´ q` s q j (3.63) t r d p r ´ q i , q e p s q j u ` q y n p σ q τ “ p δ i,j ` ´ δ i,j q r ´ ÿ t “ r η p t q i θ p p r ´ t ´ q` s q j (3.64)Thanks to Theorem 3.4 that R p y n p σ q , τ q comes equipped with the canonical grading R p y n p σ q , τ q “ À i ě R p y n p σ q , τ q i which places θ p r q i , η p r q i in degree r and the Poisson bracket in degree ´
1. An-other crucial feature is the loop filtration R p y n p σ q , τ q “ Ť i ě F i R p y n p σ q , τ q which places θ p r q i , η p r q i in degree r ´ ´
1. Both of these structures are naturally inherited from y n p σ q .The following is our main structural result on the Dirac reduction of the shifted Yangian. Theorem 3.7.
The Dirac reduction R p y n p σ q , τ q is Poisson generated by elements t η p r q i | ď i ď n, r P Z ą u Y t θ p r q i | ď i ă n, r ´ s i,i ` P Z ą u (3.65) together with the following relations t η p r q i , η p s q j u “ (cid:32) η p r q i , θ p s q j ( “ p δ i,j ´ δ i,j ` q r ´ ÿ t “ η p t q i θ p r ` s ´ ´ t q j (3.67) (cid:32) θ p r q i , θ p s q i ( “ for r ` s ´ odd (3.68) (cid:32) θ p r q i , θ p s q i ( “ s ´ ÿ t “ r θ p t q i θ p m ´ t q i ` p´ q s ` s i,i ` m ÿ t “ η p m ´ t q i ` r η p t q i for r ă s, r ` s ´ “ m (3.69) (cid:32) θ p r ` q i , θ p s q i ` ( ´ (cid:32) θ p r q i , θ p s ` q i ` ( “ θ p r q i θ p s q i ` (3.70) (cid:32) θ p r q i , θ p s q j ( “ for | i ´ j | ą ! θ p r q i , (cid:32) θ p s q i , θ p t q j () ` ! θ p s q i , (cid:32) θ p r q i , θ p t q j () “ for | i ´ j | “ and r ` s odd (3.72) ! θ p r q i , (cid:32) θ p s q i , θ p t q j () ` ! θ p s q i , (cid:32) θ p r q i , θ p t q j () “ (3.73) 2 p´ q s ` s i,i ` ´ δ i,j ` m ´ ÿ m “ m ÿ m “ η p p m ´ m ´ qq i ` r η p m q i θ p p m ´ m q` t q j ` p´ q s ` s i,i ` ´ δ i ` ,j m ´ ÿ m “ m ´ m ´ ÿ m “ r η p m q i η p m q i ` θ p p m ´ m ´ m ´ q` t q j for | i ´ j | “ and r ` s “ m even where we adopt the convention η p q i “ r η p q i “ and the elements t r η p r q i | r P Z ě u are defined viathe recursion r η p r q i : “ ´ r ÿ t “ η p t q i r η p r ´ t q i . (3.74) Proof.
First we show that the elements r η p r q i : “ r d p r q i ` q y n p σ q τ P R p y n p σ q , τ q satisfy the recursion(3.74). By Theorem 3.4 the subalgebra of y n p σ q generated by t d p r q i | i “ , ..., n, r ą u is a gradedpolynomial ring with d p r q i in degree r . By induction r d p r q i lies in degree r . This forces r d p r ´ q i P q y n p σ q for all r ą q y n p σ q τ , confirming the claim.We now deduce relations (3.66)–(3.73) from (3.36)–(3.47). First of all (3.66) follows immediatelyfrom (3.36). By (3.38) and (3.39) we have t d p r q i , p e p s q j u “ p δ i,j ´ δ i,j ` q r ´ ÿ t “ d p t q i ´ e p r ` s ´ ´ t q j ´ p´ q s ` s j,j ` f p r ` s ´ ´ t q j ¯ “ p δ i,j ´ δ i,j ` q ´ r ´ ÿ t “ d p t q i p e p r ` s ´ ´ t q j ` r ´ ÿ t “ d p t ` q i q e p r ` s ´ t ´ q j ¯ . Projecting to R p y n p σ q , τ q “ y n p σ q τ { q y n p σ q τ we see that the elements (3.60) satisfy (3.67).Note that (3.37) implies that t e p r q i , f p s q i u “ t e p s q i , f p r q i u for r, s ą s i,i ` . (3.75)Together with (3.40) and (3.41) we deduce for all r ă s t p e p r q i , p e p s q i u “ t e p r q i , e p s q i u ` p´ q r ` s t f p r q i , f p s q i u ` p´ q s ` s i,i ` t e p r q i , f p s q i u ´ p´ q r ` s i,i ` t e p s q i , f p r q i u“ s ´ ÿ t “ r e p t q i e p r ` s ´ ´ t q i ´ s ´ ÿ t “ r f p r ` s ´ ´ t q i f p t q i ´ pp´ q s ` s i,i ` ´ p´ q r ` s i,i ` qt e p r q i , f p s q i u Substituting in (3.59) we see that e p t q i e p r ` s ´ ´ t q i ´ f p r ` s ´ ´ t q i f p t q i “ p ´ p´ q r ` s ´ q p e p t q i p e p r ` s ´ ´ t q i ` q y n p σ q τ . Combining with (3.37) we have now deduced (3.68) and (3.69).Using (3.2), (3.37), (3.42), (3.43) we calculate t p e p r ` q i , p e p s q i ` u ´ t p e p r q i , p e p s ` q i ` u “ t e p r ` q i , e p s q i ` u ´ t e p r q i , e p s ` q i ` u`p´ q r ` s ` ` s i,i ` pt f p r ` q i , f p s q i ` u ´ t f p r q i , f p s ` q i ` uq“ e p r q i e p s q i ` ` p´ q r ` s ` s i,i ` f p s q i ` f p r q i Substituting in (3.59) we see that this is equal to the right hand side of (3.70) modulo q y n p σ q τ . Finally take | i ´ j | “
1. Expanding in terms of generators (3.35) and applying relations (3.37),(3.46), (3.47) together with the Jacobi identity and (3.75) we obtain t p e p r q i , t p e p s q i , p e p t q j uu ` t p e p s q i , t p e p r q i , p e p t q j uu“ p´ q s ` t ` s i,i ` ` s j,j ` t e p r q i , t f p s q i , f p t q j uu ` p´ q r ` t ` s i,i ` ` s j,j ` t e p s q i , t f p r q i , f p t q j uu`p´ q r ` s i,i ` t f p r q i , t e p s q i , e p t q j uu ` p´ q s ` s i,i ` t f p s q i , t e p r q i , e p t q j uu“ p´ q s ` t ` s i,i ` ` s j,j ` p ` p´ q r ` s qtt e p r q i , f p s q i u , f p t q j u ´ p´ q r ` s i,i ` p ` p´ q r ` s qtt e p r q i , f p s q i u , e p t q j u If r ` s is odd this vanishes, which proves (3.72). Now assume that r ` s “ m is even. Using(3.37) and Jacobi the last line of the previous equation reduces to2 p´ q s ` s i,i ` ´ (cid:32) t e p r q i , f p r q i u , q e p t q j ( p´ q s ` s i,i ` ´ (cid:32) m ´ ÿ m “ d p m ´ ´ m q i ` r d p m q i , q e p t q j ( “ p´ q s ` s i,i ` ´ m ´ ÿ m “ ` d p m ´ ´ m q i ` t r d p m q i , q e p t q j u ` t d p m ´ ´ m q i ` , q e p t q j u r d p m q i ˘ . Using the fact that d p r ´ q i P q y n p σ q τ for all i, r we simplify this expression modulo q y n p σ q τ to get2 p´ q s ` s i,i ` ´ ˜ m ´ ÿ m “ d p m ´ m ´ q i ` t r d p m ` q i , q e p t q j u ` m ´ ÿ m “ t d p p m ´ m q´ q i ` , q e p t q j u r d p m q i ¸ ` q y n p σ q τ Finally using (3.63) and (3.64) this expression coincides with the right hand side of (3.73).We have shown that the generators (3.65) satisfy (3.66)–(3.73), and it remains to show thatthese are a complete set of relations.Let p R p y n p σ q , τ q denote the Poisson algebra with generators (3.65) and relations (3.66)–(3.73).We have shown that there is a Poisson homomorphism p R p y n p σ q , τ q (cid:16) R p y n p σ q , τ q sending theelements (3.65) to the elements (3.60) with the same names. To complete the proof we show thatthis map sends a spanning set to a basis.We define a loop filtration on p R p y n p σ q , τ q “ Ť i ě F i p R p y n p σ q , τ q by placing θ p r q i , η p r q i in degree r ´ ´
1. Examining the top filtered degree pieces of the relations(3.66)–(3.73) with respect to the loop filtration, we see from Theorem 3.3 that there is a surjectivePoisson homomorphism S p c n p σ q τ q (cid:16) gr p R p y n p σ q , τ q . We deduce that gr p R p y n p σ q , τ q is generatedas a commutative algebra by elements θ p r q i,j ` F r ´ p R p y n p σ q , τ q , η p r q i ` F r ´ p R p y n p σ q , τ q with indexesvarying in the same ranges as (3.61). Again the elements θ p r q i,j P p R p y n p σ q , τ q are defined via therecursion (3.62). By a standard filtration argument we deduce that the elements of p R p y n p σ q , τ q with the same names as (3.61) generate p R p y n p σ q , τ q as a commutative algebra. We have shownthat p R p y n p σ q , τ q (cid:16) R p y n p σ q , τ q maps a spanning set to a basis. This completes the proof. (cid:3) Let gr R p y n p σ q , τ q denote the graded algebra for the loop filtration. In the last paragraph of theproof of Theorem 3.7 we obtained the following important result. Corollary 3.8.
There is a Poisson isomorphism S p c n p σ q τ q „ ÝÑ gr R p y n p σ q , τ q given by θ i ; r ´ ÞÝÑ θ p r q i ` F r ´ y n p σ q for ď i ă n, r P Z ą s i,i ` ; η i ;2 r ´ ÞÝÑ η p r q i ` F r ´ y n p σ q for ď i ď n, r P Z ą . (3.76) 4. Finite W -algebras for classical Lie algebras In this section C continues to be an algebraically closed field of characteristic zero, and we fixthe following additional notation: ‚ N ą ε P t˘ u such that ε N “ ‚ g “ gl N p C q and g Ď g is a classical Lie subalgebra such that g – so N if ε “ , sp N if ε “ ´ . ‚ G “ GL N p C q and G Ď G is the connected algebraic subgroup satisfying g “ Lie p G q . ‚ κ : g ˆ g Ñ C is the trace form associated to the natural representation C N of G .4.1. Partitions for nilpotent orbits in classical Lie algebras.
We refer to the G -orbits in g consisting of nilpotent elements as nilpotent orbits . Nilpotent G -orbits in g are classified bypartitions, corresponding to the sizes of Jordan blocks, and for each nilpotent G -orbit O λ Ď g theintersection O λ X g is a union of either one or two nilpotent G -orbits. Therefore an approximateclassification of nilpotent G -orbits is achieved by describing the partitions λ $ N for which O λ X g ‰ H . The set of such partitions is denoted P ε p N q , and they are characterised as follows. Lemma 4.1.
Let λ “ p λ , ..., λ n q $ N . Then λ P P ε p N q if and only if there exists an involution i ÞÑ i on the set t , ..., n u such that: (1) λ i “ λ i for all i “ , ..., n ; (2) i “ i if and only if (cid:15) p´ q λ i “ ; (3) i P t i ´ , i, i ` u . (cid:3) Whenever we choose λ P P ε p N q we will assume that a choice of involution has been fixed inaccordance the Lemma. We also adopt the convention that our partitions are non-decreasing λ ď ¨ ¨ ¨ ď λ n . For completeness we mention that λ P P ε p N q corresponds to one orbit unless ε “ λ are even, which case there are two G -orbits of nilpotent elements of g withJordan blocks given by λ and these are permuted by the outer automorphism group of g , see[CM93, Section 5.1] for example.4.2. The Dynkin pyramid and the centraliser.
Choose λ “ p λ , ..., λ n q P P p N q . We recalledthe notion of a shift matrix in Section 3.2, following [BK06]. The symmetric shift matrix for λ isthe even, symmetric shift matrix σ “ p s i,j q ď i,j ď n defined by s i,j : “ | λ i ´ λ j | ď i ď n and 1 ď j ď λ i we let b i,j denote the j th box in the i th row and we identify C N with the vector space spanned by the symbols b i,j . The general linear Lie algebra g “ gl N has a basis(4.2) t e i,j ; k,l | ď i, k ď n, ď j ď λ i , ď l ď λ k u where e i,j ; k,l b r,s “ δ k,r δ l,s e i,j ; r,s , so that r e i ,j ; k ,l , e i ,j ; k ,l s “ δ k ,i δ l ,j e i ,j ; k ,l ´ δ k ,i δ l ,j e i ,j ; k ,l (4.3)We pick the nilpotent element in g p q by the rule e : “ n ÿ i “ λ i ´ ÿ j “ e i,j ; i,j ` (4.4)It has Jordan blocks of sizes λ , ..., λ n . We also define a semisimple element h P gl N by h : “ n ÿ i “ λ i ÿ j “ p λ i ´ ´ j q e i,j ; i,j (4.5)It is easy to see that the pair t e, h u can be completed to an sl triple t e, h, f u . We refer to thegrading g “ À i P Z g p i q induced by ad p h q as the Dynkin grading for e . It is well-known that this isa good grading in the sense of [BG07]. It satisfiesdeg p e i,j ; k,l q “ p l ´ j q ` λ i ´ λ k . (4.6)For 1 ď i, k ď n and r “ s i,k , s i,k ` , ..., s i,k ` min p λ i , λ k q ´
1, we define elements c p r q i,k “ ÿ r “ p l ´ j q` λ i ´ λ k e i,j ; k,l (4.7)The following fact is well known. See [BK06, Lemma 7.3] where the notation c p r q i,j differs fromours by a shift in r . Lemma 4.2.
The centraliser g e has basis t c p r q i,k | ď i, k ď n, r ´ s i,k “ , , ..., min p λ i , λ k q ´ u (4.8) with Lie bracket (4.9) r c p r q i,j , c p s q k,l s “ δ j,k c p r ` s q i,k ´ δ i,l c p r ` s q k,j . Furthermore g e is a Dynkin graded Lie subalgebra with c p r q i,k lying in degree r . (cid:3) Symplectic and orthogonal subalgebras.
In this Section we assume that λ “ p λ , ..., λ n q P P ε p N q and that the involution on t , ..., n u coming from Lemma 4.1 is trivial.Consider the matrix J : “ n ÿ i “ λ i ÿ j “ p´ q j e i,j ; i,λ i ` ´ j (4.10)This block diagonal matrix can be described as follows. For each index i “ , ..., n there is a blockof size λ i , each of these blocks has entries ˘ ε “ J is symmetric and when ε “ ´ J ´ “ εJ. (4.11) Now define an involution of gl N by the rule τ : X ÞÑ ´ J ´ X J J (4.12) Lemma 4.3.
For all admissible indexes we have: (i) τ p e i,j ; k,l q “ p´ q j ´ l ´ e k,λ k ` ´ l ; i,λ i ` ´ j ; (ii) τ p e q “ e and τ p h q “ h ; (iii) τ p c p r q i,k q “ p´ q r ´ λk ´ λi ´ c p r q k,i . Proof.
Using (4.11) and multiplying matrices we have τ p e i,j ; k,l q “ ´ ε p´ q λ k ` j ´ l e k,λ k ` ´ l ; i,λ i ` ´ j and so (i) follows from Lemma 4.1(2). Now (ii) and (iii) follow by applying (i) to (4.4), (4.5) and(4.7). (cid:3) We decompose g “ g ‘ g into eigenspaces for τ so that g is the space of τ -invariants. Thenwe have g – so N if ε “ sp N if ε “ ´ . (4.13)By Lemma 4.3(ii) we have t e, h u Ď g , and g is a g -module. Write G for the connectedcomponent of the group of elements g P GL N such that gJ g J “ J . This is a classical groupsatisfying Lie p G q “ g .As τ preserves g p´ q we also have that g p´ q “ g p´ q ‘ g p´ q . Writing f “ f ` f with f P g p´ q and f P g p´ q and using the fact that h P g we deduce that h “ r e, f s and r e, f s “ e in g this yields f “
0. We conclude that the sl -triple t e, h, f u is contained in g .Our next result explains the relationship between the shifted current algebras and the centralisersdescribed above. Lemma 4.4. (1)
There is a surjective Lie algebra homomorphism c n p σ q Ý (cid:16) g e ; e i,j t r ÞÝÑ c p r q i,j where c p r q i,j : “ for r ě s i,j ` λ . The kernel is Poisson generated by e , t r with r ě λ . (2) If σ is even and symmetric the homomorphism from (1) restricts to c n p σ q τ (cid:16) g e . (i) For ε “ the kernel is Poisson generated by t η r ´ | r ´ ě λ u . (ii) For ε “ ´ the kernel is Poisson generated by those same elements along with t θ i ; λ i ` s i,i ` | i “ , ..., n ´ u . Proof.
Part (1) is explained in [GT19c, Lemma 2.6]. For the first claim in part (2) compare (3.22)with Lemma 4.3(iii) to see that the map is τ -equivariant.We go on to describe the kernel in (2). First of all observe that the elements listed there are τ -fixed elements of the kernel of c n p σ q (cid:16) g e ; for ε “ ´ g e has a spanning set consisting of elements of the form c p r q i,j ` τ p c p r q i,j q . By Lemma 4.2 we can show thatthe elements in (2) generate the kernel by checking that the ideal i which they generate contains t η i ; r | i “ , ..., n, r ě λ i u Y t θ i ; r | i “ , .., n ´ , r ě s i,i ` ` λ i u . (4.14) First take ε “ λ is odd, by Lemma 4.1. Using relation (3.67) we see that i contains t η λ , θ r u “ θ λ ` r for r ě s , , whilst (3.70) shows that η λ ` r ´ η λ ` r P i for all r ě
0. Inparticular η λ ` r P i for r ě
0. By Theorem 3.3 the subalgebra of c n p σ q τ generated by θ i,r , η i ; r with 2 ď i is isomorphic to a current Lie algebra of smaller rank so the description of the kernelfollows by induction.For ε “ ´ λ is even and the ideal i contains t η λ ` , θ s u “ θ λ ` ` s for s ě s , . Togetherwith θ λ ` s , this gives all elements of the form θ r listed in (4.14). Now the induction proceedsin the same was as the case ε “ (cid:3) Generators of the W -algebra. Here we recall formulas for generators of S p gl N , e q , due toBrundan and Kleshchev [BK06]. Their notation is slightly different, however it is a simple exerciseto translate between the two settings. In this Section we continue to assume that the involutionon t , ..., n u coming from Lemma 4.1 is trivial. This implies that λ is even.Now for 1 ď i, j ď n , 0 ď x ă n and r ą
0, we let(4.15) t p r q i,k ; x : “ r ÿ s “ p´ q r ´ s ÿ p i m ,j m ,k m ,l m q m “ ,...,s p´ q q “ ,...,s ´ | k q ď x u e i ,j ; k ,l ¨ ¨ ¨ e i s ,j s ; k s ,l s P S p g pě qq where the sum is taken over all indexes such that for m “ , ..., s ď i , ..., i s , k , ..., k s ď n, ď j m ď λ i m and 1 ď l m ď λ k m (4.16)satisfying the following six conditions:(a) ř sm “ p l m ´ j m ` λ i m ´ λ k m q “ p r ´ s q ;(b) 2 l m ´ j m ` λ i m ´ λ k m ě m “ , . . . , s ;(c) if k m ą x , then l m ă j m ` for each m “ , . . . , s ´ k m ď x then l m ě j m ` for each m “ , . . . , s ´ i “ i , k s “ k ;(f) k m “ i m ` for each m “ , . . . , s ´ r ą s P t , ..., r u we make the notation X p r,s q i,k for the set of orderedsets p i m , j m , k m , l m q sm “ in the range (4.16) satisfying the conditions (a)–(f) and consider the map υ p i m , j m , k m , l m q : “ p k s ` ´ m , λ k s ` ´ m ` ´ l s ` ´ m , i s ` ´ m , λ i s ` ´ m ` ´ j s ` ´ m q (4.17)defined on ordered sets in the range (4.16). Lemma 4.5.
The map υ defines a bijection X p r,s q i,k - ÝÑ X p r,s q k,i . Proof.
The index bounds (4.16) ensure that X p r,s q i,k is finite. Since υ is the identity map it willsuffice to show that υ is well defined X p r,s q i,k Ñ X p r,s q k,i . We fix p i m , j m , k m , l m q sm “ P X p r,s q i,k and checkthat conditions (a)–(f) hold for υ p i m , j m , k m , l m q . For example, s ÿ m “ ` p λ i s ` ´ m ` ´ j s ` ´ m q ´ p λ l s ` ´ m ` ´ k s ` ´ m q ` p λ k s ` ´ m ´ λ i s ` ´ m q ˘ “ s ÿ m “ p l m ´ j m ` λ i m ´ λ k m q “ p r ´ s q This shows that (a) for p i m , j m , k m , l m q sm “ implies (a) for υ p i m , j m , k m , l m q sm “ . Conditions (b)–(f)can be checked similarly. (cid:3) Now we define a map θ : g e Ñ S p g pě qq by c p r q i,i ÞÝÑ t p r ´ q i,i ; i ´ for i “ , ..., n and 0 ď r ă λ i ; c p r q i,k ÞÝÑ t p r ´ q i,k ; i for 1 ď i ă k ď n and r ´ s i,k “ , ..., min p λ i , λ k q ´ c p r q k,i ÞÝÑ t p r ´ q k,i ; i for 1 ď i ă k ď n and r ´ s j,k “ , ..., min p λ i , λ k q ´ . (4.18)Comparing (4.6) with Lemma 4.2 and Lemma 4.3 we see that τ induces involutions on g e and S p g pě qq , and these will all be denoted τ . The hardest part of the following result is the assertionthat the image of the map θ described in (4.18) lies in S p g , e q , which follows from a result ofBrundan and Kleshchev. We add to this the observation that θ is τ -equivariant. Proposition 4.6. θ : g e Ñ S p g , e q is τ -equivariant. In particular we have τ p t p r ´ q i,k ; m q “ p´ q r ´ λk ´ λi ´ t p r ´ q k,i ; m for ď i ď k ď n and all m, r. (4.19) Proof.
The assertion that the image of θ lies in S p g , e q is an immediate consequence of [BK06,Corollary 9.4], upon taking the top graded term with respect to the Kazhdan filtration of thefinite W -algebra (see also [GT19b, §
4] for a short summary). Thanks to Lemma 4.3(iii) the τ -equivariance will follow from (4.19). Fix indexes 1 ď i ď k ď n, ď x ă n and 0 ă r . Choose s P t , ..., r u . Let p i m , j m , k m , l m q sm “ P X p r,s q i,i . Thanks to Lemma 4.3(i) we have τ p s ź m “ e i m ,j m ; k m ,l m q “ p´ q ř sm “ p j m ´ l m ` q s ź t “ e υ p i m ,j m ,k m ,s m q (4.20)Conditions (a), (e) (f) imply that ř m p j m ´ l m ` q “ λ i ´ λ ks ´ r. Hence the sign on the right handside of (4.20) is equal to p´ q r ´ λk ´ λi . Furthermore condition (f) ensures that m “ , ..., s ´ | k m ď x u “ m “ , ..., s | i m ď x u . Combining these observations together with Lemma 4.5 wehave τ p t p r q i,k ; x q “ r ÿ s “ p´ q r ´ s ÿ p i m ,j m ,k m ,l m q m P X p r,s q i,k p´ q m “ ,...,s ´ | k m ď x u τ p ź m e i m ,j m ,k m ,l m q“ p´ q r ´ λk ´ λi r ÿ s “ p´ q r ´ s ÿ p i m ,j m ,k m ,l m q m P X p r,s q k,i p´ q m “ ,...,s ´ | k m ď x u ź m e i m ,j m ,k m ,l m “ p´ q r ´ λk ´ λi t p r q k,i ; i ´ This completes the proof. (cid:3)
Remark . Comparing the linear terms appearing in (4.15) with the basis (4.7) for g e one cancheck that θ is a Poincar´e–Birkhoff–Witt map satisfying the properties of Theorem 2.4.4.5. The semiclassical Brundan–Kleshchev homomorphism.
Recall that in (3.57) we de-fined a Poisson involution of y n p σ q which is denoted τ . Proposition 4.8.
There is a unique τ -equivariant surjective Poisson homomorphism ϕ : y n p σ q Ý (cid:16) S p g , e q (4.21) determined by d p r q i ÞÝÑ t p r q i,i ; i ´ for ď i ď n, r P Z ą ; e p r q i ÞÝÑ t p r q i,i ` i for ď i ă j ď n, r ´ s i,j P Z ą ; f p r q i ÞÝÑ t p r q i ` ,i ; i for ď i ă j ď n, r ´ s j,i P Z ą . (4.22) The kernel is the Poisson ideal generated by t d p r q | r ą λ u . The canonical grading on y n p σ q inTheorem 3.4 corresponds to the Kazhdan grading on S p g , e q .Proof. The fact that (4.22) defines a graded Poisson homomorphism follows from [BK06, The-orem 10.1] upon taking the associated graded map with respect to the canonical and Kazhdanfiltrations. The τ -equivariance can be checked by comparing Proposition 4.6 with formulas (3.57).Another consequence of [BK06, Theorem 10.2] is that the elements t d p r q | r ą λ u lie in the kernelof the homomorphism. Write I for the Poisson ideal generated by these elements. CombiningTheorem 2.4 and Lemma 4.2 we can show that I “ Ker ϕ by demonstrating that the quotient y n p σ q{ I is generated as a commutative algebra by t e p r q i,j | ď i ă j ď n, s i,j ă r ď s i,j ` λ min p i,j q u Y t d p r q i | ď i ď n, ă r ď λ i uYt f p r q i,j | ď i ă j ď n, s j,i ă r ď s j,i ` λ min p i,j q u . (4.23)By Theorem 3.4 we can identify S p c n p σ qq “ gr y n p σ q as Poisson algebras, with respect to theloop filtration. The associated graded ideal gr I contains elements t d i ; r | r ě λ u , and so applyingLemma 4.4(1) we see that gr y n p σ q{ I is generated by by the top graded components of the elements(4.23). Applying a standard argument on filtered algebras we deduce that y n p σ q{ I is generated bythe requisite elements. (cid:3) Recall that the Dirac reduction functor is right exact. Applying Theorem 2.6 we obtain anapproximate description of the W -algebra for g . Theorem 4.9.
There is a surjective Poisson homomorphism ϕ : R p y n p σ q , τ q (cid:16) S p g , e q . ‚ If ε “ the kernel of ϕ is Poisson generated by t η p r q | r ą λ u . ‚ If ε “ ´ the kernel of ϕ is Poisson generated by ! η p r q i | r ą λ ) Y ! λ i { ÿ t “ η p t q i θ p λ i ´ t ` s , ` q i | i “ , ..., n ´ ) . (4.24) Proof.
The elements d p r q with 2 r ą λ lie in the kernel of y n p σ q Ñ S p g , e q and project to η p r q under y n p σ q τ (cid:16) R p y n p σ q , τ q . Similarly for ε “ ´ t d p λ i ` q i , q e p s q i u ` y n p σ q τ “ ř λ i { t “ η p t q i θ p λ i ´ t ` s , ` q i lie in the kernel.To complete the proof we use a loop filtration argument identical to the proof of Proposition 4.8to show that the ideal generated by the stated elements is large enough. For this one can applyCorollary 3.8 and Lemma 4.4(2), and we leave the details to the reader (cid:3) Thanks to Lemma 4.2 the Lie algebra g is spanned by elements c p r q i,j ` τ p c p k q i,j q . Using (4.15) alongwith Theorem 2.4 and Theorem 4.9 we see that S p g , e q is generated (as a commutative algebra)by the images of elements t η p r q i | ď i ď n, ď r ď λ i u Y t θ p r q i,j | ď i ă j ď n, s i,j ă r ď s i,j ` λ i u (4.25) By slight abuse of notation we denote the images by the same names.
Corollary 4.10. If m is the unique graded maximal ideal of S p g , e q then we have a Poissonisomorphism S p g e q „ ÝÑ gr m S p g , e q to the m -adic graded algebra defined by c p r q i,i ` ` τ p c p r q i,i ` q ÞÝÑ θ p r ` q i ` m for ď i ă n, s i,j ď r ă s i,j ` λ i ;(4.26) c p r q i,i ` τ p c p r q i,i q ÞÝÑ η p r ` q i ` m for ď i ď n, ď r ă λ i . (4.27) Proof.
The graded algebra gr m S p g , e q is Poisson, and it is straightforward to check, using The-orem 3.7, that (4.26) gives a Lie algebra homomorphism g e (cid:16) gr m S p g , e q . By the universalproperty of the Lie–Poisson algebra we get a Poisson homomorphism S p g e q (cid:16) gr m S p g , e q . ByTheorem 2.4 we see that this map is an isomorphism. (cid:3) Part One dimensional representations of W -algebras Generalities on Poisson schemes
Conic degenerations of affine schemes.
We begin the second Part of the paper by record-ing two basic results, allowing us to compare the number of irreducible components of certain com-plex schemes of finite type. The first records an obvious bound on the dimensions of componentsof subschemes, whilst the second allows us to bound the number of components of a deformationof a reduced conic affine scheme.We define P be the set of all finite sequences of all non-negative integers of arbitrary length.We define a dominance (partial) order on P as follows. If d “ p d , ..., d n q and d “ p d , ..., d m q thenwe say that d ą d if there is an index j “ , ..., min p n, m q such that d i “ d i for i “ , ..., j and d j ` ą d j ` . Here we adopt the convention d m ` “ d n ` “ X is a complex scheme of finite type then we write X “ Ť li “ X i for the decomposition intoirreducible components, ordered so that dim X i ě dim X j for i ă j . We define the dimensionvector of X to be the sequence d p X q : “ p dim X , dim X , ..., dim X l q P P .The following useful fact can be checked by considering the dimension vector of all proper closedsubschemes of a scheme X . Lemma 5.1.
Let
X, Y be complex schemes of finite type with a closed embedding Y Ñ X . Then d p X q ě d p Y q with equality if and only if the underlying reduced schemes are isomorphic. Let A be a finitely generated C -algebra. We say that A “ Ť i ě F i A is a standard filtration of A if there is a surjection C r x , ..., x n s (cid:16) A for some n , along with integers m , ..., m n ě F m A is spanned by the image of t x k ¨ ¨ ¨ ` x k n n | k m ` ¨ ¨ ¨ ` k n m n ď m u . If X “ Spec p A q is theaffine scheme the we refer to C X “ Spec p gr A q as the asymptotic cone of X . Lemma 5.2. If C X is reduced then Comp p X q ď Comp p C X q .Proof. If gr A is reduced then so is A . We begin by choosing a presentation for the standardfiltration. Let R “ C r x , ..., x n s “ À i ě R i be a graded polynomial ring with x i in degree m i . WriteF i R “ À ij “ R j . We let φ : R Ñ A be the homomorphism inducing the standard filtration, set I : “ Ker p φ q and denote the minimal prime ideals over I by p , ..., p m Ď A , so that I “ Ş i p i . Using [MR01, Proposition 7.6.13], for example, we see that there is a natural isomorphism gr A – R { gr I and we view X and C X as subschemes of Spec R “ A n .For g P F i R we write ¯ g “ g ` F i ´ R P gr R . For any ideal J of R we write V p J q Ď A n for thecorresponding variety of closed points. We have inclusions gr p J q gr p K q Ď gr p J K q Ď gr p J q X gr p K q for any ideals J, K Ď R (see [Pr02, § V p gr I q “ m ď i “ V p gr p i q . (5.1)We prove the contrapositive of the lemma, so assume that m “ Comp p X q ą Comp p C X q .Then, a fortiori, we must have V p gr p j q Ď Ť i ‰ j V p gr p i q for some j , say j “
1. Equivalently Ş i ‰ ? gr p i Ď ? gr p , which implies Ş i ‰ gr p i Ď ? gr p . Since I Ĺ p p ¨ ¨ ¨ p m it follows thatgr I Ĺ gr p p ¨ ¨ ¨ p m q and so we may choose g , ..., g m with g i P p i such that g : “ g ¨ ¨ ¨ g m satisfies¯ g R gr I . On the other hand, we have ¯ g P gr p p ¨ ¨ ¨ p m q Ď Ş mi “ gr p p i q Ď Ş mi “ a gr p p i q “ a gr p I q ,where the last equality follows from (5.1). Thus we conclude that ¯ g P a gr p I qz gr I , so that C X isnot reduced. This completes the proof. (cid:3) Completions and nilpotent elements of graded algebras. If p P Spec p A q then, asusual, A p denotes the localisation and A ^ p the completion at the maximal ideal of A p . The kernelof A Ñ A p is the set of elements annihilating some element of A z p , whilst the kernel of A Ñ A ^ p is Ş k ą p k . When p is a maximal ideal it follows from Krull’s intersection theorem that these kernelsare equal (see the remark following [AM69, Theorem 10.17]).Now let A “ À i ě A i be a finitely generated, connected graded algebra with unique maximalgraded ideal m . The connected grading induces a one parameter family of automorphisms of A ,and a C ˆ -action on m-Spec p A q contracting to the unique fixed point m . For a P A make thenotation Ξ a : “ ! m P m-Spec p A q | a R Ker p A Ñ A ^ m q ) . Lemma 5.3.
The following hold: (1) Ş m Ş i ą m i “ where the intersection is taken over all maximal ideals of A . (2) If a P A i for some i then Ξ a is closed and conic for the contracting C ˆ -action.Proof. Since A is finitely generated over C the intersection Ş m P m-Spec p A q m “ Rad p A q is a gradedideal, however for every i ą A i X m i ` “ m P m-Spec p A q , we remarked above that the kernel of A Ñ A ^ m is equal to the kernel of A Ñ A m . Writing Ann p a q “ t b P A | ab “ u , it follows that Ker p A Ñ A ^ m q “ t a P A | ab “ b R m u “ t a P A | Ann p a q Ę m u . We have shown that Ξ a “ t m P m-Spec p A q | Ann p a q Ď m u , which is conic and closed since Ann p a q is a graded ideal (see [Pr02, 5.3] for more detail). (cid:3) Lemma 5.4.
Suppose that A “ À i ą A i is a finitely generated, connected graded algebra withunique maximal graded ideal m . The following are equivalent: (i) A is reduced. (ii) A ^ m is reduced. (iii) A ^ m is reduced for every maximal ideal m P Spec p A q .Furthermore if gr m A is reduced then these equivalent conditions hold. Proof.
We prove (ii) ñ (i) by contraposition. So suppose that 0 ‰ f P Rad p A q is a nonzeroelement. Without loss of generality we may assume f is homogeneous. By Lemma 5.3(1) we haveKer p A Ñ ś m P m-Spec p A q A ^ m q “
0, which means that f maps to a nonzero element of A ^ m for some m ,implying m P Ξ f . Using Lemma 5.3(2) we see Ξ f is conic and closed, so it must contain m , whichimplies that A ^ m has nilpotent elements. This proves (ii) ñ (i).To see (i) ñ (iii) we first of all observe that the property of being reduced is preserved bylocalisation for any commutative ring, and then apply [Ma86, §
32, Remark 1]. Clearly (iii) ñ (ii).We prove that that if A is non-reduced then gr m A is non-reduced. Let 0 ‰ f P Rad p A q and suppose (without loss of generality) that f is homogeneous. There are two cases: either f P m i z m i ` for some i ą
0, or f P Ş i ą m i . In the first case f ` m i ` is a nonzero nilpotentelement of gr m A , which proves that gr m A is non-reduced.We now show that the second case cannot happen. Suppose f P Ş i ą m i “ Ker p A Ñ A ^ m q , orequivalently, m R Ξ f . By Lemma 5.3(2) we must have Ξ f “ H which implies that f P Ker p A Ñ A ^ m q for all maximal ideals m . By Lemma 5.3(1) we see f P Ş m Ş i ą m i “
0. This contradictioncompletes the proof. (cid:3)
Poisson schemes. If A is a Poisson algebra then the abelianisation A ab is the largest Poissoncommutative quotient of A . It is constructed by factoring by the ideal generated by t A, A u .When X is a complex manifold it can be stratified into immersed submanifolds known as sym-plectic leaves (see [We83] for an introduction). Therefore the closed points of a regular complexPoisson scheme can be decomposed into a disjoint union of symplectic leaves. More generally theclosed points of a singular affine Poisson scheme can be decomposed into symplectic leaves by aniterative procedure [BG03, 3.5].The stratification by leaves can be coarsened to a stratification by rank, which is especiallytransparent in the case of affine schemes: if A is finitely generated by elements x , ..., x n andcarries a Poisson structure, then we can form the matrix π “ pt x i , x j uq ď i,j ď n and consider thesubschemes X Ď X Ď ¨ ¨ ¨ X where X k is defined by the ideal of A generated by all p k ` qˆp k ` q minors of π .It follows from [BG03, Proposition 3.6] that the locally closed set X k z X k ´ is the union of allsymplectic leaves of dimension k , and it is known as the k th rank stratum . In particular X isthe union of symplectic leaves of dimension zero. Note that X “ Spec p A ab q by definition. Theseobservations prove the following lemma. Lemma 5.5.
The following reduced subschemes of X coincide: (1) Spec p A ab q red ; (2) The union of all zero dimensional symplectic leaves of
Spec p A q . (cid:3) If A is a Poisson algebra and I is an ideal then the completion A ^ I can be equipped with a Poissonstructure in a unique way such that A Ñ A ^ I is Poisson. In particular, for a : “ p a i ` I i q i ą , b : “p b i ` I i q i ą P A ^ I Ď ś i P Z ą A { I i we define t a, b u “ p c i ` I i q i ą where c i “ t a j , b k u for any choiceof j, k ě i `
1. The definition of A ^ I ensures that this does not depend on j, k modulo I i . When I is Poisson this structure has an easier description: here every A { I i is a Poisson algebra and thebracket on A ^ I coincides with the inverse limit in the category of Poisson algebras.The next lemma states that Poisson abelianisation and completion commute. Lemma 5.6.
Let A be finitely generated and Poisson, and pick m P m-Spec p A ab q . Then there isa natural isomorphism p A ^ m q ab „ ÝÑ p A ab q ^ m . Proof.
According to Lemma 5.5 the point m is a zero dimensional symplectic leaf, and it followsthat m is a Poisson ideal. In particular, A ^ m is a projective limit of Poisson algebras. We let B denote the ideal of A generated by the Poisson brackets t A, A u , let p B m be the ideal of A ^ m generatedby t A ^ m , A ^ m u and let B ^ m be the ideal A ^ m b A B , which we identify with an ideal of A ^ m using [AM69,Proposition 10.13]. Considering the exact sequence of A -modules B Ñ A Ñ A ab , we deduce p A ab q ^ m – A ^ m { B ^ m from [AM69, Proposition 10.12].We must show that B ^ m “ p B m . We certainly have an inclusion B ^ m Ď p B m and so the claim willfollow from the universal property of p A ^ m q ab if we can show that A ^ m { B ^ m is an abelian Poissonalgebra. As we noted earlier, the latter is isomorphic to p A ab q ^ m . Since this is a projective limit ofabelian Poisson algebras the claim follows. (cid:3) Sheets and induction
Lusztig–Spaltenstein induction, sheets and slices.
Let G be a complex connected re-ductive group and P Ď G a parabolic subgroup with Levi decomposition P “ LN , and write l “ Lie p L q . If v Ď g is any quasi-affine subvariety then we write v reg for the set of elements v P v such that dim Ad p G q v attains the maximal value.For any choice of nilpotent orbit O Ď l and z P z p l q it is not hard to see that Ad p G qp z ` O ` n q contains a dense G -orbit. This orbit is (Lusztig–Spaltenstein) induced from p l , O , z q and is denotedInd gl p z ` O q . Remarkably the induced orbit depends only on the G -conjugacy class of p l , O , z q ,not on the choice of P admitting L as a Levi factor [Lo16, Lemma 4.1]. We call this G -orbitan induction datum , and when z “ p l , O q is the induction datum,suppressing z .If an orbit cannot be obtained by induction from a proper Levi subalgebra then it is called rigid ,otherwise it is called induced . Rigid orbits are necessarily nilpotent. We say that an inductiondatum p l , O q is rigid if O is a rigid orbit in l . When G is almost simple of classical type theconjugacy classes of Levi subalgebras and nilpotent orbits can be described combinatorially, andinduction can be totally understood in terms of partitions [CM93, § p l , O q define D p l , O q : “ Ad p G qp O ` z p l q reg q S p l , O q : “ D p l , O q reg . Theorem 6.1.
Then the sheets of g are the sets S p l , O q where p l , O q varies over conjugacy classesof rigid data. Furthermore dim S p l , O q “ dim z p l q ` dim Ad p G q e . Classical finite W -algebras and Slodowy slices. Resume the notation κ, t e, h, f u , χ usedin Section 2.2, so g “ À i P Z g p i q is the ad p h q -grading. Using κ we can identify g with g ˚ as G -modules, so that S p g q is identified with the coordinatering C r g s . Then S p g q g pă –1 q χ is identified with the defining ideal of e ` g pă –1 q K “ t x P g | κ p x ´ e, g pă –1 qq “ u “ e ` g pď q . It follows from [GG02, Lemma 2.1] that the adjoint action map of G restricts to an isomorphism G pă q ˆ e ` g f „ ÝÑ e ` g pď q . Via this isomorphism we identify S p g , e q “ C r e ` g f s .Let γ : C ˆ Ñ G be the cocharacter with differential d γ p q “ h . There is a C ˆ -action on g givenby t ÞÑ t Ad p γ p t q ´ q which restricts to a contracting C ˆ -action on e ` g f . The grading on S p g , e q induced by this action coincides with the Kazhdan grading on S p g , e q . Since S p g , e q is Poissongraded in degree ´ C ˆ -action permutes the symplectic leaves of e ` g f .6.3. The Katsylo variety and the tangent cone.
The Slodowy slice e ` g f reflects the localgeometry of g in a neighbourhood of the adjoint orbit of e . Let S , ..., S l be the sheets of g containing e . Recall from the introduction that we define the Katsylo variety to be e ` X : “ p e ` g f q X ď i “ S i . (6.1)Let m denote the maximal ideal of C r e ` X s corresponding to e . One of the main tools in thispaper is the tangent cone TC e p e ` X q “ Spec p gr m C r e ` X sq , which we equip with the reducedscheme structure. Proposition 6.2. (1) Spec S p g , e q abred “ e ` X as reduced schemes. (2) Let g be a classical Lie algebra. We have bijections Comp p l ď i “ S i q ´ ÝÑ Comp p e ` X q ´ ÝÑ Comp TC e p e ` X q . The first bijection reduces dimension by dim Ad p G q e whilst the second is dimension pre-serving.Proof. By Lemma 5.5 we know that the reduced scheme associated to Spec S p g , e q ab is the union ofzero dimensional symplectic leaves of e ` g f . By [GG02, 3.2] we know that the symplectic leaves of e ` g f are the irreducible components of the non-empty intersections O X p e ` g f q where O Ď g is anadjoint orbit. Since e ` g f intersects orbits transversally it follows that the zero dimensional leavesare precisely the components of O X p e ` g f q where dim O “ dim Ad p G q e . Since the contracting C ˆ -action permutes the leaves of e ` g f , these are precisely the orbits lying in the sheets containing e . This proves (1).To prove the first bijection in (2) it suffices to show that for every sheet S i the intersection S i X p e ` g f q is irreducible. It was proven in [Im05, Theorem 6.2] that S i X p e ` g f q can beexpressed as the image of a morphism from an irreducible variety, which gives the first bijection.The main result of loc. cit. shows that the varieties S i are smooth. Since they are smoothly equivalent to S i X p e ` g f q the latter are also smooth, and soTC e p e ` X q “ l ď i “ T e p S i X e ` g f q . Now the second bijection will follow if we can show that T e p S i X e ` g f q is never contained inT e p S j X e ` g f q for i ‰ j which is a consequence of the fact that the sheets S , ..., S l intersecttransversally at e [Bu]. Finally the claims about dimensions follow from the fact that e ` g f istransversal to adjoint orbits. (cid:3) Abelian quotients of finite W -algebras The finite W -algebra. We now recall the definition of the (quantum) finite W -algebra.Once again G is a connected complex reductive group with simply connected derived subgroup.A nondegenerate trace form on g is denoted κ . We pick an sl -triple p e, h, f q in g “ Lie p G q andconsider the grading g “ À i P Z g p i q given by ad p h q -eigenspaces. Write χ : “ κ p e, ¨q P g ˚ . Recall thenotation g pă –1 q and G pă q .The generalised Gelfand–Graev module is defined to be Q : “ U p g q{ U p g q g pă –1 q χ “ U p g q b U p g pă´ qq C χ where C χ is the one dimensional representation of g pă –1 q afforded by χ . The finite W -algebra isthe quantum Hamiltonian reduction U p g , e q : “ Q G pă q . It inherits a natural algebra structure from U p g q via ¯ u ¯ u “ u u where ¯ u denotes the projectionof u P U p g q to Q , and ¯ u , ¯ u P Q G pă q (see [GG02], for example). The Kazdan filtration on U p g q isdefined by placing g p i q in degree i `
2. This descends to a non-negative filtration on Q and U p g , e q and we write gr U p g , e q for the associated graded algebra. The graded algebra gr U p g , e q naturallyidentifies with a subalgebra of gr Q “ S p g q{ S p g q g pă –1 q χ , and we have the following fundamentalfact (see [Pr02, Proposition 6.3], [GG02, Theorem 4.1]). Lemma 7.1. gr U p g , e q “ S p g , e q as Kazhdan graded algebras. One of the main objects of study in this paper is the maximal abelian quotient U p g , e q ab , whichis defined to be the quotient by the derived ideal which is generated by all commutators r u, v s with u, v P U p g , e q .The Kazhdan filtration descends to U p g , e q ab and we write gr U p g , e q ab for the associated Kazhdangraded algebra. Our main results describe the structure of the following affine schemes E p g , e q : “ Spec U p g , e q ab , C E p g , e q : “ Spec gr p U p g , e q ab q . The closed points of the first of these parameterises the one dimensional representations of U p g , e q ,whilst the second is the asymptotic cone of the first (cf. Section 5.1). Bounding the asymptotic cone.
The next result is equivalent to (1.3), and we view it as aversion of Premet’s theorem [Pr10, Theorem 1.2] for the asymptotic cone of E p g , e q . Our methodsare adapted from his work. Theorem 7.2.
There is a surjective homomorphism S p g , e q ab (cid:16) gr U p g , e q ab which induces an isomorphism of reduced algebras. First of all we prove the existence of the surjection
Lemma 7.3.
There is a surjective homomorphism S p g , e q ab (cid:16) gr U p g , e q ab .Proof. Let θ : g e Ñ S p g , e q be a PBW map coming from Theorem 2.4 and let Θ : g e Ñ U p g , e q bea filtered lift of θ . Using the Leibniz rule for the biderivation r¨ , ¨s we see that the derived ideal D of U p g , e q is generated by the set tr Θ p u q , Θ p v qs | u, v P g e u . Similarly the bracket ideal B in S p g , e q is generated by tt θ p u q , θ p v qu | u, v P g e u . If u, v are homogeneous for the ad p h q -gradingon g then the image of r Θ p u q , Θ p v qs in the graded algebra gr U p g , e q “ S p g , e q is t θ p u q , θ p v qu ,using the identification of Lemma 7.1. This shows that B Ď gr D . Hence we have gr U p g , e q ab “ S p g , e q{ gr D (cid:16) S p g , e q{ B “ S p g , e q ab . (cid:3) In order to show that this surjection gives an isomorphism of reduced algebras we use reductionmodulo p . This process was pioneered by Premet in the theory of W -algebras and used system-atically to great effect in subsequent work [Pr07, Pr10, To17, PT20]. Rather than repeating thedetails in full we state the important properties of the modular reduction procedure which we needfor our proof.Let Π be the set of bad primes for the root system of G . Pick a Chevalley Z -form g Z Ď g andwrite g Z “ h Z ‘ À α P Φ Z x α . Let T Ď G denote the complex torus with Lie algebra h Z b Z C . When k is an algebraically closed field we let G k denote the reductive algebraic group with Lie algebra g k : “ g Z b Z k and let T k denote the algebraic torus in G k with Lie algebra h Z b Z k .It follows from the observations of [PT20, § O Ď g there is anelement e P g Z X O and a cocharacter λ P X ˚ p T q such that for all algebraically closed fields k ofcharacteristic p R Π.(1) The Bala–Carter labels of the adjoint orbits of e and e k : “ e b P g k coincide.(2) The differential d λ p C q equals C h where h lies in an sl -triple containing e .(3) After canonically identifying to cocharacter lattices X ˚ p T q “ X ˚ p T k q the grading on g k induced by λ is good for e , meaning e k P g k p q and g e k Ď g k pě q .Fix an orbit O Ď g and a choice of e, λ as above. Let k be algebraically closed, and assumechar p k q “ p is large enough so that there is a G -equivariant trace form κ k . Write χ k for theelement of g ˚ k corresponding to e k via κ k . Using the grading g k “ À i P Z g k p i q coming from λ welet v denote a homogeneous complement to r g k , e k s in g k . If S , ..., S l denote the sheets of g k containing e k then we define e k ` X k “ p e k ` v q X l ď i “ S i . In the following we write e ` X C for the complex Katsylo variety (6.1), and write X _ k Ď g ˚ k for theimage of X k under the G -equivariant isomorphism g k Ñ g ˚ k coming from the trace form. Lemma 7.4.
For char p k q " there are dimension preserving bijections Comp C E p g , e q ´ ÝÑ Comp k E p g k , e k q (7.1) Comp X C ´ ÝÑ Comp X k . (7.2) Proof.
The first bijection (7.1) can be obtained by reciting the proof of [Pr10, Lemma 3.1] verbatim,whilst the second (7.2) was constructed in the proof of [Pr10, Theorem 3.2]. (cid:3)
Since g k “ Lie p G k q is algebraic there is a G k -equivariant restricted structure on g k . The p -centre Z p p g k q is a central subalgebra of U p g k q which identifies with k rp g ˚ k q p q s , the coordinate ring on theFrobenius twist of the dual space of g k , as G k -algebras.The modular finite W -algebra is defined in precisely the same manner as the complex analogue: U p g k , e k q “ Q G k pă q k where Q k “ U p g k q{ U p g k q g k pă –1 q χ and g k pă –1 q χ “ t x ´ χ k p x q | x P g k pă –1 qu [Pr10, GT19a]. If φ : U p g k q Ñ Q k is the natural projection then the p -centre of U p g k , e k q is definedto be p φZ p p g k qq G k pă q ; see [GT19a, Section 8] for more detail. It follows from Lemma 8.2 of loc.cit. that Z p p g k , e k q “ k rp χ k ` v _ q p q s where v _ “ κ k p v , ¨q Ď g ˚ , and the maximal ideals of the p -centre will be referred to as p -characters. For η P χ k ` v _ we define the reduced finite W -algebra U η p g k , e k q to be the quotient of U p g k , e k q by the ideal generated by the corresponding maximalideal of Z p p g k , e k q (identifying χ k ` v _ with its Frobenius twist as sets). Lemma 7.5.
There is a finite, dominant morphism k E p g k , e k q Ñ X k . Proof.
Consider the homomorphism Z p p g k , e k q Ñ U p g k , e k q ab and denote the kernel by K . Themaximal ideals of Z p p g k , e k q{ K correspond to the p -characters η P χ k ` v _ such that U η p g k , e k q admits a one dimensional module. Using Premet’s equivalence [Pr02, Theorem 2.5(ii) and Propo-sition 2.6] we see that these are the p -characters η P p χ k ` v _ q p q such that U η p g k q admits a moduleof dimension p dim Ad ˚ p G k q χ k . By the main result of [PT20] this is precisely the set of η such thatdim Ad ˚ p G k q η “ dim Ad ˚ p G k q χ k . We claim that this set is equal to p χ k ` X _ k q p q . In the currentsetting, χ k ` v _ admits a contracting k ˆ -action defined in the same manner as the one parametergroup of automorphisms appearing in Section 6.2, using the cocharacter appearing in (3) above.This contracting action preserves the sheets of g ˚ k and this implies the claim.Let R denote the reduced algebra corresponding Z p p g , e q{ K . Since Z p p g , e q “ k rp χ ` v _ q p q s asKazhdan filtered algebras we see that R “ k rp χ ` X _ k q p q s is also an identification of Kazhdan filteredalgebras. Since χ ` X _ k is stable under the contracting k ˆ -action inducing the Kazhdan grading,it follows that R – gr R . We conclude that there is a finite injective algebra homomorphism R ã Ñ gr U p g , e q ab , which completes the proof. (cid:3) Proof of Theorem 7.2.
Recall that finite morphisms are closed, and so a finite dominant morphismis surjective. Now it follows from Lemma 7.5 that there is a surjective map Comp k E p g k , e k q Ñ Comp X k such that for every Z P Comp X k there exists an element of Comp k E p g k , e k q mappingto Z of the same dimension. Combining with Lemma 7.4 we deduce that there is surjectionComp C E p g , e q (cid:16) Comp p e ` X C q which respects dimensions in the same manner. In particular we have d p C E p g , e qq ě d p e ` X C q , where d denotes the dimension vector defined in Section 5.1. Thanksto Lemma 5.1, Proposition 6.2(1) and Lemma 7.3 the embedding C E p g , e q ã Ñ Spec S p g , e q ab “ e ` X C is an isomorphism on the underlying reduced schemes. (cid:3) The next result follows immediately from the theorem.
Corollary 7.6. If S p g , e q ab is reduced then so are gr U p g , e q ab and U p g , e q ab . (cid:3) The abelian quotient of the classical W -algebra Rigid, singular and distinguished partitions.
We introduce three classes of partitionwhich correspond to important families of nilpotent orbit in classical Lie algebras of type
B, C, D .Let ε “ ˘ N ą ε N “
1. Recall the notation P ε p N q from Section 4.1.According to Lemma 4.1 the partitions λ “ p λ , ..., λ n q P P ε p N q are characterised by the existenceof an involution i ÞÑ i satisfying the following conditions: (1) λ i “ λ i , (2) ε p´ q λ i “ i ‰ i , and (3) i P t i ´ , i, i ` u . Whenever we choose λ P P ε p N q we fix such an involution. Wealso adopt the convention that λ “ λ n ` “ 8 .For λ P P ε p N q a is a pair of indexes p i, i ` q such that: ‚ ε p´ q λ i “ ε p´ q λ i ` “ ´ ‚ λ i ´ ă λ i ď λ i ` ă λ i ` .We note that λ n ă λ n ` always holds because of our convention λ n ` “ 8 . We write ∆ p λ q for theset of 2-steps for λ . A 2-step p i, i ` q P ∆ p λ q is called bad if one of the following conditions holds: ‚ λ i ´ λ i ´ P Z ą ; ‚ λ i ` ´ λ i ` P Z ą .We note that the second condition never holds for i “ n ´
1, whilst if the first condition holds for i “ ε “ ´
1. If the first of these two conditions holds then we refer to i ´ bad boundary of the 2-step whilst if the second condition holds then i ` singular if it admits a bad 2-step, and non-singular otherwise.We say that a partition is rigid if the following two conditions hold: ‚ ∆ p λ q “ H ; ‚ λ i ´ λ i ´ ă i “ , ..., n .We say that a partition is distinguished if the following two conditions hold: ‚ λ i ă λ i ` for all i “ , ..., n ; ‚ i “ i for all i . Lemma 8.1. [CM93, PT14]
Let e be a nilpotent element with Jordan block sizes given by thepartition λ P P ε p N q in a classical Lie algebra preserving a non-degenerate form Ψ on C N such that Ψ p u, v q “ ε Ψ p v, u q for u, v P C N . The following hold: (i) e lies in a unique sheet of g if and only if λ is non-singular. (ii) e is distinguished in the sense of Bala–Carter theory if and only if λ is distinguished. (iii) e is rigid in the sense of Lusztig–Spaltenstein induction if and only if λ is rigid. (cid:3) The Kempken–Spaltenstein algorithm.
Keep fixed a choice of λ “ p λ , ..., λ n q P P ε p N q .We say that i is an admissible index for λ if Case 1 or Case 2 occurs: Case 1: λ i ´ λ i ´ ą Case 2: p i ´ , i q P ∆ p λ q and λ i ´ “ λ i . When i is an admissible index we define λ p i q P P (cid:15) p N ´ p n ´ i ` qq as follows: Case 1: λ p i q “ p λ , λ , ..., λ i ´ , λ i ´ , λ i ` ´ , ..., λ n ´ q ; Case 2: λ p i q “ p λ , λ , ..., λ i ´ , λ i ´ ´ , λ i ´ , λ i ` ´ , ..., λ n ´ q Now we extend this definition inductively from indexes to sequences. We say that p i q is an admissi-ble sequence for λ if i is an admissible index. We inductively extend these definitions to sequencesof indexes by saying that i “ p i , ..., i l q is an admissible sequence for λ and define λ i “ p λ p i ,...,i l ´ q q p i l q if i “ p i , ..., i l ´ q is an admissible sequence for λ and i l is an admissible index for λ i . A sequence i is called maximal admissible if λ i does not admit any admissible indexes. These definitions firstappeared in [PT14], where the opposite ordering of partitions was used.We define admissible multisets to be the multisets taking values in t , ..., n u obtained fromadmissible sequences by forgetting the ordering. If i is an admissible sequence then write r i s forthe corresponding admissible multiset.For ε “ ˘ g be a classical Lie algebra in accordance with Lemma 8.1, and let O Ď g be anilpotent orbit with partition λ . According to Proposition 8 and Corollary 8 of [PT14], for eachaddmisible multiset r i s , there is a procedure for choosing:(1) a Levi subalgebra l i – gl i ˆ ¨ ¨ ¨ gl i l ˆ g , where g is a classical Lie algebra of the sameDynkin type as g and natural representation of dimension N ´ ř i j .(2) a nilpotent orbit O i “ t u ˆ ¨ ¨ ¨ t u ˆ O λ i where O λ i Ď g has partition λ i .These choices satisfy O “ Ind gl p O i q . Furthermore r i s ÞÑ p l i , O i q sets up a one-to-one correspon-dence between maximal admissible multisets and rigid induction data for O . Combining withTheorem 6.1 this proves. Proposition 8.2.
The sheets of g containing e are in bijection with maximal admissible multisetsfor λ . The dimension of sheet corresponding to i is | i | ` dim p Ad p G q e q . (cid:3) Distinguished elements and induction.
The proof of the main theorem reduces to thedistinguished case, and the following result is one of the crucial steps.
Lemma 8.3.
Let O Ď g be a nilpotent orbit with partition λ P P ε p N q in a classical Lie algebra ofrank r . There exists a sequence of indexes i , ..., i l and a classical Lie algebra ˜ g of rank r ` ř j p n ´ i j ` q of the same Dynkin type as g such that: (1) gl n ´ i ` ˆ ¨ ¨ ¨ ˆ gl n ´ i l ` ˆ g embeds as a Levi subalgebra l Ď ˜ g ; (2) Identifying O with the as a nilpotent orbit t u ˆ ¨ ¨ ¨ ˆ t u ˆ O Ď l we have that Ind ˜ gl p O q isa distinguished orbit in ˜ g .Furthermore we can assume that the first part of the partition of Ind ˜ gg p O q is arbitrarily large.Proof. If i “ , ..., n is an index such that i “ i ´ λ by p λ , ..., λ i ´ , λ i ´ ` , λ i ` , λ i ` ` , ..., λ n ` q . By iterating this procedure we can eventually obtain a partition with i “ i for all i . If i “ , ..., n is an index such that λ i “ λ i ´ then we replace our new partition by p λ , ..., λ i ´ , λ i ` , λ i ` ` , ..., λ n ` q . Iterating this procedure we eventually replace λ with a distinguished partition ˜ λ P P ε p N ` ř i P i p n ´ i ` qq where i Ď t , ..., n u is some multiset consisting of indexes which were chosen in the aboveiterations. By applying the latter procedure at index i “ λ as largeas we choose.Let ˜ g be the classical Lie algebra of the same type as g with natural representation of dimension N ` ř j P i p n ´ j ` q . By Lemma 8.1 every orbit with partition ˜ λ is distinguished. Since ˜ λ i “ λ the remarks of the previous Section show that there is a unique orbit r O Ď ˜ g with partition λ anda unique induction datum p l , O q such that l and O satisfy the properties described in the currentproposition, and r O “ Ind ˜ gl p O q . (cid:3) The combinatorial Katsylo variety.
In this section we introduce an affine algebraic variety X λ , determined by a choice of partition, which we call the combinatorial Katsylo variety. Let λ “ p λ , ..., λ n q P P ε p N q such that i “ i for all i and λ ą
1. In particular λ is distinguished.This condition ensures that λ i ´ λ i ´ P Z ą for all i “ , ..., n and that p i, i ` q P ∆ p λ q for i “ , ..., n ´
1. We set s : “ t λ u ; s i : “ λ i ´ λ i ´ for i “ , ..., n. (8.1)Introduce a set of variables S λ : “ t x i,r | ď i ď n, ď r ď s i u Y t y j | p j, j ` q P ∆ p λ qu (8.2)and we consider the following collection of quadratic elements Q λ in the polynomial ring C r S λ s Q λ : “ t x i, y i | i “ , ..., n ´ , λ i ´ ´ λ i P Z ą uY t x i ` , y i , y i y i ` | i “ , ..., n ´ u . (8.3)Finally we define X λ to be the algebraic variety with coordinate ring C r X λ s : “ C r S λ s{p Q λ q . (8.4)In order to understand these varieties it suffices to consider a minimal partition. For this purposewe define a partition α “ p α , ..., α n q by α “ p , , ..., n q if ε “ ´ p , , ..., n ` q if ε “ . (8.5)It follows directly from the definitions that C r X λ s – C r X α s b C r x i,r | ď i ď n, ă r ď s i s ,which implies that X λ – X α ˆ A t λ n { u ´ n . (8.6)For our purposes it is now sufficient to study X α .Let P n be the set of subsets S Ď t , ..., n ´ u such that for all i “ , ..., n ´ i P S or i ` P S or both. For S P P n we let S c : “ t , ..., n ´ uz S be the complementary set. We define a map from P n to the set of subsets of t x , , ..., x n, , y , ..., y n ´ u Ď C r X α s by ι : S ÞÑ t y i | i P S u Y t x i | i P S c or i ´ P S c u . (8.7) Lemma 8.4. S ÞÑ p ιS q is a bijection from P n to the set of minimal prime ideals of C r X α s and dim C r X α s{p ιS q “ |t x , ..., x n , y , ..., y n ´ uz ιS | (8.8) Proof.
The torus T : “ p C ˆ q n ´ acts on C r x , , ..., x n, , y , ..., y n ´ s by automorphisms rescalinggenerators, and this action descends to C r X α s . Since T is connected it preserves the irreduciblecomponents of X α , hence preserves the minimal primes of C r X α s . Since the T -weight spaces on C r x , , ..., x n, , y , ..., y n ´ s are one dimensional, it follows that each minimal prime p is generatedby p X t x , , ..., x n, , y , ..., y n ´ u .By inspection we see that for each S P P n the set ιS contains at least one of the factors of eachquadratic relation (8.3) in C r X α s and is minimal with respect to this property. This implies that ι maps P n to minimal primes.The injectivity of S ÞÑ p ιS q is clear and it remains to show that every minimal prime p is inthe image. Suppose that p is generated by some subset S Ď t x , , ..., x n, , y , ..., y n ´ u . Using theprimality of p it is straightforward to check that S : “ t ď i ď n ´ | y i P S u P P n and p ιS q Ď p .Now by minimality we have p “ p ιS q . (cid:3) Proposition 8.5.
There is a bijection i ÞÑ X i λ between maximal admissible multisets for λ andirreducible components of X λ such that dim X i λ “ | i | . Proof.
It is not hard to see that the admissible multisets for λ are of the form i Y j where i is anadmissible multiset for α and j is the multiset taking values in t , ..., n u such that j occurs withmultiplicity p λ j ´ λ j ´ q{ ´
1. Combining this with (8.6) it suffices to prove the current propositionin the case λ “ α .By Lemma 8.4 the irreducible components of X α are parameterised by P n . In order to provethe first claim of the theorem we construct a bijection from P n to the set of maximal admissiblemultisets.The admissible multisets for α have multiplicities ď
2. Thanks to [PT14, Lemma 6] we knowthat a maximal admissible multiset is totally determined by the collection of indexes in t , ..., n u which occur with multiplicity 2: this is the set of indexes such that Case 2 occurs at some pointin the KS algorithm. Therefore the collection of indexes of multiplicity 2 is only constrained bythe fact that if i has multiplicity 2 then neither i ´ i ` S P P n then there is a unique maximal admissible sequence i for α such that the indexes in t , ..., n uzp S ` q occur with multiplicity 2. This gives the desired bijection.It remains to prove the claim regarding dimensions. Let S Ď P n and enumerate t , ..., n ´ uz S “p i , ..., i l q . By the remarks of the previous paragraph the maximal admissible sequence associatedto S P P n is determined as follows: first we let i “ p i ` , i ` , i ` , i ` , ..., i l ` , i l ` q andthen we let i be the maximal admissible sequence obtained from i by applying Case 1 of the KSalgorithm to α i as many times as possible. The set of indexes where Case 1 can be applied to α i are precisely t ď i ď n | α i i ´ α i i ´ “ u “ t ď i ď n | i ‰ i l and i ´ ‰ i l for all l u (8.9) and it follows that the length of the maximal admissible sequence associated to S is n ´ ´ | S | ` ď i ď n | i, i ´ P S c u . (8.10)According to (8.7) and Lemma 8.4 the dimension of the component of X α corresponding to S is2 n ´ ´ | S | ´ ď i ď n | i P S c or i ´ P S c u (8.11)It is straightforward to see that (8.10) and (8.11) are equal. (cid:3) The semiclassical abelianisation I: the distinguished case.
Let λ “ p λ , ..., λ n q P P ε p N q be a distinguished partition satisfying λ ą
1. The indexes s , ..., s n ´ defined in (8.1) allow us toconstruct an n ˆ n symmetric shift matrix via σ “ p s i,j q ď i,j ď n by setting s i ` ,i “ s i,i ` : “ s i ` for i “ , ..., n ´
1. By (3.2) these values determine the entire matrix, which coincides with (4.1).Next we describe generators of gr m S p g , e q ab and certain relations between them. Lemma 8.6.
Let m Ď S p g , e q ab be unique maximal Kazhdan graded ideal. Then gr m S p g , e q ab isgenerated by elements t ¯ θ p s i ` ` q i | i “ , ..., n ´ u Y t ¯ η p r q i | i “ , ..., n, r “ , ..., s i u (8.12) These elements satisfy the following relations ¯ θ p s i ` ` q i ¯ θ p s i ` ` q i ` “ for i “ , ..., n ´ θ p s ` q η p s q “ for ε “ ´ θ p s i ` ` q i p ¯ η p s i q i ´ ¯ η p s i q i ´ q “ for i “ , ..., n ´ θ p s i ` ` q i p ¯ η p s i ` q i ` ´ ¯ η p s i ` q i ` q “ for i “ , ..., n ´ where the elements η p r q i are taken to be zero if they lie outside the range prescribed by (8.12) .Proof. By Theorem 3.7 and Corollary 4.9 we see that S p g , e q is generated (as a commutativealgebra) by elements (3.61). We write ¯ θ p r q i,j : “ θ p r q i,j ` m and ¯ η p r q i : “ η p r q i ` m . By Corollary 4.10 wesee that gr m S p g , e q is generated by elements t ¯ η p r q i | ď i ď n, ă r ď λ i u Y t ¯ θ p r q i,j | ď i ă j ď n, s i,j ă r ď s i,j ` λ i u , and these elements satisfy the relations of g e .If we identify a subset of these elements which span a vector space complement to r g e , g e s in g e then it follows that these elements generate gr m S p g , e q ab . Now [PT14, Theorem 6] implies thatthe elements (8.12) span such a complement, although we warn the reader that the notation thereis slightly different.In order to complete the proof we show that they satisfy the quadratic relations (8.13)–(8.16).Relation (8.13) follows directly from (3.70).Now suppose that i “ ε “ ´
1, so that λ is even. Using (3.67) and (4.24) we seethat the ideal of S p g , e q generated by Poisson brackets contains the element λ { ÿ t “ η p t q θ p λ ´ t ` s , ` q ´ λ { ´ ÿ t “ η p t q t η p q , θ p λ ´ t ` s , q u “ η p λ q θ p s , ` q . Note that λ “ s in this case by Lemma 4.1. Projecting this expression into gr m S p g , e q we seethat (8.14) lies in the kernel of gr m S p g , e q (cid:16) gr m S p g , e q ab . The relations (8.15), (8.16) can be deduced similarly by considering the quadratic terms appearing in (3.67), (3.73) and projectinginto gr m S p g , e q . Since the arguments for these two cases are almost identical we just sketch (8.16).Fix i “ , ..., n ´
2. Using (3.73) we consider the expression (cid:32) θ p s i ` ` q i ` , t θ p s i ` ` q i ` , θ p s i ` ` q i u ( .There is a unique linear term θ p s i ` ` s i ` ` q i corresponding to m “ m ´ m “
0. By (3.67)this term lies in the ideal of y n p σ q generated by the brackets, so we may subtract it to obtain anelement of the bracket ideal. Now consider the quadratic terms. If we choose any term such that m ă m then this term has a factor of θ p p m ´ m q` s i ` ` q i and so these terms lie in the bracketideal again by (3.67). After subtracting the the only remaining quadratic terms are η p s i ` q i ` θ p s i ` ` q i and ˜ η p s i ` q i ` θ p s i ` ` q i which correspond to m “ m “ m “ m “ m ´ η p s i ` q i ` “ ´ η p s i ` q i ` modulo m , and so η p s i ` q i ` θ p s i ` ` q i ` ˜ η p s i ` q i ` θ p s i ` ` q i ` m “ ¯ θ p s i ` ` q i p ¯ η p s i ` q i ` ´ ¯ η p s i ` q i ` q . The proof of (8.15) is almost identical, instead examining (cid:32) θ p s i ` ` q i ` , t θ p s i ` ` q i ` , θ p s i ` ` q i u ( . (cid:3) Theorem 8.7.
There is an isomorphism gr m S p g , e q ab „ ÝÑ C r TC e p e ` X qs . In particular S p g , e q ab is reduced.Proof. Let A denote the algebra with generators (8.12) and relations (8.13)–(8.16). Before weproceed we outline the argument. First of all we will show that A – C r X λ s where X λ is thecombinatorial Katsylo variety from Section 8.4. By Lemma 8.6 we have that A (cid:16) gr m S p g , e q ab and since the reduced algebra of S p g , e q ab is C r e ` X s by Proposition 6.2(1), we see that C r X λ s (cid:16) C r TC e p e ` X qs . We then combine deductions of the previous sections to see that the componentsof X λ and TC e p e ` X q have the same dimensions. Since X λ is reduced we deduce that we haveisomorphisms C r X λ s – gr m S p g , e q ab – C r TC e p e ` X qs , from which the theorem follows. Step (i):
First we define an isomorphism A – C r X λ s . Notice that the algebra endomorphismof A defined by ¯ η p s i ´ q i ÞÑ ¯ η p s i ´ q i ` ¯ η p s i ´ q i for all i “ , ..., n , and acting identically on the othergenerators is a unimodular subsitution. By slight abuse of notation we denote the images of the thegenerators under the inverse automorphism by the same symbols. With this new set of generatorsthe relations now take the form¯ θ p s i ` q i ¯ θ p s i ` ` q i ` “ i “ , ..., n ´ θ p s ` q ¯ η p s q “ ε “ ´ θ p s i ` q i ¯ η p s i q i “ i “ , ..., n ´ θ p s i ` q i ¯ η p s i ` q i ` “ i “ , ..., n ´ A „ ÝÑ C r X λ s defined by¯ η p r q i ÞÝÑ x i,s i ` ´ r ;¯ θ p s i ` ` q i ÞÝÑ y i . Step (ii):
Now we use the tangent cone TC e p e ` X q . By Proposition 6.2(1) and Lemma 8.6 wehave C r X λ s (cid:16) gr m S p g , e q ab (cid:16) C r TC e p e ` X qs (8.17) Applying Proposition 6.2(2), Proposition 8.2 and Proposition 8.5 we have d p X λ q “ d p e ` X q “ d p TC e p e ` X qq . Now by Lemma 5.1 we see that the surjections (8.17) are isomorphisms on the underlying varietiesof closed points. Since C r X λ s is reduced these maps are algebra isomorphisms and gr m S p g , e q ab is reduced. Applying Lemma 5.4 we see that S p g , e q ab is reduced, as claimed. (cid:3) The semiclassical abelianisation II: the general case.
The following result uses thelocal geometry of Poisson manifolds to reduce the claim to the distinguished case, and then applyTheorem 8.7.
Theorem 8.8. S p g , e q ab is reduced.Proof. Part (i): We let O Ď g be the adjoint orbit of e in g , and let ˜ g and r O be the Lie algebraand distinguished nilpotent orbit introduced in Lemma 8.3. Pick ˜ e P ˜ g and an sl -triple p ˜ e, ˜ h, ˜ f q .Write r G for the simply connected, connected complex algebraic group with ˜ g “ Lie p r G q . Recallthat l “ gl n ´ i ` ˆ ¨ ¨ ¨ ˆ gl n ´ i l ` ˆ g embeds as a Levi subalgebra of ˜ g . If we choose e P O Ď g Ď l then O identifies with the adjoint L -orbit and we have the following isomorphism S p l , e q – S p gl i q b ¨ ¨ ¨ b S p gl i l q b S p g , e q Therefore S p g , e q ab is reduced if and only if S p l , e q ab is reduced. This holds if and only only if p S p l , e q ^ m q ab is reduced, where m is the maximal ideal of e , thanks to Lemma 5.4 and Lemma 5.6. Part (ii):
Now pick a regular element s P z p l q so that ˜ g s – l . Write x “ s ` e P ˜ g . We will explainbelow that x can be viewed as a point of ˜ e ` ˜ g ˜ f “ Spec S p ˜ g , ˜ e q , and by slight abuse of notationwe will also write m for the maximal ideal of x in S p ˜ g , ˜ e q . We claim that S p l , e q ^ m – S p ˜ g , ˜ e q ^ m asPoisson algebras. Note that p S p ˜ g , ˜ e q ^ m q ab is reduced by Lemma 5.4 and Theorem 8.7. Thanks topart (i) the current proof will follow from the claim. Part (iii):
Pick an sl -triple p e, h, f q for e inside l , then define y “ s ` f P ˜ g . By sl -theory theaffine variety x ` ˜ g y is a transverse slice to the adjoint orbit Ad p r G q x at the point x . Also pick an sl -triple p ˜ e, ˜ h, ˜ f q in ˜ g . After replacing the latter triple by some conjugate we can actually assumethat x P ˜ e ` ˜ g ˜ f . By construction we have Ind ˜ gl p O q “ Ad p r G q ˜ e . It follows that x lies in a sheet of ˜ g containing ˜ e , and so we have x P ˜ e ` r X , where the latter variety is defined in parallel with (6.1).Now we choose small neighbourhoods U x Ď ˜ e ` ˜ g ˜ f and V x Ď x ` ˜ g y of x in the complex topologies.Similarly let W e Ď e ` l f be a small neighbourhood of e . It is well-known that U x , V x , W e all carrytransverse Poisson structures on the ring of analytic functions, see [LPV13, 5.3.3] for example.Write C an p M q for the ring of analytic functions on a complex manifold M . There is a naturalrestriction map C r e ` l f s Ñ C ab p W e q which is a Poisson homomorphism. Thanks to [Ser55,Proposition 3] this induces an isomorphism S p l , e q ^ m – C an p W e q ^ e of complete Poisson algebras.Similarly we have a Poisson isomorphism S p ˜ g , ˜ e q ^ m – C an p U x q ^ x .Now the claim will follow from the existence of an isomorphism C an p U x q ^ x – C an p W e q ^ e of completePoisson algebras. First of all we observe that x P ˜ e ` r X implies that both ˜ e ` ˜ g ˜ f and x ` ˜ g y aretransverse slices to Ad p r G q x at x . Therefore by [LPV13, Proposition 5.29] we have an isomorphism U x Ñ V x of analytic Poisson manifolds sending x to x . Furthermore by [DSV07, Proposition 2.1]we see that there is a similar isomorphism V x Ñ W e sending x to e . This completes the proof. (cid:3) The abelianisation of the finite W -algebra via deformation theory. Here we prove thefirst main theorem (Theorem 1.1), which states that Premet’s component map (1.2) is a bijection.First of all we apply Proposition 7.6 and Theorem 8.8 to see that both gr U p g , e q ab and U p g , e q ab are reduced. Now apply Lemma 5.2 and Theorem 7.2 to see that Comp E p g , e q ď Comp C E p g , e q “ Comp p e ` X q . Since Premet’s map (1.2) is surjective and restricts to a dimension preserving bijection from somesubset, the theorem follows.
Remark . Premet asked whether the abelian quotient U p g , e q ab of a finite W -algebra is re-duced [Pr10, Question 3.1]. Combining Proposition 7.6 with Theorem 8.8 we are able to give anaffirmative answer in the case of classical Lie algebras.The problem of understanding whether U p g , e q ab is reduced for exceptional Lie algebra is rathersubtle. The methods of this paper will certainly not work in general: in the introduction to [Pr14]it is explained that there are four orbits in exceptional Lie algebra such that the associated finite W -algebra is known to admit precisely two one dimensional representations (these correspondto the first four columns of [PT20, Table 1]). For these W -algebras it is not hard to see thatthe reduced algebra associated to U p g , e q ab is isomorphic to C r x s{p x ´ q as filtered algebras,with the generator x in some positive degree. This ensures that gr U p g , e q ab is not reduced, andProposition 7.6 implies that S p g , e q ab admits nilpotent elements.9. The orbit method
Losev’s orbit method map.
In this Section we suppose that G is semisimple and simplyconnected. Using the Killing form we identify g with g ˚ for the rest of the paper. Let us recall theconstruction of the orbit method map g { G Ñ Prim U p g q from [Lo16].We say that a Poisson algebra has degree ´ t¨ , ¨u : A i ˆ A j Ñ A i ` j ´ . In this case afiltered quantization of A is a pair p A , ι q consisting of a non-commutative filtered algebra with r F i A , F j A s Ď F i ` j ´ A and an isomorphism ι : gr A Ñ A of filtered Poisson algebras. FilteredPoisson deformations are defined similarly, the only difference being that A is a filtered Poissonalgebra.Let l “ Lie p L q be a Levi subalgebra of g , O Ď l a nilpotent adjoint orbit, and z P z p l q anelement of the centre. Recall that p l , O , z q is referred to as an induction datum. If P is a choiceof parabolic with Levi decomposition p “ Lie p P q “ l ˙ n , then the generalised Springer map is G ˆ P p z ` O ` n q Ñ g (9.1)In general this is a finite morphism onto the image, which contains a unique dense adjoint orbit.This orbit does not depend on P , only on its Levi factor [Lo16, Lemma 4.1], and it is denotedInd gl p z ` O q . Recall that p l , O , z q is known as an induction datum for Ind gl p z ` O q . We say thatthe datum is proper if l ‰ g . If the generalised Springer map (9.1) is birational then we say thatthe induction datum is birational . If O does not admit any proper, birational induction data thenit is said to be birationally rigid . The datum p l , O , z q is said to be a birationally minimal if:(1) O is birationally rigid in l ;(2) p l , O , z q is a birational orbit datum for O . According to [Lo16, Theorem 4.4] there is a unique birationally minimal induction datum p l , O , ξ q associated to each orbit O P g ˚ { G .Suppose that O is birationally rigid in l . Then according to [Lo16, Proposition 4.2] there is anon-empty open subset z p l q reg Ď z p l q such that (9.1) is birational for all z P z p l q reg . This subsetdoes not depend on the choice of P , only on L , and we warn the reader that this notation differentto that used in Section 6.1. The birational sheet corresponding to p l , O q is the union of all orbitsof maximal dimension in Ad p G qp z p l q reg ` O ` n q .Pick l , P, n as above, and a birationally rigid orbit O Ď l . By [Ja04, Proposition 8.3] theaffinization X “ Spec C r O s is isomorphic to the normalisation of O . Consider the map G ˆ P p X ˆ n q Ñ Ind gl p O q . The preimage of the induced orbit is a finite covering which we denote G { H Ñ Ind gl p O q . The affinization X “ Spec C r G { H s is a normal conical Poisson variety indegree -2.Writing P “ z p l q there is an important Poisson deformation X P Ñ P of X and a filteredquantization A P of C r X P s over the same base, described in [Lo16, § § P is actually a certain cohomology group, and we use [Lo16, Proposition 4.7] to identify it with z p l q throughout this paper. By the results of [Lo16, §
4] the Poisson deformation can be describedvery explicitly using Lusztig–Spaltenstein induction: the group P acts on P ˆ X ˆ n as aboveand taking global sections of the structure sheaf on r X P : “ G ˆ P p P ˆ X ˆ n q we obtain C r X P s .In words, X P is the affinization of r X P . The C r P s -algebra structure arises from the projection r X P Ñ P . The Poisson structure is equipped with a Hamiltonian G -action and the comoment map C r g s Ñ C r X P s lifts to a quantum comoment map U p g q Ñ A P . Lemma 9.1.
The quantum comoment map induces a finite morphism P Ñ Spec Z p g q .Proof. Taking the associated graded algebra it suffices to show that the comoment map C r g s Ñ C r X P s induces a finite morphism z p l q Ñ g { G “ Spec C r g s G . The latter map is just the composition z p l q ã Ñ t (cid:16) t { W “ Spec C r g s G where t is a maximal torus and W the Weyl group, hence finite. (cid:3) Now we can define the orbit method map . Pick an orbit O P g { G . This corresponds to a uniquebirationally minimal induction datum p l , O , z q , and we pick a parabolic P for L . Let A O be thequantization of C r G ˆ P p X ˆ n qs with parameter z . Then Losev defines J : g { G ÝÑ Prim U p g q O ÞÝÑ
Ker p U p g q Ñ A O q . (9.2)Recall the notation Prim O U p g q from the introduction. The following refinement of Theorem 1.2,uses the notation of Section 7. Theorem 9.2.
Let e P O Ď g be an element of a nilpotent orbit in a classical Lie algebra. Thefollowing sets coincide: (1) The image of J intersected with Prim O U p g q . (2) The ideals
Ann U p g q p Q b U p g ,e q C η q where C η P U p g , e q -mod is one dimensional. The proof of the theorem will occupy the rest of the section. First of all we describe theassociated variety of the primitive ideal J p O q . This was stated in [Lo16, § Lemma 9.3.
For O P g ˚ { G we let S be a sheet of g ˚ containing O . Then VA p J p O qq “ S X N p g ˚ q . Proof.
Let p l , O , z q be the birationally minimal orbit datum inducing to O . For t P C we canconsider the orbit O p t q which is dense in the image of the generalised Springer map G ˆ P p tz ` O ` n q Ñ g . By definition we have O p q “ O and O p q “ Ind gl p O q , whilst dim O p t q is constant.It follows that O and Ind gl p O q lie in a sheet. Since O is nilpotent and every sheet contains aunique nilpotent orbit, S X N p g q “ Ind gl p O q for every sheet containing O .Now write X for the dense orbit in G ˆ P p X ˆ n q . Since gr is exact on strictly filtered vectorspaces, the graded ideal gr J p O q is the kernel of the comoment map C r g s Ñ C r X s . This factorsthrough the comoment map for C r Ind gl p O qs , which shows that gr J p O q is the defining ideal ofInd gl p O q . This completes the proof. (cid:3) The Namikawa–Weyl group and the Poisson automorphism group.
Keep the nota-tion of the previous section. There are two groups at play in the theory of universal quantizations.The first of these is the Namikawa–Weyl group W , described in [Lo16, § P , C r X P s and A P compatibly, and it follows from op. cit. that the algebras C r X P s W and A W P satisfy universal properties for Poisson deformations and filtered quantizationsrespectively (see also [ACET20] for an elementary discussion of the universal properties). To beslightly more precise it is known that every other deformation and quantization are obtained fromthese algebras by base change.Now let A be the group of Poisson automorphisms of X . As explained in [Lo16, § C r X P s W and A W P imply that there is a natural action of A on C r X P s W by graded Poisson automorphisms and on A W P by filtered algebra automorphisms. These actionspreserve C r P s W and the induced actions on P { W are called the Poisson and quantum actionsrespectively.When X is the affinization of a finite cover G { H Ñ Ad p G q e there is a map Γ “ G e {p G e q ˝ Ñ A defined by letting γ p G e q ˝ act on gH by gγ ´ H . In fact one can show that the image of this mapconsists of all G ˆ C ˆ -equivariant Poisson automorphisms of G { H . Lemma 9.4.
Let p l i , O i q , i “ , be induction data giving covers X i , i “ , of the induced orbit.Suppose l Ď l and Ind l l p O q “ O . Write W for the Namikawa–Weyl group of X . If G { H Ď X is the dense orbit then the image of H in Γ fixes pointwise the image of z p l q in z p l q{ W under thePoisson action.Proof. Choose e P Ind gl i p O i q X p O i ` n i q . If ˜ e i is an element of the fibre of G ˆ P i p Spec C r O i s ˆ n i q Ñ Ind gl i p O i q over e then we can take H i “ G ˜ e i . We have H i Ď P i because ˜ e i lies in the fibre of r X i Ñ G { P i over the identity coset. The right action of G e on G descends to an action on r X z p l q .For h P H we see that the action on the central fibre over 0 P z p l q i coincides with the action viaPoisson automorphisms, and that the action on r X z p l q induces the Poisson action on the quotient z p l q{ W . The inclusion H Ď P now ensures that H stabilises the fibres of X z p l q over z p l q henceinduces the trivial action on the image of z p l q Ñ z p l q{ W . (cid:3) Primitive ideals arising from generalised Springer maps.
For every birationally rigidorbit O Ď l in a Levi subalgebra there is a corresponding orbit cover, and a universal quantization of the affinization. Every fibre of this quantization gives rise to primitive ideal of U p g q , takingthe kernel of the quantum comoment map, and the collection of ideals obtained this way will bedenoted C Ď Prim U p g q . We also consider the subset R Ď C obtained by only considering the rigidorbits O Ď l in Levi subalgebras. Proposition 9.5.
Suppose that g is classical. Then the image of J is equal to C and C “ R .Proof. The image of J is contained in C by construction and so we prove the opposite inclusion.Let O Ď l be birationally rigid and let X be the dense orbit in G ˆ P p Spec C r O s ˆ n q . Let A z p l q be the quantization of the Poisson deformation of X . Pick z P z p l q . We shall show thatKer p U p g q Ñ A z q lies in the image of J . If the generaised Springer map for the datum p l , O , z q is an isomorphism then Ker p U p g q Ñ A z q lies in the image of J by construction. So assume thatthis map is not birational.One can check that G ˆ P p z ` O ` n q Ñ Ind gl p O q is birational if and only if the map G z ˆ P z p z ` O ` n z q Ñ Ind g z l p O q is birational (see [Am20, Remark 3.2] for more detail). Thereforeby assumption the latter map is not birational. Define l : “ g z , O : “ Ind l l p O q . Then aftermaking an appropriate choice of parabolic P we have a factorisation G ˆ P p Spec C r O s ˆ n q (cid:16) G ˆ P p Spec C r O s ˆ n q (cid:16) Ind gl p O q . Let H , H Ď G e be the subgroup such that G { H i is thedense orbit in G ˆ P i p Spec C r O i s ˆ n i q . Since g is classical the component group Γ of G e is abelian[Ja04, § H in Γ is normal. It follows that G { H Ñ G { H is a Galois coverwith Galois group H { H , and that C r G { H s H – C r G { H s (invariants under the right action).Now by Lemma 9.4 and [Lo16, Proposition 3.21] the group H acts on the filtered quantization A z of G { H with parameter z , by filtered automorphisms, and these automorphisms lift the actionon C r G { H s . Since the action factors through a finite group the filtration splits for H and wededuce that gr p A H z q – C r G { H s . In other words, A H z is a filtered quantization of G { H . Wededuce that A H z is isomorphic to a fibre of the universal quantization of G { H . Since the imageof the comoment map g Ñ C r G { H s is H -invariant we conclude that the kernels of U p g q Ñ A z and U p g q Ñ A H z are equal. However G { H (cid:16) G { H (cid:16) O and G { H is a cover of the inducedorbit of strictly lower degree than G { H . By induction we know that this kernel lies in the imageof J , which therefore equals C .To complete the proof it remains to show that C Ď R . We take a birationally rigid orbit O Ď l which is not rigid. Let p l , O q be a rigid induction datum such that the sheet corresponding to p l , O q contains the birational sheet corresponding to p l , O q . Such a choice is always possiblebecause the closure of a birational sheet is irreducible. We may assume that l Ď l so that z p l q Ď z p l q , and we retain the notation H , H as above. Now take the quantization A z p l q ofthe cover corresponding to p l , O q . There is a map C r z p l qs Ñ C r z p l qs given by restriction offunctions. The base change A z p l q b C r z p l qs C r z p l qs is a flat C r z p l qs -algebra carrying an action of H by algebra automorphisms fixing the base pointwise. Since the action of H factors througha finite group the previous paragraph shows that p A z p l q b C r z p l qs C r z p l qsq H is a quantization of aflat deformation of C r G { H s . If A W z p l q denotes the quantization of C r G { H s constructed in [Lo16, §
3] then the universal property of Theorem 3.4 of loc. cit. implies that there is a unique map C r z p l qs W Ñ C r z p l qs such that A W z p l q b C r z p l q{ W s C r z p l qs „ ÝÑ p A z p l q b C r z p l qs C r z p l qsq H Furthermore the morphism is G -equivariant and so we get induced maps from Z p g q to C r z p l q{ W s and C r z p l qs which form a commutative triangle with C r z p l qs W Ñ C r z p l qs . Using Lemma 9.1 wesee C r z p l qs is finite over Z p g q , hence it is also finite over C r z p l qs W . In particular z p l q Ñ z p l q{ W is a finite morphism, hence closed and surjective. This shows that the primitive ideals whichappear as kernels of the moment maps for fibres of A W z p l q already appear as kernels for fibres of A z p l q . It follows that C Ď R and this completes the proof. (cid:3) Quantum Hamiltonian reduction of universal quantizations.
Now we keep fixed e P N p g q , the adjoint orbit O : “ Ad p G q e , an sl -triple t e, h, f u for e and write G e p q for the point-wise stabiliser of the triple. Also g e p q “ Lie G e p q is consistent with the grading notation fromSection 2.2. The component group is Γ : “ G e {p G e q ˝ – G e p q{ G e p q ˝ .The category HC p g q of Harish-Chandra U p g q -bimodules consists of U p g q -bimodules, finitelygenerated on both sides, which are locally finite for ad p g q . It follows from [GG02] that U p g , e q admits G e p q -action by filtered automorphisms, and a quantum comoment map g e p q ã Ñ U p g , e q .Therefore one can define the category HC G e p q U p g , e q of G e p q -equivariant Harish-Chandra U p g , e q -bimodules to be the category of U p g , e q -bimodules, finitely generated on both sides, with a com-patible G e p q -action which differentiates to the ad p g e p qq -action (see [Lo11] for example).Losev has constructed a pair of adjoint functors p‚ : , ‚ : q with ‚ : : HC U p g q Ô HC G e p q U p g , e q : ‚ : which behave nicely when applied to quantizations of nilpotent orbit covers. The definition firstappeared in [Lo11] and depends on the decomposition theorem from [Lo10a] which we will notdescribe here. Instead we recall from [Lo11, § ‚ : is isomorphic to a quantum Hamiltonianreduction M : – p M { M g pă –1 q χ q ad p g pă qq , which is in the spirit of the definitions used in this paper.Let p l , O q be a rigid induction datum for O leading to a finite cover X “ G { H (cid:16) O . Let A denote the family of filtered quantizations of C r X s over P “ z p l q . Since A P HC U p g q , there is anatural G e p q -action on A : P HC G e p q U p g , e q . Lemma 9.6.
The algebra A : is commutative and A G e p q: “ p A : q G e p q – C r P s .Proof. For z P P write I z Ď C r P s for the corresponding maximal ideal, and p´q b C r P s C z forthe specialisation of C r P s -modules. Since C r P s Ď A is central, the image of the map C r P s Ñ A { A g pă –1 q χ lies in the space of ad p g pă qq -invariants, giving C r P s Ñ A : .We claim that A : b C r P s C z – p A b C r P s C z q : . Since ‚ : is exact [Lo11, Lemma 3.3.2] there is asurjective map A : (cid:16) p A b C r P s C z q : which factors to A : b C r P s C z (cid:16) p A b C r P s C z q : . Write U p g q φ p z q for the central reduction at φ p z q P Spec Z p g q where φ : P Ñ Spec Z p g q is induced by the quantumcomoment map. By Lemma 9.1 it will suffice to check that A : b U p g q U p g q φ p z q – p A b U p g q U p g q φ p z q q : .Using the fact that ‚ : is a tensor functor [Lo11, Theorem 1.3.1(3)] this will follow from the claimthat p U p g q φ p z q q : – U p g , e q φ p z q . Using [Gi09, Lemma 4.4.1] we can deduce the claim from thesemiclassical limit: we must show that p C r N p g qs{ C r N p g qs g pă –1 q χ q ad p g pă qq – C r N p g q X p e ` g f qs ,which follows directly from [GG02, Lemma 2.1].Now write A ab : for the abelian quotient of A : , defined by the ideal generated by commutators.Write C for the cokernel of the projection A : Ñ A ab : in the category of C r P s -modules. By [Lo16,Lemma 5.2(1)] A : b C r P s C z is commutative for all z . Since the functor p´q b C r P s C z is right exact on C r P s -modules we have C b C r P s C z “ z . This forces C “
0, which implies that A : iscommutative.Since the image of C r P s Ñ A lies in the centre it is G -invariant, and so the image of C r P s Ñ A : is G e p q -invariant. Consider the cokernel C of C r P s Ñ A G e p q: in the category of C r P s -modules. Using [Lo16, Lemma 5.2(1)] and a similar argument to the previous paragraph we seethat C b C r P s C z “ z , and deduce C “
0. This completes the current proof. (cid:3)
Proof of Theorem 9.2.
Let O P g be nilpotent and let Z denote the union of all sheets containing O . It follows from Lemma 9.3 that the image of J intersected with Prim O U p g q is precisely J p Z q .Let S , ..., S l be the sheets of g containing O and let p l , O q , ..., p l l , O l q be the correspondingrigid induction data. If we write P i : “ z p l i q then each datum p l i , O i q gives rise to a cover X i of O and a quantization A P i of the Poisson deformation X P i Ñ P i of X i . These each come equippedwith comoment maps U p g q Ñ A P i . For z P P i write A P i ,z for the specialisation at z . We write R O for the set of ideals t K i,z : “ Ker p U p g q Ñ A P i ,z q | i “ , ..., l, z P P i u . Since A P i ,z quantizes anirreducible variety it follows that K i,z is completely prime, and by the Dixmier–Mœglin equivalence R O Ď Prim U p g q [Di96, 8.5.7]. The argument in Lemma 9.3 shows that actually R O Ď Prim O U p g q and, applying Proposition 9.5, the current proof will be complete if we can show that R O is equalto the set of annihilators Ann U p g q p Q b U p g ,e q C η q of one dimensional U p g , e q -modules C η .Consider the B : “ ś i C r P i s -algebra A : “ ś i A P i together with the product of quantum como-ment maps U p g q Ñ A . By [Lo16, Lemma 5.1] we get a map U p g , e q Ñ A : which factors throughthe G e p q -invariants.Both algebras are filtered and we consider the graded homomorphism of Rees algebras R h U p g , e q Ñ R h p A G e p q: q – R h B (apply Lemma 9.6). Consider the ideal D “ p h ´ r a, b s | a, b P U p g , e qq Ď R h U p g , e q , which contains the commutator ideal. Write U p g , e q ab h for the quotient R h U p g , e q{ D .We obtain a morphism of schemes over A C “ Spec C r h s φ : C ˆ l ď i “ P i “ Spec B Ñ Spec U p g , e q ab h . We claim that for i “ , ..., l the image φ p C ˆ P i q is not contained in Ť j ‰ i φ p C ˆ P j q .Suppose the opposite. Let S “ e ` g f be the Slodowy slice for e . We have Spec U p g , e q ab0 “ Spec S p g , e q ab – C r S X Ť i S i s as reduced schemes, by Proposition 6.2. Let r S denote the preimageof S under the moment map r X P i Ñ g . The image of the morphism φ : Ť i P i Ñ Spec U p g , e q ab0 identifies with the union of images of the maps r S Ñ S and the latter is equal to S X Ť i S i . Ourassumption that φ p C ˆ P i q is contained in the image of the union of the other components implies S X S i “ φ p P i q Ď Ť j ‰ i φ p P j q “ S X Ť j ‰ i S j , which is impossible. The contradiction confirmsthe claim.Finally we consider the morphism φ over p h ´ q P Spec C r h s . It follows from Lemma 9.1 that φ is a finite morphism, hence closed. By the previous paragraph the dimensions of the irreduciblecomponents of φ p Ť i P i q are given by dim z p l i q for i “ , ..., l . By Theorem 1.1 these are thedimensions of the irreducible components of E p g , e q – Spec U p g , e q ab1 and so by Lemma 5.1 we have φ surjective.This shows that for each i, z there is a one dimensional representation C η of U p g , e q such that p K i,z q : “ Ann U p g ,e q C η . The functor ‚ : admits a right adjoint ‚ : , first constructed in [Lo10a]. By [Lo11, Theorem 1.3.1(1)] both ‚ : , ‚ : are left exact, i.e. tranform kernels into kernels. Itfollows that pp K i,z q : q : “ p Ker p U p g , e q Ñ p A P i ,z q : q : “ Ker p U p g q Ñ pp A P i ,z q : q : q . Thanks to[Lo16, Lemma 5.2(3)] this is equal to K i,z . Finally by [Lo10a, Theorem 1.2.2(ii)] we have K i,z “p Ann U p g ,e q C η q : “ Ann U p g q Q b U p g ,e q C η . The proof of the theorem is complete. (cid:3) References [Am20]
F. Ambrosio , Birational sheets in reductive groups,
Math. Z. (2020). https://doi.org/10.1007/s00209-020-02597-3.[ACET20]
F. Ambrosio, G. Carnovale, F. Esposito, L. Topley , Universal filtered quantizations of nilpotentSlodowy slices, arXiv:2005.07599 (2020).[AM69]
M. F. Atiyah, I. G. Macdonald
Introduction to commutative algebra. Addison-Wesley PublishingCo., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.[Bo81]
W. Borho , ¨Uber Schichten halbeinfacher Lie-Algebren.
Invent. Math. (1981), 283–317.[BG18] J. Brown, S. M. Goodwin , On the variety of 1-dimensional representations of finite W -algebras inlow rank. J. Algebra (2018), 499–515.[BG03]
K. Brown & I. Gordon , Poisson orders, symplectic reflection algebras and representation theory.
J.Reine Angew. Math. (2003), 193–216.[BG07]
J. Brundan and S. M. Goodwin , Good grading polytopes.
Proc. London Math. Soc. (2007),155–180.[BK05] J. Brundan & A. Kleshchev , Parabolic presentations of the Yangian Y p gl n q . Comm. Math. Phys. (2005), no. 1, 191–220.[BK06]
J. Brundan, A. Kleshchev , Shifted Yangians and finite W -algebras. Adv. Math. (2006), 136–195.[BT18]
J. Brundan and L. Topley , The p -centre of Yangians and shifted Yangians. Mosc. Math. J. (2018), 617–657.[Bu] M. Bulois , Geometry of sheets in ordinary and symmetric Lie algebras. in preparation. [CM93]
D.H. Collingwood and
W. McGovern , “Nilpotent orbits in semisimple Lie algebras”. Van NostrandReinhold, New york, 1993.[Dir50]
P. A. M. Dirac , Generalized Hamiltonian dynamics. Canad. J. Math. 2 (1950), 129–148.[Di96]
J. Dixmier , Enveloping algebras. Revised reprint of the 1977 translation. Graduate Studies in Mathe-matics, 11. American Mathematical Society, Providence, RI, 1996.[DSV07]
P. Damianou, H. Sabourin, P. Vanhaecke,
Transverse Poisson structures to adjoint orbits insemisimple Lie algebras.
Pacific J. Math. (2007), no. 1, 111–138.[GG02]
W. L. Gan, V. Ginzburg , Quantization of Slodowy slices.
Internat. Math. Res. Notices (2002),243–255.[Gi09] V. Ginzburg , Harish–Chandra bimodules for quantized Slodowy slices.
Represent. Theory (2009),236–271.[GT19a] S. M. Goodwin & L. Topley , Modular finite W -algebras, Int. Math. Res. Not. IMRN (2019), no. 18,5811–5853.[GT19b] , Minimal-dimensional representations of reduced enveloping algebras for gl n . Compos. Math . (2019), no. 8, 1594–1617.[GT19c] , Restricted shifted Yangians and restricted finite W -algebras. arXiv:1903.03079 (2019).[Hum75] J. E. Humphreys , “Linear algebraic groups”. Graduate Texts in Mathematics, No. 21. Springer-Verlag,New York-Heidelberg, 1975.[Im05]
A. Im Hof , “The Sheets of a Classical Lie Algebra”. Inauguraldissertation zur Erlangung der W¨urdeeines Doktors der Philosophie, 2005, available at http://edoc.unibas.ch [Ja04]
J.C. Jantzen , “Nilpotent orbits in representation theory”, in: B. Orsted (ed.), “Representation andLie theory”, Progr. in Math., , 1–211, Birkh¨auser, Boston 2004. [Ka82] P. Katsylo , Sections of sheets in a reductive algebraic Lie algebra.
Math. USSR-Izv. (1983), 449—458.[LPV13] C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke , “Poisson structures”. Grundlehren der Math-ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 347. Springer, Heidel-berg, 2013.[Lo10a] I. Losev, Quantized symplectic actions and W -algebras. J. Amer. Math. Soc. (2010), no. 1, 35–59.[Lo10b] , Quantizations of nilpotent orbits vs 1-dimensional representations of W -algebras. arXiv:1004.1669 , 2010.[Lo11] , Finite dimensional representations of W -algebras. Duke Math. J. (2011), no.1, 99–143.[Lo15] , Dimensions of irreducible modules over W-algebras and Goldie ranks.
Invent. Math. (2015),no. 3, 849—923.[Lo16] , Deformations of symplectic singularities and Orbit method for semisimple Lie algebras. arXiv:1605.00592 , 2016.[LS79]
G. Lusztig & N. Spaltenstein , Induced unipotent classes,
J. London Math. Soc. (2) , (1979),41–52.[Ma86] H. Matsumara , “Commutative ring theory”, Cambridge University Press, Cambridge, 1986.[MR01]
J. C. McConnell & J.C. Robson “Non-commutative Noetherian rings”. Revised edition. GraduateStudies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001.[Pr02]
A. Premet
Special transverse slices and their enveloping algebras. With an appendix by Serge Skryabin.
Adv. Math. (2002), no. 1, 1–55.[Pr07] , Primitive ideals, non-restricted representations and finite W -algebras. Mosc. Math. J. (2007)743–762.[Pr10] , Commutative quotients of finite W-algebras. Adv. Math.
225 (2010), no. 1, 269–306.[Pr14] , Multiplicity-free primitive ideals associated with rigid nilpotent orbits.
Transform. Groups (2014), no. 2, 569–641.[PT14] A. Premet, L. Topley , Derived subalgebras of centralisers and finite W-algebras.
Compos. Math. (2014), no. 9, 1485–1548.[PT20] , Modular representations of Lie algebras of reductive groups and Humphreys’ conjecture.arXiv:2010.10800, 2020.[Ra01]
E. Ragoucy , Twisted Yangians and folded W -algebras. Internat. J. Modern Phys. A (2001), no.13, 2411–2433.[Ser55] J.-P. Serre , G´eom´etrie alg´ebrique et g´eom´etrie analytique. (French) Ann. Inst. Fourier (Grenoble) 6(1955/56), 1–42.[Ser06] , Lie algebras and Lie groups. 1964 lectures given at Harvard University. Lecture Notes in Math-ematics, 1500. Springer-Verlag, Berlin, 2006.[Sl80]
P. Slodowy , “Simple Singularities and Simple Algebraic Groups”. Lecture Notes in Mathematics 815,Springer-Verlag, Berlin Heidelberg, 1980.[To17]
L. Topley , A Morita theorem for modular finite W-algebras,
Math. Z. (2017), 685–705.[Vo94]
D. Vogan , “The orbit method and unitary representations for reductive Lie groups”. Based on Lecturesto the European School of Group Theory, Sandbjerg, Denmark, 1994. „ dav/dmkrev.pdf .[We83] A. Weinstein , The local structure of Poisson manifolds.
J. Differential Geom. (1983), no. 3, 523–557. Email: