One dimensional representations of finite $W$-algebras, Dirac reduction and the orbit method

In this paper we study the variety of one dimensional representations of a finite $W$-algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a conjecture of Losev describing the image of his orbit method map. In order to do so we first establish new Yangian-type presentations of semiclassical limits of the $W$-algebras attached to distinguished nilpotent elements in classical Lie algebras, using Dirac reduction.


1.1.
Let G be a complex connected reductive algebraic group with Lie algebra g, nilpotent cone N pgq. Identify g with g˚by a choice of non-degenerate G-invariant trace form on g. The primitive spectrum Prim U pgq is the set of all primitive ideals of the enveloping algebra, equipped with the Jacobson topology. These ideals are classically studied via their invariants, and the most important of these are the associated variety and Goldie rank. The associated variety VApIq is defined to be the vanishing locus in g of the associated graded ideal gr I Ď Spgq " Crgs with respect to the PBW filtration. A celebrated theorem of Kostant states that N pgq is the vanishing locus of the positive degree invariant polynomials Spgq G whilst Joseph's irreducibility theorem states that VApIq 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 rank is defined to be the uniform dimension of the primitive quotient U pgq{I.
It is natural to consider the decomposition Prim U pgq " Ť O Prim O U pgq, were the union is taken over all nilpotent G-orbits O and Prim O U pgq " tI P Prim U pgq | VApIq " Ou. Now fix an orbit O Ď N pgq and e P O, and let U pg, eq denote the finite W -algebra, first associated to pg, eq by Premet [Pr02]. The reductive part of the centraliser G e p0q acts naturally on U pg, eq by algebra automorphisms, and this induces an action of the component group Γ " G e p0q{G e p0q˝on the category of finite dimensional modules. Losev famously gave a new construction of U pg, eq via deformation quantization [Lo10a] and used this to show that Prim O U pgq is in bijection with U pg, eq -mod f.d. {Γ [Lo11]. The one dimensional representations of U pg, eq play an especially important role here for two reasons: on one hand the images under Skryabin's equivalence are all completely prime, and therefore play a key role in Joseph's theory of Goldie rank polynomials [Lo15], and on the other hand they classify quantizations of G-equivariant coverings of O [Lo10b].
1.2. The above narrative leads us to consider the affine scheme Epg, eq :" Spec U pg, eq ab associated to the maximal abelian quotient. By Hilbert's nullstellensatz the closed points classify the one dimensional representations of U pg, eq. The work of Losev and Premet [Lo10a,Pr14] shows that Epg, eq is nonempty and in [Pr10,PT14] the first steps were made towards a full description of the variety of closed points.
Recall that the sheets of g are the maximal irreducible subsets consisting of orbits of constant dimension. They are classified via the theory of decomposition classes which, in turn, are classified by the Lusztig-Spaltenstein induction data. One of the main themes of [PT14], which we build upon in this paper, is the interplay between the sheets of g and the structure of Epg, eq.
In the case where g is classical, by which we mean a simple Lie algebra of type A, B, C or D, we described a combinatorial procedure for enumerating the sheets of g containing a given orbit O, and we named it the Kempken-Spaltenstein (KS) algorithm. This algorithm played a key role in [PT14, Theorem 1], which states that for g classical the variety Epg, eq is an affine space if and only if e lies in a unique sheet of g. The first goal of this paper is to elucidate the structure of Epg, eq when g is classical and e is singular, which means that it lies in multiple sheets.
Let S 1 , ..., S l be the set of all sheets containing O P N pgq{G. If e`g f denotes the Slodowy slice to O at e then we define the Katsylo variety e`X :" pe`g f q X In [Ka82] Katsylo used this variety to construct a geometric quotient of the variety Ť l i"1 S i . Perhaps the first indication that e`X should influence the representation theory of U pg, eq appeared in [Pr10]. Premet used reduction modulo p to show that there is a surjective map on the sets of irreducible components Comp Epg, eq Ý↠ Comppe`Xq (1.2) which restricts to a dimension preserving bijection on some subset of Comp Epg, eq. The following is our first main result.
Theorem 1.1. When g is a simple Lie algebra of classical type, the map (1.2) is a dimension preserving bijection.
Thanks to [Pr10,Corollary 3.2] the variety Epg, eq is irreducible when g " sl n , and so Theorem 1.1 follows in this case, using the properties of (1.2) listed above. Hence we focus on types B, C and D in this paper. For these classical types, 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 8.3. Thus Theorem 1.1 provides an effective method for computing dimensions of all components of Epg, eq. We note that these dimensions were calculated in low ranks in [BG18].
1.3. In [Lo22] Losev demonstrated that for every conic symplectic singularity the functor of filtered quantizations of Poisson deformations admits an initial object (see [ACET20] for more detail). We call such an initial object a universal quantization. Using this result, he then showed that every coadjoint orbit uniquely 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 a map J : g˚{G Ñ Prim U pgq, which is known to be an embedding whenever g is classical [Lo22,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 orbit method can be found in [Vo94], whilst Losev's construction is surveyed in Section 9.2 of the current paper.
It is important to understand and characterise the primitive ideals appearing in the image of the orbit method map for g. Losev has conjectured that they are precisely the annihilators of simple Whittaker modules coming from one dimensional representations of W -algebras. In the final Section of this paper we deduce his conjecture from Theorem 1.1.
Theorem 1.2. For g classical, the image of J consists of primitive ideals obtained from one dimensional representations of W -algebras under Skryabin's equivalence.
1.4. It is worth commenting on Theorems 1.1 and 1.2 in the context of exceptional Lie algebras: there are 6 rigid orbits for which the finite W -algebra admits two 1-dimensional representations (see [PT21, Table 1] and [Pr14] for more detail). In these cases Epg, eq has two points and e`X consists of a single point. This shows that Theorem 1.1 cannot hold outside classical types. Similarly, for each rigid orbit Losev's orbit method map attaches a unique primitive ideal of U pgq with associated variety equal to the closure of G¨e (see Lemma 9.10) and so these same examples show that Theorem 1.2 also fails outside classical types.

1.5.
There are several new tools involved in the proof of Theorem 1.1, and we now briefly describe the most important ones. One of the basic ideas comes from deformation theory. The finite W -algebra is a filtered quantization of the transverse Poisson structure on the Slodowy slice Cre`g f s and so there are two natural degenerations associated to Epg, eq. On the one hand, we may degenerate U pg, eq to the classical finite W -algebra Spg, eq -Cre`g f s and then abelianise, which leads to the spectrum of the maximal Poisson abelian quotient Fpg, eq :" Spec Spg, eq ab . On the other hand we may abelianise and then degenerate, which leads us to the asymptotic cone of Epg, eq, denoted CEpg, eq :" Specpgr U pg, eq ab q.
One of the general results of this paper states that there is a closed immersion of schemes inducing a bijection on closed points, which we prove using reduction to prime characteristic CEpg, eq ãÝÑ Fpg, eq. (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 Fpg, eq is e`X (Proposition 6.2). Therefore the main theorem will follow if we can show that Comp Epg, eq is no larger than Comp CEpg, eq. It is an elementary fact from commutative algebra that Comp Epg, eq ď Comp CEpg, eq provided CEpg, eq is reduced, and so our approach is to show that gr U pg, eq ab has no nilpotent elements. By (1.3) it suffices to show that Spg, eq ab is reduced.
In this paragraph we take g classical. By passing to the completion of Spg, eq ab at the maximal graded ideal and using the fact that the Slodowy slice is transverse to every point of e`X we are able to reduce the problem of showing that Spg, eq ab is reduced to the case where e is distinguished.
To be more precise, Spg, eq ab is reduced if and only if the completion at the maximal graded ideal is so, and we show that this completion is reduced if and only if pSpg,ẽqxq ab is reduced, where x is a 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].

1.6.
Now Theorem 1.1 will follow if we can show that Spg, eq ab is reduced for distinguished elements e. We introduce a new method to attack this problem. If X is a complex Poisson scheme of finite type and H is a reductive group acting rationally by Poisson automorphisms then the invariant subscheme X H can be equipped with a Poisson structure via Dirac reduction. The reduced Poisson algebra will be denoted RpCrXs, Hq. Now if g " LiepGq for G reductive and H Ď Autpgq is a reductive group fixing an sl 2 -triple te, h, f u Ď g then we prove the following isomorphism of Poisson algebras RpSpg, eq, Hq " ÝÑ Spg H , eq.
We apply this isomorphism in the case where g " gl n and H " Z{2Z is generated by some involution τ . Then g τ " so n or g " sp n . It follows from the work of Brundan and Kleshchev that Spg, eq is a quotient of a (semiclassical) shifted Yangian y n pσq depending on e. If e is distinguished then we can identify an involution τ on y n pσq defined so that the Poisson homomorphism y n pσq ↠ Spg, eq is τ -equivariant. As a consequence we can apply the Dirac reduction procedure to the shifted Yangian y n pσq and thus obtain a presentation of the Poisson structure on Spg, eq. We mention that obtaining presentations of finite W -algebras outside of type A is one of the key open problems in the field.
Let m 0 be the maximal graded ideal. Finally we use the presentation of Spg, eq ab to calculate generators and certain relations of gr m 0 Spg, eq ab . We see that the reduced algebra of gr m 0 Spg, eq ab is naturally identified with the coordinate ring on the tangent cone CrTC e pe`Xqs. Michaël Bulois has recently demonstrated that the sheets of g containing e are transversal at e (work in preparation [Bu]), which allows us to calculate the dimensions of the irreducible components of TC e pe`Xq, once again they are determined by the KS algorithm. Using a combinatorial argument we then show that the relations mentioned above give a full presentation for gr m 0 Spg, eq ab . It follows quickly that both gr m 0 Spg, eq ab and Spg, eq ab are reduced, which allows us to conclude the proof of Theorem 1.1.

1.7.
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 combine Theorem 3.7 and Propositions 4.8 and 4.10.
Theorem 1.3. Let g " so N or sp N and let e be a distinguished nilpotent element with partition λ " pλ 1 , ..., λ n q. Then Spg, eq is generated as a Poisson algebra by elements tη p2rq i | 1 ď i ď n, 0 ă ru Y tθ prq i | 1 ď i ă n, λ i`1´λi 2 ă ru (1.5) together with the following relations tη p2rq i relations. The main results on the Dirac reduction of y n pσq are presented in Subsection 3.4, including the canonical grading, the loop filtration, the PBW theorem and the presentation by generators and relations. All of the results about Rpy n pσq, τ q are ultimately deduced from similar results on y n pσq. In Section 4 we recall the definition of Brundan-Kleshchev's isomorphism y n pσq ↠ Spg, eq and show by an explicit calculation that this is τ -equivariant for a suitable choice of involution on y n pσq. By (1.4) this leads to a surjection Rpy n pσq, τ q ↠ Spg τ , eq and in Subsection 4.4 we describe a full set of Poisson generators for the kernel.
Part II: In Section 5 we gather together some important general facts about degenerations and completions of schemes, as well as reviewing the theory of rank stratification and symplectic leaves of a Poisson scheme. In Section 6 we explain how Spg, eq ab is related to Katsylo variety e`X, and use the aforementioned results of Bulois to enumerate the irreducible components of TC e pe`Xq and calculate their dimensions. In Section 7 we introduce quantum finite Walgebras and prove the existence of the closed immersion (1.3). The proof of the latter uses 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 Fpg, eq red " Cre`Xs from Section 6. Finally in Section 8 we describe the Kempken-Spaltenstein algorithm, as well as its relationship with sheets, and then use this to construct an algebraic variety X λ associated to a distinguished nilpotent orbit O with partition λ, which we call the combinatorial Katsylo variety. In Theorem 8.8 we use the presentation of Spg, eq ab obtained in part I of the paper to demonstrate that CrX λ s ↠ Spg, eq ab ↠ CrTC e pe`Xqs, and we show that these are isomorphisms by comparing the dimensions of the irreducible components. In particular this implies that Spg, eq ab is reduced for e distinguished. Theorem 8.9 reduces the general case to the distinguished case. Finally we conclude the proof of Theorem 1.1 in Subsection 8.8, making use of deformation techniques gathered in Section 5.
In Subsection 9.1 we recall the classification of Poisson deformations and their quantizations for conic symplectic singularities, due to Losev and Namikawa. In Subsection 9.2 we recall Losev's theory of birational induction, describe the orbit method map and formulate a slight refinement of Theorem 1.2. In Subsection 9.3 we recall some of the key properties of Losev's dagger functor from [Lo11] and explain how they treat universal quantizations of affinisations of orbit covers. Finally in Subsection 9.4 we relate the orbit method map to U pg, eq ab and prove Theorem 1.2. Notation and conventions. The following notation will be used throughout the paper. All algebras and vector spaces are defined over C, except in Section 7 where we use reduction modulo a large prime. We use capital letters G, H, ... for algebraic groups and gothic script g, h, ... for their Lie algebras. If A is an algebra and X Ď A then pXq will denote the two-sided ideal of A generated by X.

Contents
If g is a Lie algebra then U pgq denotes the enveloping algebra and Spgq the symmetric algebra equipped with its Lie-Poisson structure: this is the unique Poisson bracket on Spgq extending the Lie bracket on g Ď Spgq. The associated graded algebra of an almost commutative, filtered associative algebra is equipped with 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.
Almost all schemes appearing in this paper will be affine schemes 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. In fact, the consideration of nilpotent elements will be vital to our main results. Occasionally we write MaxpAq for the variety of closed points in the prime spectrum of a commutative algebra A. If X is a noetherian scheme then we write ComppXq for the set of irreducible components.
Acknowledgements. I would like to offer thanks to Simon Goodwin, Ivan Losev, Dmytro Matvieievskyi, Sasha Premet and Matt Westaway for useful comments on the first version of this paper. I would also like to thanks Lukas Tappenier for careful reading, and spotting various typos. I'm especially grateful to Ivan for suggesting some of the constructions used in the proof of Theorem 1.2, and to Sasha to whom this paper is dedicated -his insightful teaching first introduced me to these fascinating problems. I would also like to thank the referees for their many comments which have improved the exposition of this paper significantly. I have also benefited from many interesting conversations and email correspondence with Jon Brundan, Michaël Bulois, Paul Levy, Anne Moreau, and Daniele Valeri, and am grateful for all their help. Some of these results were announced at the conference "Geometric and automorphic aspects of W -algebras", Lille 2019. This research is supported by the UKRI Future Leaders Fellowship project "Geometric representation theory and W -algebras", grant numbers MR/S032657/1, MR/S032657/2, MR/S032657/3. Part 1. Presentations of classical W -algebras 2. Dirac reduction for classical finite W -algebras 2.1. Invariants via Dirac reduction. Throughout this Section H will be a reductive algebraic group, although in our later applications H will be a cyclic group of order 2. When H acts locally finitely and rationally on some vector space A we write A`:" A H for the invariants and write Af or the unique H-invariant complement to A`in A.
Let X be an complex affine Poisson variety and suppose z 1 , ..., z n P CrXs such that the determinant of the matrix ptz i , z j uq 1ďi,jďn is a unit. In his seminal paper [Dir50] Dirac defined a new Poisson bracket on CrXs, such that the z i are Casimirs, thus equipping CrXs{pz 1 , ..., z n q with a Poisson structure. In fact this is a special case of the following procedure: say that I Ď CrXs is a Dirac ideal if N CrXs pIq ↠ CrXs{I surjects where N CrXs pIq :" tf P CrXs | tf, Iu Ď Iu denotes the Poisson idealiser. Then the subscheme associated to I inherits a Poisson structure from X, and we call this induced structure the Dirac reduction [LPV13, §5.4.3].
Now let H be a reductive group acting rationally on X by Poisson automorphisms. We regard the set of invariants X H as a (not necessarily reduced) affine scheme such that the structure sheaf has global sections CrX H s :" CrXs{I H and I H :" ph¨f´f | h P H, f P CrXsq. Although I H is usually not a Poisson ideal, it is always a Dirac ideal (this is a corollary of Lemma 2.1) so that X H acquires the structure of a Poisson scheme.
The following suggests an alternative approach of the Poisson structure on X H , better-suited to calculations.
Lemma 2.1. The map CrXs H Ñ CrX H s is surjective and Proof. If f P CrXs H , g P CrXs and h P H then tf, h¨g´gu " h¨tf, gu´tf, gu and so CrXs H Ď N CrXs pI H q is a Poisson subalgebra. Since H is reductive and acts rationally we can decompose If τ P AutpAq is a semisimple Poisson automorphism of finite order, then we often abuse notation writing RpA, τ q for RpA, Hq where H is the group generated by τ .
Since H is reductive the functor of H-invariants is exact on the category of locally finite, rational H-modules. This implies that whenever A is a Poisson algebra as above and I Ď A is an H-stable Poisson ideal we have RpA, Hq Ý↠ RpA{I, Hq. (1) V is a direct sum of locally finite H-modules with H-stable decomposition V " V H ' V´.
(2) SpV q is a Poisson algebra with H acting by Poisson automorphisms.
Then the natural map SpV H q Ñ RpSpV q, Hq is an isomorphism of commutative algebras.
Proof. Since SpV q " SpV H q'pV´q and SpV H q is H-fixed we must have SpV q´Ď pV´q. Combining with V´Ď SpV q´we deduce that I H " pV´q which proves the map SpV H q Ñ RpSpV q, Hq is a commutative algebra isomorphism. □ Remark 2.3. Let g be a Lie algebra and H a reductive group of automorphisms of g acting locally finitely. In this case the composition Spg H q Ñ Spgq H Ñ RpSpgq, Hq is a Poisson homomorphism and so Lemma 2.2 shows that Spg H q " ÝÑ RpSpgq, Hq as Poisson algebras.
2.2. Classical finite W -algebras. For the rest of the section we fix a connected reductive algebraic group G such that the derived subgroup is simply connected, and write g " LiepGq. Let κ : gˆg Ñ C be a choice of non-degenerate trace form on g which is preserved by Autpgq. Pick a nilpotent element e P g and write χ :" κpe,¨q P g˚. Pick an sl 2 -triple te, h, f u and write g " À iPZ gpiq for the grading by adphq-eigenspaces. Throughout the paper we use the notation gpď iq " À jďi gpjq and similar for gpă iq. Since e P gp2q we see that χ restricts to a character on gpă -1q. Make the following notation gpă -1q χ :" tx´χpxq | x P gpă -1qu Ď Spgq. The nilpotent Lie algebra gpă 0q is algebraic and we write gpă 0q " Lie Gpă 0q.
The Kazhdan grading is defined on Spgq by placing gpiq in Kazhdan degree i`2. Notice that gpă -1q χ generates a homogeneous ideal of Spgq and that Spgq{Spgqgpă -1q χ inherits a connected grading in non-negative degrees, with Spg, eq embedded as a graded subalgebra, with Poisson bracket in degree´2.
Since the grading is good for e the map g e Ñ Spgq{Spgqgpă -1q χ is injective.
(1) There exists a Kazhdan graded map θ : g e Ñ Spg, eq such that θpxq´x P m 2 0 . (2.5) Furthermore θ can be chosen to be equivariant with respect to any reductive group of Poisson automorphisms acting rationally on Spg, eq by graded automorphisms.
(2) If θ is any map satisfying (2.5) then the induced map Spg e q Ñ Spg, eq is an isomorphism of commutative algebras.
Proof. It follows from [GG02, Lemma 2.1] that the restriction homomorphism Spgq{Spgqgpă -1q χ -Crχ`gpă -1q K s Ñ Crχ`g f s -Spg e q gives a G e p0q-equivariant isomorphism Spg, eq Ñ Spg e q of commutative algebras, where gpă -1q K :" tη P g˚| ηpgpă -1qq " 0u. Taking the inverse isomorphism restricted to g e Ď Spg e q gives the desired map θ. If H Ď G e p0q is any reductive group of Poisson isomorphisms then θ can be replaced with an H-equivariant map using the standard trick of projecting onto isotypic components of Spg, eq for the H-action. Now if θ is any map satisfying (2.5) then Spg e q Ñ Spg, eq and it suffices to show that this map is surjective. If x 1 , ..., x r P g e is a homogenous basis then [Ja04, Lemma 7.1] shows that θpx 1 q, ..., θpx r q generate a graded radical ideal of finite codimension. The only such ideal is the maximal graded ideal of Spg, eq and this implies that θpx 1 q, ..., θpx r q generate Spg, eq as a commutative algebra. □ 2.3. Dirac reduction of classical W -algebras. Now let H Ď Autpgq a reductive subgroup which fixes our choice of sl 2 -triple pointwise. Make the notation g`:" g H and let g´denote an H-invariant complement to g`in g. Also write G`Ď G for the connected component of the subgroup consisting of elements g P G such that Adpgq commutes with the action of H. Our next goal is to relate the Dirac reduction RpSpg, eq, Hq with the classical W -algebra Spg`, eq. Later in the paper we will only exploit this relationship in the case where g " gl N and H is a cyclic group of order 2. For the sake of simplicity, we state and prove our results in a much greater generality.
(1) The restriction of κ to g`is non-degenerate and g K " g´.
Proof. Since g´is spanned by elements th¨x´x | h P H, x P gu (see Lemma 2.1) it follows from a short calculation that κpg`, g´q " 0. Since κ is non-degenerate and g " g`' g´, part (1) follows.
Since κ is a non-degenerate trace form, we may pick a representation ρ : g Ñ glpV q such that κpx, yq " Trpρpxqρpyqq. If n Ď g`is a nilpotent ideal then by Engel's theorem there exists k ą 0 such that ρpnq k " 0. Hence pρpxqρpnqq k " 0 for all x P g`and n P n, so κpx, nq " 0. Since κ is non-degenerate the nilradical of g`is trivial.
If Ad G : G Ñ GLpgq is the adjoint representation then we consider ρ`:" Ad GLpgq˝A d G : G Ñ GLpEndpgqq and let ρ : G Ñ GLpW q be any faithful representation admitting ρ`as a direct summand. If we identify G (resp. g) with its image in GLpW q (resp. glpW q) via ρ (resp. d 0 ρ), and identify H with a subset of Endpgq Ď W , then G`is precisely the subgroup of GLpW q fixing H, and similar for g`. Now apply [Hum75, Theorem 13.2] to see that g`" LiepG`q. Since the nilradical of g`is trivial, G`is reductive thanks to [Hum75, Theorem 13.5]. □ Since g`is a reductive subalgebra of g containing pe, h, f q we can consider Spg`, eq. Furthermore H preserves gpă -1q χ and the induced action on Spgq{Spgqgpă -1q χ stabilises the Gpă 0q-invariants, so that H acts by Poisson automorphisms on Spg, eq.
We are now ready to formulate one of our first main theorems, stated in (1.4).
Theorem 2.6. Let g be the Lie algebra of a reductive group. The projection map ϕ defined in (2.13) descends to a Poisson isomorphism RpSpg, eq, Hq " ÝÑ Spg`, eq. (2.6) The proof of Theorem 2.6 will be given at the end of the current section. First we prepare for the proof with two Lemmas and a Proposition.
Thanks to Lemma 2.5(1) we know that g´is the orthogonal complement of g`with respect to κ. This implies (2.7) χpg´q " 0.
Proof. Observe that g`and g´are graded subspaces of g. In both parts (1) and (2) it is easy to see that the right hand side is contained in the left (for part (2) one should use (2.7)). Therefore to prove (1) and (2) it remains to show that we have the reverse inclusion. Let x P gpă -1q and f P Spgq, and let f " f``f´and x " x``x´be the decompositions over Spgq " Spg`q ' Spgqg´and g " g`' g´respectively. By (2.7) the projections of f px´χpxqq to Spg`q and Spgqg´are f`px`´χpx`qq and f`x´`f´px´χpxqq respectively. Therefore if f px´χpxqq lies in Spg`q we must have f`x´`f´px´χpxqq " 0 and f px´χpxqq " f`px`´χpx`qq, proving (1).
As a step towards proving (2) we claim that the projection of Spgq onto Spgqg´across the decomposition Spgq " Spg`q'Spgqg´preserves Spgqgpă -1q χ . To see this observe that Spgqgpă -1q χ is spanned by elements of the form f px´χpxqq with f P Spgq and x P gpă -1q. It suffices to show that the projection of such an element lies in Spgqgpă -1q χ . Expanding f " f``f´across Spgq " Spg`q ' Spgqg´and writing x " x``x´P g`' g´we see that the required projection is f x´`f´px`´χpx`qq P Spgqgpă -1q χ thanks to (2.7), which proves the claim.
Let F P Spgqg´X Spgqgpă -1q χ . Then F " ř i f i px i´χ px i qq for some elements f i P Spgq and x i P gpă -1q. The fact that F P Spgqg´implies that the projection of F to Spg`q is zero. Therefore if we replace each f i px i´χ px i qq with its projection to Spgqg´then we leave the equality F " ř i f i px i´χ px i qq unchanged. It follows that every element of Spgqg´X Spgqgpă -1q χ can be written as a sum of elements f px´χpxqq P Spgqg´with f P Spgq, x P gpă -1q (here we use the claim from the previous paragraph). This observation reduces (2) to showing that every such element f px´χpxqq P Spgqg´X Spgqgpă -1q χ lies in Spgqpg´pă -1q`g´g`pă -1q χ q. We let f px´χpxqq lie in the required space and write x " x``x´where x˘P g˘pă -1q. Then f px´´χpx´qq P Spgqg´pă -1q by (2.7), whilst f px`´χpx`qq P Spgqg´implies that f P Spgqg´so that f px`´χpx`qq P Spgqpg´g`pă -1q χ q. This proves (2). □ Lemma 2.8. The ideal Spgqgpă -1q χ is the direct sum of its intersections with Spg`q and Spgqg´. Therefore Spgq " Spg`q ' Spgqg´gives a G`pă 0q-module decomposition Proof. We begin by proving the claim The G`pă 0q-decomposition (2.8) is an immediate consequence.
The right hand side of (2.9) is clearly contained in the left hand side and so our proof will focus on the reverse inclusion. Let f P Spgq and x´χpxq P gpă -1q χ . Since gpă -1q " g`pă -1q'g´pă -1q we can consider two cases: (i) if x P g´then by (2.7) we have f px´χpxqq P Spgqg´X Spgqgpă -1q χ ; (ii) if x P g`then we can write f " f``f´P Spg`q ' Spgqg´, in which case f`px´χpxqq P Spg`qg`pă -1q χ Ď Spg`q X Spgqgpă -1q χ and f´px´χpxqq P Spgqg´X Spgqgpă -1q χ . Now the Lemma follows from Lemma 2.7. □ In what follows we will use the notation Spg, eq " Spg, eq`' Spg, eq´for the decomposition into trivial and non-trivial H-modules, generalising our notation for Spgq. Consider the two sets N :" tf P Spgq | g¨f´f P Spgqgpă -1q χ for all g P Gpă 0qu; (2.10) N`:" tf P Spg`q | g¨f´f P Spg`qg`pă -1q χ for all g P G`pă 0qu.
(i) N X Spgq`Ď Spgq and N`Ď Spg`q are Poisson subalgebras; (ii) ϕpN X Spgq`q Ď N`and the map ϕ : N X Spgq`Ñ N`is a Poisson homomorphism; (iii) The map π : Spgq Ñ Spgq{Spgqgpă -1q χ restricts to a surjective Poisson homomorphism N X Spgq`↠ Spg, eq`; (iv) The map π`: Spg`q Ñ Spg`q{Spg`qg`pă -1q χ restricts to a surjective Poisson homomorphism N`↠ Spg`, eq; (v) The kernel of the map N X Spgq`↠ Spg, eq`is contained in the kernel of the map π`˝ϕ. Thus ϕ induces a Poisson homomorphism Spg, eq`↠ Spg`, eq which we also denote by ϕ; (vi) Spg, eq`X Spg, eqSpg, eq´is contained in the kernel of Spg, eq`↠ Spg`, eq, inducing a surjective Poisson homomorphism ϕ : RpSpg, eq, Hq ↠ Spg`, eq. (2.14) Proof. Since H acts on Spgq by Poisson automorphisms the invariant subspace Spgq`is a Poisson subalgebra. Therefore the proofs of the two claims in (i) are identical, and we will only prove that N`Ď Spg`q is a Poisson subalgebra. It is evidently closed under multiplication so we only need to show that it is closed under the bracket. Suppose that f 1 , f 2 P N`, that g P G`pă 0q and that g¨f i´fi " h i P Spg`qg`pă -1q χ . We have g¨tf 1 , f 2 u " tg¨f 1 , g¨f 2 u " tf 1`h1 , f 2`h2 u and so (i) will follow if we can show that tf 1 , h 2 u, th 1 , f 2 u, th 1 , h 2 u P Spg`qg`pă -1q χ . Since χ vanishes on gpă´2q we have rg`pă -1q χ , g`pă -1q χ s Ď g`pă -1q χ , therefore th 1 , h 2 u P Spg`qg`pă -1q χ by the Leibniz rule. To complete the proof of (i) we observe that N`X Spg`qg`pă -1q χ is a Poisson ideal of N`, which follows quickly from (2.12).
We now address (ii). Let f P N X Spgq`and write f " f``f´according to the decomposition Spgq " Spg`q ' Spgqg´. If g P G`pă 0q then g¨f`´f`P Spg`q X Spgqgpă -1q χ " Spg`qg`pă -1q χ by Lemma 2.7 and Lemma 2.8. This shows that ϕpf q " f`P N .
Finally, to see that ϕ : N X Spgq`Ñ N`is a Poisson homomorphism it suffices to show that N X Spgq`X Spgqg´is a Poisson ideal of N X Spgq`. Recall that g´" th¨x´x | h P H, x P gu and Spgq´" th¨x´x | h P H, x P Spgqu. If f P Spgq`then tf, h¨x´xu " h¨tf, xu´tf, xu P Spgqf or any x P Spgq´. Lemma 2.2 shows that the ideals generated by g´and Spgq´coincide, hence tSpgq`, g´u Ď Spgqg´, which shows that Spgq`XSpgqg´is a Poisson ideal of Spgq`. This completes the proof of (ii).
The map π restricts to a surjection N ↠ Spg, eq by definition. Since the latter map is Hequivariant we get N X Spgq`Ñ Spg, eq`, which proves (iii), whilst (iv) is proven similarly.
We move on to (v). The kernel of N X Spgq`Ñ Spg, eq`is N X Spgq`X Spgqgpă -1q χ . This is mapped to Spg`q X Spgqgpă -1q χ " Spg`qg`pă -1q χ by ϕ, thanks to Lemma 2.7 and Lemma 2.8. Finally Spg`qg`pă -1q χ lies in the kernel of π`, which proves (v). Now we take f P N X Spgq`such that πpf q P Spg, eqSpg, eq´. Using Lemma 2.8 again we see that Spgq{Spgqgpă -1q χ decomposes as the direct sum of the image of Spg`q and the image of Spgqg´. Therefore Spg, eq´, and the ideal which it generates, are contained in the image of Spgqg´. It follows immediately that Hence ϕpf q is in the kernel of π`, completing (vi). This concludes the proof. □ Proof of Theorem 2.6. Thanks to Proposition 2.9,(vi) we have ϕ : RpSpg, eq, Hq ↠ Spg`, eq. Pick an H-equivariant map θ : g e Ñ Spg, eq satisfying the properties of Theorem 2.4, and define θ`: g è Ñ Spg`, eq via θ`pxq :" ϕpθpxq`Spg, eq`X Spg, eqSpg, eq´q Property (2.5) for θ implies (2.5) for θ`. By Theorem 2.4(2) this implies that ϕ is surjective. Applying Proposition 2.2 we see that RpSpg, eq, Hq is a polynomial algebra generated by the image of θpg è q under the map Spg, eq`Ñ RpSpg, eq, Hq. It follows that ϕ maps a basis of RpSpg, eq, Hq to a basis of Spg`, eq, hence it is an isomorphism. □ Remark 2.10. It is natural to search for an analogue of Dirac reduction for quantum finite Walgebras. Although the naive approach to quantising Spg, eq works extremely well (see Section 7.1), the naive approach to quantising the Dirac reduction fails.

Dirac reduction for shifted Yangians
3.1. 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 be constructed as the Lie subalgebra of the free algebra CxXy generated by the vector space spanned by X. When L is a Lie algebra generated by X we say that L has relations Y Ď L X if Y generates the kernel of L X ↠ L.
The free Poisson algebra generated by X is the initial object in the category of (complex) Poisson algebras generated by X. It can be constructed as the symmetric algebra SpL X q together with its Poisson structure. If there is a Poisson surjection SpL X q ↠ A then we say that A is Poisson generated by X. It is important to distinguish this from A being generated by X as a commutative algebra, as both notions will occur frequently.
We say that a (complex) Poisson algebra A has Poisson generators X and relations Y Ď SpL X q if there is a surjective Poisson homomorphism SpL X q ↠ A and the kernel is the Poisson ideal generated by Y .
Let X be a set and Y Ď L X Ď SpL X q. Write I (resp. J) for the ideal of SpL X q (resp. L X ) generated by Y . It is easy to see that the natural map SpL X q Ñ SpL X q{I induces an isomorphism 3.2. Chevalley-Serre presentations for shifted current Lie-Poisson algebras. Throughout this section we fix an integer n ą 0. Following [BK06,BG07] a shift matrix is an nˆn array σ " ps i,j q 1ďi,jďn of non-negative integers with zero on the diagonal, satisfying s i,k " s i,j`sj,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 its transpose. These shift matrices serve two key purposes: they classify certain good gradings for nilpotent elements in general linear Lie algebras [BG07], and they provide one of the ingredients in the definition of shifted Yangians which give presentations of finite W -algebras in type A [BK06]. The symmetric shift matrices correspond to Dynkin gradings.
All examples of shift matrices used in the applications of our results are symmetric, and so for convenience we assume that every shift matrix in this paper is symmetric. Theorems 3.2 and 3.4 do not require this hypothesis.
The current algebra is the Lie algebra c n :" gl n b Crts. It has a basis consisting of elements te i,j t r | 1 ď i, j ď n, r ě 0u where we write x b t r " xt r for x P gl n and r ě 0. For any shift matrix σ " ps i,j q 1ďi,jďn we define the shifted current algebra c n pσq to be the subalgebra spanned by Lemma 3.1. Let σ be a symmetric shift matrix. The Lie subalgebra u n pσq Ď c n pσq spanned by elements (3.3) with i ă j is generated as a complex Lie algebra by subject to the relations " e i;r , e j;s ‰ " 0 for |i´j| ‰ 1, (3.5) " e i;r`1 , e i`1;s ‰´" e i;r , e i`1;s`1 ‰ " 0, (3.6) " e i;r 1 " e i;r 2 , e j;r 3 ‰ ı`" e i;r 2 " e i;r 1 , e j;r 3 ‰ ı " 0 for all |i´j| " 1. Proof. Write 0 for the nˆn zero matrix. It follows from (3.2) that the linear map u n pσq Ñ u n p0q defined by e i,j t r Þ Ñ e i,j t r´s i,j is a Lie algebra isomorphism and so it suffices to prove the current lemma when σ " 0.
Let p u n be the Lie algebra with generators (3.4) and relations (3.5)-(3.7), with σ " 0. We inductively define elements e i,j;r P p u n by setting e i,i`1;r :" e i;r and e i,j;r :" re i,j´1;r , e j´1,j;0 s for 1 ď i ă j ď n. There is a homomorphism p u n ↠ u n p0q given by e i;r Þ Ñ e i,i`1 t r and, in order to show that it is an isomorphism, we show that p u n is spanned by the elements te i,j;r | 1 ď i ă j ď n, 0 ď ru.
Following (1)-(7) in the proof of [BK05, Lemma 5.8] verbatim we have for all i, j, k, l, r, s re i,j;r , e k,l;s s " δ j,k e i,l;r`s´δi,l e k,j;r`s . Define an ascending filtration on p u n " Ť dą0 F d p u n satisfying F d p u n " ř d 1`d2 "d rF d 1 p u n , F d 2 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 " 1 holds by definition. For d 1`d2 " d ą 1 we know by the inductive hypothesis that F d 1 p u n and F d 2 p u n are spanned by elements e i,j;r . Using (3.8) we complete the induction, which finishes the proof. □ Theorem 3.2. Let σ be a symmetric shift matrix and let c n pσq denote the shifted current algebra. Then Spc n pσqq is Poisson generated by Proof. By (3.1) it suffices to show that c n pσq is generated as a Lie algebra by (3.9) subject to relations (3.10)-(3.15). Let pp c n pσq, t¨,¨uq denote the Lie algebra with these generators and relations. Define a map from the set (3.9) to c n pσq by d i;r Þ Ñ e i,i t r , e i;r Þ Ñ e i,i`1 t r , f i;r Þ Ñ e i`1,i t r . One can easily verify using (3.8) that this extends to a surjective Lie algebra homomorphism p c n pσq ↠ c n pσq. To show that this is an isomorphism it suffices to show that the elements te i,j;r | 1 ď i, j ď n, s i,j ď ru Ď p c n pσq (3.20) defined inductively by setting e i,i`1;r :" e i,r , e i`1,i;r :" f i;r and e i,j;r :" te i;s i,i`1 , e i`1,j;r´s i,i`1 u for i ă j; (3.21) form a spanning set. Using (3.10),(3.12), (3.13) and a simple inductive argument one can see that p c n pσq is a direct sum of three subalgebras: the diagonal subalgebra, spanned by the elements d i;r , and the upper and lower triangular subalgebras uǹ pσq and uń pσq 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 uǹ pσq and uń pσq are spanned by the elements defined in (3.21) and (3.22) respectively. Since the argument is identical for uǹ pσq and uń pσq we only need to consider the former, where the claim follows from Lemma 3.1 □ The following theorem is one of the key stepping stones for understanding the Dirac reduction of the shifted Yangian. The algebra described here is the twisted shifted current Lie-Poisson algebra.
Theorem 3.3. If σ is any symmetric shift matrix then Spc n pσqq admits a Poisson automorphism (3.23) The Dirac reduction RpSpc n pσq, τ q " Spc n pσq τ q is Poisson generated by for |i´j| " 1, r 1`r2 even.
Proof. Relation (3.8) implies that τ gives a Poisson automorphism. By Remark 2.3 we can identify RpSpc n pσq, τ q with Spc n pσq τ q, so it suffices to check that Spc n pσq τ q has the stated Poisson presentation. Use (3.1) to reduce the claim to a statement about the presentation of c n pσq τ .
Consider the Lie algebra p c n pσq τ which is generated by the set (3.24) subject to relations (3.25)-(3.31). Make the notation θ i,i`1 t r :" e i,i`1 t r`τ pe i,i`1 t r q P c n pσq τ . We define a map from the set (3.24) to c n pσq τ by sending η i;2r´1 Þ Ñ e i,i t 2r´1 P c n pσq τ , and sending θ i;r Þ Ñ θ i,i`1 t r P c n pσq τ . One can check that this determines a surjective Lie algebra homomorphism p c n pσq τ ↠ c n pσq τ , indeed, checking that relations (3.25)-(3.31) hold amongst the corresponding elements of c n pσq 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 ě s i,j we inductively define elements where θ i,i`1;r :" θ i;r . It remains to check that p c n pσq τ is spanned by the elements We define a filtration p c n pσq τ " Ť ią0 F i p c n pσq τ by placing the generators (3.24) in degree 1 and satisfying F d p c n pσq τ " ř d 1`d2 "d tF d 1 p c n pσq τ , F d 2 p c n pσq τ u. By convention F 0 p c n pσq τ " 0. The associated graded Lie algebra grp c n pσq τ " À ią0 F i p c n pσq τ { F i´1 p c n pσq τ is generated by elements η i;2r´1 :" η i t 2r´1`F 0 p c n pσq τ for 1 ď i ď n, 0 ď r (3.34) θ i;r :" θ i t r`F 0 p c n pσq τ for 1 ď i ă n, s i,i`1 ď r. Let a be an abelian Lie algebra with basis td i t r | 1 ď i ď n, 0 ă ru. Let u n pσq be the Lie subalgebra of c n pσq described in Lemma 3.1. Then a ' u n pσq is a Lie algebra with a an abelian ideal. Comparing the top components (3.25)-(3.31) to (3.5)-(3.7) we see that there is surjective Lie algebra homomorphism a ' u n pσq ↠ grp c n pσq τ defined by d i t r Þ Ñ η i;2r´1 and e i,i`1 t r Þ Ñ θ i,r . The algebra a ' u n pσq has basis consisting of element d i t r , e j,k t s where i " 1, ..., n, r ą 0, 1 ď j ă k ď n, s ě s j,k . It follows that grp c n pσq τ is spanned by elements η i;2r´1 , θ j,k;s where the indexes vary in the ranges specified in (3.33), and θ j,k;s defined inductively from (3.35), analogously to (3.32). We deduce that p c n pσq τ is spanned by the required elements, which completes the proof. □ 3.3. The semiclassical shifted Yangian. In this section we fix n ą 0, a shift matrix σ of size n and ε P t˘1u. The (semiclassical) shifted Yangian y n pσq is the Poisson algebra generated by the set for all admissible i, j, r, s, t. In these relations, the notation d In order to describe the structure of y n pσq as a commutative algebra we make the notation The shifted Yangian admits a Poisson grading y n pσq " À rě0 y n pσq r , which we call the canonical grading. It places d in degree r and the bracket lies in degree´1, meaning t¨,¨u : y n pσq rˆyn pσq s Ñ y n pσq r`s´1 . There is also an important Poisson filtration y n pσq " With respect to this filtration, the bracket is in degree 0 so that F r y n pσqˆF s y n pσq Ñ F r`s y n pσq. The associated graded Poisson algebra is denoted gr y n pσq.
The following theorem is a semiclassical analogue of [BK06, Theorem 2.1], which we ultimately deduce from the noncommutative setting.
Theorem 3.4. Let σ be a symmetric shift matrix. There is a Poisson isomorphism Spc n pσqq " ÝÑ gr y n pσq defined by As a consequence y n pσq is isomorphic to the polynomial algebra on infinitely many variables Proof. Comparing the top graded components of the relations (3.37)-(3.48) with respect to the loop filtration, with relations (3.10)-(3.19), it is straightforward to see that (3.54) gives a surjective Poisson homomorphism. To prove the theorem we demonstrate that the ordered monomials in the elements (3.55) are linearly independent.
Consider the set ] the shifted Yangian Y n pσq is defined as a quotient of the free algebra CxXy by the ideal 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 " Then we place a filtration on the enveloping algebra U pLq so that L i lies in degree i`1. By the PBW theorem for U pLq we see that gr U pLq -SpLq.
The universal property of U pLq ensures that there is a surjective algebra homomorphism U pLq ↠ CxXy and by [Ser06, I, Ch. IV, Theorem 4.2] this is an isomorphism. Identifying these algebras, the filtration on U pLq descends to Y n pσq " Ť iě0 F 1 i Y n pσq, and this resulting filtration is commonly referred to as the canonical filtration [BK06,§5]. The associated graded algebra gr Y n pσq is equipped with a Poisson structure in the usual manner. Comparing relations (3.37)-(3.48) with the top graded components of relations [BK06, (2.4)-(2.15)] we see that the Poisson surjection SpLq ↠ gr Y n pσq factors through SpLq Ñ y n pσq. As a result there is a surjective Poisson homomorphism π : y n pσq ↠ gr Y n pσq given by e By the definition of the filtration and the elements (3.52), (3.53) we have are linearly independent in Y n pσq and so we deduce that the images of these monomials in gr Y n pσq are linearly independent. This completes the proof. □ We record two formulas for future use, which are semiclassical analogues of [BT18, (4.32)] Lemma 3.5. The following hold for i " 1, ..., n, j " 1, ..., n´1, r ą 0 and s ą s j,j`1 : Proof. We only sketch (3.56), as the proof of (3.57) is almost identical. The argument is by induction based on (3.49). Note that r d " 1 we see that all quadratic terms cancel, whilst the linear term equates to pδ i,j`1´δi,j q r d pr´1´tq i , which concludes the induction. □ 3.4. The Dirac reduction of the shifted Yangian. In this section we suppose that σ is symmetric. Examining the relations (3.37)-(3.48) we see that there is unique involutive Poisson automorphism τ of y n pσq determined by (3.58) Our present goal is to give a complete description of the Dirac reduction Rpy n pσq, τ q.
For i " 1, ..., n and r ą s i,i`1 we write (3.60) Now let q y n pσq be the ideal of y n pσq generated by Also write q y n pσq τ :" q y n pσq X y n pσq τ . By Lemma 2.2 we see that Rpy n pσq, τ q " y n pσq τ {q y n pσq τ is Poisson generated by elements with Poisson brackets induced by the bracket on y n pσq. Furthermore Rpy n pσq, τ q is generated as a commutative algebra by elements where the θ prq i,j :" e prq i,j`p´1 q r`s i,j f prq i,j`q y n pσq τ P Rpy n pσq, τ q. Using an inductive argument and (3.38) we see the elements θ prq i,j can also be defined via the following recursion , θ ps j´1,j`1 q j´1 u for 1 ď i ă j ď n, s i,j ă r. (3.63) The next lemma can be deduced from (3.39), (3.40), (3.56), (3.57).
Lemma 3.6. The following equalities hold in Rpy n pσq, τ q: Thanks to Theorem 3.4 we see that Rpy n pσq, τ q comes equipped with the canonical grading Rpy n pσq, τ q " À iě0 Rpy n pσq, τ q i which places θ prq i , η prq i in degree r and the Poisson bracket in degree´1. Another crucial feature is the loop filtration Rpy n pσq, τ q " Ť iě0 F i Rpy n pσq, τ q which places θ prq i , η prq i in degree r´1 and the bracket in degree 0. Both of these structures are naturally inherited from y n pσq.
We define the following useful symbol for r, s P Z ϖ r,s :" p´1q r´p´1 q s " if r even, s odd, 0 if r`s even, 2 if r odd, s even. (3.66) The following is our main structural result on the Dirac reduction of the shifted Yangian.
Theorem 3.7. Let σ be a symmetric shift matrix, let y n pσq denote the corresponding semiclassical shifted Yangian and let τ denote the automorphism of y n pσq introduced in (3.58). The Dirac reduction Rpy n pσq, τ q is Poisson generated by elements together with the following relations for |i´j| " 1 and r`s " 2m even where we adopt the convention η i`q y n pσq τ P Rpy n pσq, τ q satisfy the recursion (3.75). By Theorem 3.4 the subalgebra of y n pσq generated by td prq i | i " 1, ..., n, r ą 0u is a graded polynomial ring with d prq i in degree r. By induction r d prq i lies in degree r. This forces r d p2r´1q i P q y n pσq for all r ą 0 and so (3.49) equals (3.75) modulo q y n pσq τ , confirming the claim.
Using i`1 u " te Substituting in (3.60) we see that this is equal to the right hand side of (3.71) modulo q y n pσq τ . Finally take |i´j| " 1. Expanding in terms of generators (3.36) and applying relations (3.38), (3.47), (3.48) together with the Jacobi identity and (3.76) we obtain If r`s is odd this vanishes, which proves (3.73). Now assume that r`s " 2m is even. Using (3.38) and Jacobi the last line of the previous equation reduces to Using the fact that d p2r´1q i P q y n pσq τ for all i, r we simplify this expression modulo q y n pσq τ to get i¸`q y n pσq τ Finally using (3.64) and (3.65) this expression coincides with the right hand side of (3.74).
We have shown that the generators (3.67) satisfy (3.68)-(3.74), and it remains to show that these are a complete set of relations.
Let p Rpy n pσq, τ q denote the Poisson algebra with generators (3.67) and relations (3.68)-(3.74). We have shown that there is a Poisson homomorphism p Rpy n pσq, τ q ↠ Rpy n pσq, τ q sending the elements (3.67) to the elements (3.61) with the same names. To complete the proof we show that this map sends a spanning set to a basis.
We define a loop filtration on p Rpy n pσq, τ q " in degree r´1 and the bracket in degree 0. Examining the top filtered degree pieces of the relations (3.68)-(3.74) with respect to the loop filtration, we see from Theorem 3.3 that there is a surjective Poisson homomorphism Spc n pσq τ q ↠ gr p Rpy n pσq, τ q. We deduce that gr p Rpy n pσq, τ q is generated as a commutative algebra by elements θ prq i,j`F r´2 p Rpy n pσq, τ q, η p2rq i`F 2r´2 p Rpy n pσq, τ q with indexes varying in the same ranges as (3.62). Again the elements θ prq i,j P p Rpy n pσq, τ q are defined via the recursion (3.63). By a standard filtration argument we deduce that the elements of p Rpy n pσq, τ q with the same names as (3.62) generate p Rpy n pσq, τ q as a commutative algebra. We have shown that p Rpy n pσq, τ q ↠ Rpy n pσq, τ q maps a spanning set to a basis, which concludes the proof. □ Let gr Rpy n pσq, τ q denote the graded algebra for the loop filtration. In the last paragraph of the proof of Theorem 3.7 we obtained the following important result.
Corollary 3.8. If σ is a symmetric shift matrix and y n pσq is the semiclassical shifted Yangian then there is a Poisson isomorphism Spc n pσq τ q " ÝÑ gr Rpy n pσq, τ q given by  In this section we continue to work over C and we fix the following notation: ‚ N ą 0 is an integer and ε P t˘1u such that ε N " 1. ‚ g " gl N pCq and g`Ď g is a classical Lie subalgebra such that ‚ G " GL N pCq and G`Ď G is the connected algebraic subgroup satisfying g`" LiepG`q.
‚ κ : g`ˆg`Ñ C is the trace form associated to the natural representation C N of G.
4.1. The symmetric shift matrix and the centraliser of a nilpotent element. Choose a partition λ $ N and assume that: all parts are even when ε "´1 and all parts are odd when ε " 1.
We recalled the notion of a shift matrix in Section 3.2, following [BK06]. The symmetric shift matrix for λ is the shift matrix σ " ps i,j q 1ďi,jďn defined by Introduce symbols tb i,j | 1 ď i ď n, 1 ď j ď λ i u and we identify C N with the vector space spanned by the b i,j . The general linear Lie algebra g " gl N has a basis We pick the nilpotent element in g by the rule e :" It has Jordan blocks of sizes λ 1 , ..., λ n . We also define a semisimple element h P gl N by h :" It is easy to see that the pair te, hu can be completed to an sl 2 triple te, h, f u. We refer to the grading g " À iPZ gpiq induced by adphq as the Dynkin grading for e. It is well-known that this is a good grading in the sense of [BG07]. It satisfies and, in particular, e P gp2q.
For 1 ď i, k ď n and r " s i,k , s i,k`1 , ..., s i,k`m inpλ i , λ k q´1, we define elements The following fact is well known. See [BK06,Lemma 7.3] where the notation c prq i,j differs from ours by a shift in r.
Lemma 4.1. The centraliser g e has basis Furthermore g e is a Dynkin graded Lie subalgebra with c prq i,k lying in degree 2r. p´1q j e i,j;i,λ i`1´j (4.10) This block diagonal matrix can be described as follows. For each index i " 1, ..., n there is a block of size λ i , each of these blocks has alternating entries˘1 on the antidiagonal and zeroes elsewhere. Thus when ε " 1 the matrix J is symmetric and when ε "´1 it is anti-symmetric. It follows that J´1 " εJ. (i) τ pe i,j;k,l q " p´1q j´l´1 e k,λ k`1´l ;i,λ i`1´j ; (ii) τ peq " e and τ phq " h; Proof. Using (4.11) and multiplying matrices we have τ pe i,j;k,l q "´εp´1q λ k`j´l e k,λ k`1´l ;i,λ i`1´j and so (i) follows from the fact that εp´1q λ i " 1 under our assumptions on ε and λ fixed at the start of Section 4.1. Now (ii) and (iii) follow by applying (i) to (4.4), (4.5) and (4.7). □ We decompose g " g`' g´into eigenspaces for τ so that g`is the space of τ -invariants. Then we have By Lemma 4.2(ii) we have te, hu Ď g`, and g´is a g`-module. Write G`for the connected component of the group of elements g P GL N such that gJg J " J. This is a classical group satisfying LiepG`q " g´.
As τ preserves gp´2q we also have that gp´2q " g`p´2q ' g´p´2q.
Writing f " f 1`f2 with f 1 P g`p´2q and f 2 P g´p´2q and using the fact that h P g`we deduce that h " re, f 1 s and re, f 2 s " 0. Since the Dynkin grading is good for e in g this yields f 2 " 0. We conclude that the sl 2 -triple te, h, f u is contained in g`.
Our next result explains the relationship between the shifted current algebras and the centralisers described above. We refer the reader to Theorem 3.3 for a recap of notation. (1) There is a surjective Lie algebra homomorphism where c prq i,j :" 0 for r ě s i,j`m inpλ i , λ j q. The kernel is Poisson generated by e 1,1 t r with r ě λ 1 .
Proof. Part (1) is explained in [GT19c, Lemma 2.6]. Compare (3.23) with Lemma 4.2(iii) to see that the map is τ -equivariant, hence it is a well-defined map c n pσq τ ↠ g è . 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 pσq ↠ g e ; for ε "´1 we use (3.12), (3.13). Now observe that g è has a spanning set consisting of elements of the form c prq i,j`τ pc prq i,j q. By Lemma 4.1 we can show that the elements in (2) generate the kernel by checking that the ideal i which they generate contains tη i;r | i " 1, ..., n, r ě λ i u Y tθ i;r | i " 1, .., n´1, r ě s i,i`1`λi u. (4.14) First take ε " 1 so that λ 1 is odd. Using relation (3.69) we see that i contains tη 1;λ 1 , θ 1;r u " θ 1;λ 1`r for r ě s 1,2 , whilst (3.71) shows that η 1;λ 2`r´η 2;λ 2`r P i for all r ě 0. In particular η 2;λ 2`r P i for r ě 0. By Theorem 3.3 the subalgebra of c n pσq 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 kernel follows by induction.
For ε "´1, λ 1 is even and the ideal i contains tη 1;λ 1`1 , θ 1;s u " θ 1;λ 1`1`s for s ě s 1,2 . Together with the additional generator θ 1;λ 1`s1,2 this gives all elements of the form θ 1;r listed in (4.14). Now the induction proceeds in the same way as the case ε " 1. □ 4.3. Generators of the W -algebra. Here we recall formulas for generators of Spgl N , eq, due to Brundan and Kleshchev [BK06,§9], see also [BK08, §3.3]. Their notation is slightly different, however it is a simple exercise to translate between the two settings. In this section we continue to assume that all parts of λ are odd when ε " 1 and all parts are even when ε "´1. Now for 1 ď i, k ď n, 0 ď x ă n and r ą 0, we let (c) if k m ą x, then l m ă j m`1 for each m " 1, . . . , s´1; (d) if k m ď x then l m ě j m`1 for each m " 1, . . . , s´1; (e) i 1 " i, k s " k; (f) k m " i m`1 for each m " 1, . . . , s´1.
Keeping r ą 0 fixed and choosing s P t1, ..., ru we make the notation X pr,sq i,k for the set of ordered sets pi m , j m , k m , l m q s m"1 in the range (4.16) satisfying the conditions (a)-(f) and consider the map υpi m , j m , k m , l m q :" pk s`1´m , λ k s`1´m`1´l s`1´m , i s`1´m , λ i s`1´m`1´j s`1´m q (4.17) defined on ordered sets in the range (4.16).  Comparing (4.6) with Lemma 4.1 and Lemma 4.2 we see that τ induces involutions on g e and Spgpě 0qq, and these will all be denoted τ . The hardest part of the following result is the assertion that the image of the map θ described in (4.18) lies in Spg, eq, which follows from a result of Brundan and Kleshchev. We add to this the observation that θ is τ -equivariant. Proof. The assertion that the image of θ lies in Spg, eq is an immediate consequence of [BK06, Corollary 9.4], upon taking the top graded term with respect to the Kazhdan filtration of the finite W -algebra (see also [GT19b,§4] for a short summary). Thanks to Lemma 4.2(iii) the τequivariance will follow from (4.19). Fix indexes 1 ď i ď k ď n, 0 ď x ă n and 0 ă r. 4.4. The semiclassical Brundan-Kleshchev homomorphism. Recall that g`Ď g " gl N is a classical Lie subalgebra constructed in Section 4.2, that e P g`is a nilpotent element, and σ is a choice of symmetric shift matrix introduced in (4.1). In Section 4.3 we observed that the automorphism τ on g which fixes g`induces a natural automorphism on Spg, eq. Furthermore in (3.58) we defined a Poisson involution of y n pσq which is also denoted τ . The relationship between these objects is described by the following result.  The kernel is the Poisson ideal generated by td prq 1 | r ą λ 1 u. If we equip y n pσq with the canonical grading doubled (see Theorem 3.4) and equip Spg, eq with the Kazhdan grading then this homomorphism is graded.
Proof. In [BK06, §2] Brundan and Kleshchev introduced the shifted Yangian Y n pσq, of which y n pσq is the semiclassical limit under the canonical filtration. In [BK06, Theorem 10.1] they construct a homomorphism to the quantum finite W -algebra (see also [GT19b,Theorem 4.3] where their result is recalled in notation similar to that of the present paper). It is straightforward to see that the doubled canonical filtration lines up with the Kazhdan filtration. The map defined by (4.22) is just the semiclassical limit of this filtered algebra homomorphism, and this proves that the map is a surjective Poisson homomorphism. The τ -equivariance can be checked by comparing Proposition 4.5 with formulas (3.58).
Another consequence of [BK06, Theorem 10.1] is that the elements td prq 1 | r ą λ 1 u lie in the kernel of φ. Write I for the Poisson ideal generated by these elements. Combining Theorem 2.4 and Lemma 4.1 we can show that I " Ker φ by demonstrating that the quotient y n pσq{I is generated as a commutative algebra by (4.23) By Theorem 3.4 we can identify Spc n pσqq " gr y n pσq as Poisson algebras, with respect to the loop filtration. The associated graded ideal gr I contains elements td i;r | r ě λ 1 u, and so applying Lemma 4.3(1) we see that gr y n pσq{I is generated by by the top graded components of the elements (4.23). Applying a standard filtration argument we deduce that y n pσq{I is generated by the requisite elements. The purpose of this section is to describe the kernel. Combined with Theorem 3.7 this will complete the proof of Theorem 1.3. First we deal with orthogonal types.
Proof. The elements η p2rq 1 with 2r ą λ 1 are certainly in the kernel of φ, thanks to Proposition 4.7. Let I denote the Poisson ideal of Rpy n pσq, τ q which they generate. It follows from Lemma 4.3(2)(i) that the associated graded algebra grpRpy n pσq, τ q{Iq with respect to the loop filtration is generated (as a commutative algebra) by the top filtered component of the following elements Hence Rpy n pσq, τ q{I is generated by these elements (4.25). Using Theorem 2.4 and Remark 4.6 we see that Spg`, eq is a polynomial algebra generated by the images of these elements under φ. It follows that the induced map Rpy n pσq, τ q{I ↠ Spg`, eq is an isomorphism, which completes the proof. □ Now we fix ε "´1 and we introduce a family of elements of Rpy n pσq, τ q; they lie in the kernel of (4.24) and provide filtered lifts of the elements appearing in Lemma 4.3(2)(ii).
Define elements t 9 θ ps i,i`1`λi q i | i " 1, ..., n´1u Ď Rpy n pσq, τ q as follows: first let 9 d pλ 1`1 q 1 :" d pλ 1`1 q 1 P y n pσq and then inductively define  Note that τ pp e ps i,i`1`1 q i q " p e ps i,i`1`1 q i and so τ p 9 d pλ i`1 q q "´9 d pλ i`1 q . Therefore we may define 9 θ pλ i`si,i`1`1 q i :" t 9 d pλ i`1 q i , q e ps i,i`1`1 q i u`q y n pσq τ P y n pσq τ {q y n pσq τ " Rpy n pσq, τ q (4.27) Example 4.9. The first occurrence of one of these new elements is easy to calculate. We have (4.28) For i ą 1 the expressions are more complicated, but it would be interesting to obtain a closed formula. We will not pursue this in the present article.
Proposition 4.10. For ε "´1 the kernel of (4.24) is Poisson generated by ) . (4.29) Proof. Let I be the Poisson ideal of Rpy n pσq, τ q generated by (4.29). Recall that the kernel of the map y n pσq Ñ Spg, eq is generated by d prq 1 with r ą λ 1 (Proposition 4.7). It follows that the elements (4.29) are projections to Rpy n pσq, τ q of certain elements of Kerpy n pσq Ñ Spg, eqq τ , and so these elements all lie in the kernel of the map (4.24). In other words φ factors through Rpy n pσq, τ q Ñ Rpy n pσq, τ q{I.
Similar to the proof of Proposition 4.8 it suffices to show that Rpy n pσq, τ q{I is generated as a commutative algebra by the elements (4.25). An easy calculation in y n pσq, using Corollary 3.8, shows that the top filtered component of 9 θ ps λ i`si,i`1`1 q i with respect to the loop filtration is θ i;λ i`si,i`1 . Now we can apply Lemma 4.3(2)(ii) to see that gr Rpy n pσq, τ q is generated by the top filtered components of the elements (4.25). The argument concludes in exactly the same manner as Proposition 4.8. □ Thanks to Lemma 4.1 the Lie algebra g è is spanned by elements c prq i,j`τ pc prq i,j q; note that τ here refers to the automorphism of gl N given in (4.12). Using (4.15) along with Theorem 2.4 and Propositions 4.8 and 4.10 we see that Spg`, eq is generated as a commutative algebra by the images under φ of elements (4.25). By slight abuse of notation we denote the images by the same names. Proof. The graded algebra gr m 0 Spg`, eq is Poisson, and it is straightforward to check, using Theorem 3.7, that (4.30) gives a Lie algebra homomorphism g è ↠ gr m 0 Spg`, eq. By the universal property of the Lie-Poisson algebra we get a Poisson homomorphism Spg è q ↠ gr m 0 Spg`, eq. By Theorem 2.4 we see that this map is an isomorphism. □ Part 2. One dimensional representations of W -algebras

Generalities on Poisson schemes
5.1. Conic degenerations of affine schemes. We begin the second Part of the paper by recording two basic results, allowing us to compare the number of irreducible components of certain complex schemes of finite type. The first records an obvious bound on the dimensions of components of subschemes, whilst the second allows us to bound the number of components of a deformation of a reduced conic affine scheme.
We define P be the set of all finite sequences of all non-negative integers of arbitrary length. We order P lexicographically as follows: if d " pd 1 , ..., d n q and d 1 " pd 1 1 , ..., d 1 m q then we say that d ą d 1 if there is an index j P t1, ..., minpn, mqu such that d i " d 1 i for i " 1, ..., j and d j`1 ą d 1 j`1 . Here we adopt the convention d 1 m`1 " d n`1 " 0 If X is a complex scheme of finite type then we write X " Ť l i"1 X i for the decomposition into irreducible components, ordered so that dim X i ě dim X j for i ă j. We define the dimension vector of X to be the sequence dpXq :" pdim X 1 , dim X 2 , ..., dim X l q P P.
The following useful fact will be used in several later arguments.
Lemma 5.1. Let X, Y be complex schemes of finite type with a closed embedding Y Ñ X. Then dpXq ě dpY 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 iě0 F i A is a standard filtration of A if there is a surjection Crx 1 , ..., x n s ↠ A for some n, along with integers m 1 , ..., m n ě 0 such that F m A is spanned by the image of tx k 1 1¨¨¨x kn n | k 1 m 1`¨¨¨`kn m n ď mu. If X " SpecpAq is the associated affine scheme the we refer to CX " Specpgr Aq as the asymptotic cone of X.
Proof. If gr A is reduced then so is A. We begin by choosing a presentation for the standard filtration. Let R " Crx 1 , ..., x n s " À iě0 R i be a graded polynomial ring with x i in degree m i . Write F i R " À i j"0 R j . We let ϕ : R Ñ A be the homomorphism inducing the standard filtration, set I :" Kerpϕq and denote the minimal prime ideals over I by p 1 , ..., 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 CX as subschemes of Spec R " A n .
For g P F i R we writeḡ " g`F i´1 R P gr R. For any ideal J of R we write V pJq Ď A n for the corresponding variety of closed points. We have inclusions grpJq grpKq Ď grpJKq Ď grpJq X grpKq for any ideals J, K Ď R (see [Pr02, §5.3]), and so We prove the contrapositive of the lemma, so assume that m " # ComppXq ą # ComppCXq. Then, a fortiori, we must have V pgr p j q Ď Ť i‰j V pgr p i q for some j, say j " 1. Equivalently Ş i‰1 ?
gr p i Ď ? gr p 1 , which implies Ş i‰1 gr p i Ď ? gr p 1 . Since I Ĺ p 2 p 3¨¨¨pm it follows that gr I Ĺ grpp 2¨¨¨pm q and so we may choose g 2 , ..., g m with g i P p i such that g :" g 2¨¨¨gm satisfies g R gr I. On the other hand, we haveḡ P grpp 2¨¨¨pm q Ď Ş m i"2 grpp i q Ď Ş m i"1 a grpp i q " a grpIq, where the last equality follows from (5.1). Thus we conclude thatḡ P a grpIqz gr I, so that CX is not reduced. This completes the proof. □

5.2.
Completions and nilpotent elements of graded algebras. If p P SpecpAq then, as usual, A p denotes the localisation and Ap the completion at the maximal ideal of A p . The kernel of A Ñ A p is the set of elements annihilating some element of Azp, whilst the kernel of A Ñ Ap is Ş ką0 p k . When p is a maximal ideal it follows from Krull's intersection theorem that these kernels are equal (see the remark following [AM69, Theorem 10.17]). Now let A " À iě0 A i be a finitely generated, connected graded algebra with unique maximal graded ideal m 0 . The connected grading induces a one parameter family of automorphisms of A, and a Cˆ-action on MaxpAq contracting to the unique fixed point m 0 . For a P A make the notation Ξ a :" .
Lemma 5.3. The following hold: (1) Ş m Ş ią0 m i " 0 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 Cˆ-action.
Proof. Since A is finitely generated over C the intersection Ş mPMaxpAq m " RadpAq is a graded ideal, however for every i ą 0 we have A i X m i`1 0 " 0 and so (1) follows. If m P MaxpAq, we remarked above that the kernel of A Ñ Am is equal to the kernel of A Ñ A m . Writing Annpaq " tb P A | ab " 0u, it follows that KerpA Ñ Amq " ta P A | ab " 0 for some b R mu " ta P A | Annpaq Ę mu. We have shown that Ξ a " tm P MaxpAq | Annpaq Ď mu, i.e. Ξ a is the subvariety of MaxpAq cut out by Annpaq, hence closed. Finally it remains to see that Ξ a is stable under the Cˆ-action, and this follows directly from two facts: Annpaq is a graded ideal whenever a is homogeneous, and Cˆ-stable sets are precisely those defined by graded ideals. □ Lemma 5.4. Suppose that A " À ią0 A i is a finitely generated, connected graded algebra with unique maximal graded ideal m 0 . The following are equivalent: (i) A is reduced.
(iii) Am is reduced for every maximal ideal m P SpecpAq.
Furthermore if gr m 0 A is reduced then these equivalent conditions hold.
Proof. We prove (ii) ñ (i) by contraposition. So suppose that 0 ‰ f P RadpAq is a nonzero element. Without loss of generality we may assume f is homogeneous. By Lemma 5.3(1) we have KerpA Ñ ś mPMaxpAq Amq " 0, which means that f maps to a nonzero element of Am 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 0 , which implies that Am 0 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 by localisation for any commutative ring, and then apply [Ma86, §32, Remark 1]. Clearly (iii) ñ (ii).
We prove that if A is non-reduced then gr m 0 A is non-reduced. Let 0 ‰ f P RadpAq and suppose (without loss of generality) that f is homogeneous. There are two cases: either f P m i In the first case f`m i`1 0 is a nonzero nilpotent element of gr m 0 A, which proves that gr m 0 A is non-reduced.
We now show that the second case cannot happen. Suppose f P Ş ią0 m i 0 " KerpA Ñ Am 0 q, or equivalently, m 0 R Ξ f . By Lemma 5.3(2) we must have Ξ f " H which implies that f P KerpA Ñ Amq for all maximal ideals m. By Lemma 5.3(1) we see f P Ş m Ş ią0 m i " 0. This contradiction completes the proof. □

Commutative quotients of Poisson schemes. If
A is a Poisson algebra then the abelianisation A ab is the largest Poisson commutative quotient of A. It is constructed by factoring by the ideal generated by tA, Au. When X is a complex manifold it can be stratified into immersed submanifolds known as symplectic leaves (see [We83] for an introduction). Therefore the closed points of a regular complex Poisson scheme can be decomposed into a disjoint union of symplectic leaves. More generally the closed points of a singular affine Poisson scheme can be decomposed into symplectic leaves by an iterative procedure [BG03, §3.5].
The stratification by leaves can be coarsened to a stratification by rank, which is especially transparent in the case of affine schemes: if A is finitely generated by elements x 1 , ..., x n and carries a Poisson structure, then we can form the matrix π " ptx i , x j uq 1ďi,jďn and consider the subschemes X 0 Ď X 1 Ď¨¨¨Ď X where X k is defined by the ideal of A generated by all pk`1qˆpk`1q minors of π.
It follows from [BG03, Proposition 3.6] that the locally closed set X k zX k´1 is the union of all symplectic leaves of dimension k. In particular X 0 is the union of symplectic leaves of dimension zero. Note that X 0 " SpecpA ab q by definition. These observations prove the following lemma.
Lemma 5.5. The following reduced subschemes of X coincide: (1) SpecpA ab q red ; (2) The union of all zero dimensional symplectic leaves of SpecpAq. □ If A is a Poisson algebra and I is a Poisson ideal then the completion AÎ can be equipped with a Poisson structure in a unique way such that A Ñ AÎ is Poisson: each A{I i is a Poisson algebra and AÎ is an 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 MaxpA ab q. Then there is a natural isomorphism pAmq ab " ÝÑ pA ab qm.
Proof. According to Lemma 5.5 the point m is a zero dimensional symplectic leaf, and it follows that m is a Poisson ideal. Hence Am is a Poisson algebra. We let B denote the ideal of A generated by the Poisson brackets tA, Au, let p B m be the ideal of Am generated by tAm, Amu and let Bm be the ideal Am b A B, which we identify with an ideal of Am using [AM69, Proposition 10.13]. Considering the exact sequence of A-modules B Ñ A Ñ A ab , we deduce pA ab qm -Am{Bm from [AM69, Proposition 10.12].
We must show that Bm " p B m . We certainly have an inclusion Bm Ď p B m and so the claim will follow from the universal property of pAmq ab if we can show that Am{Bm is an abelian Poisson algebra. Since this is isomorphic to pA ab qm, which is a projective limit of abelian Poisson algebras, the claim follows. □

Sheets and induction
6.1. Lusztig-Spaltenstein induction and sheets. Let G be a complex connected reductive group and P Ď G a parabolic subgroup with Levi decomposition P " LR, where R denotes the unipotent radical, and write l " LiepLq, r " LiepRq. 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 AdpGqv attains the maximal value. For any choice of nilpotent orbit O 0 Ď l and z P zplq it is not hard to see that AdpGqpz`O 0`r q contains a dense G-orbit. This orbit is (Lusztig-Spaltenstein) induced from pl, O 0 , zq and is denoted Ind g l,z pO 0 q. Remarkably the induced orbit depends only on the G-conjugacy class of pl, O 0 , zq, not on the choice of P admitting L as a Levi factor [Lo22, Lemma 4.1]. We call this G-orbit an induction datum, and when z " 0 we say that the orbit of pl, O 0 q is the induction datum with induced orbit Ind g l pO 0 q, 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 induction datum pl, O 0 q is rigid if O 0 is a rigid orbit in l. When G is almost simple of classical type the conjugacy classes of Levi subalgebras and nilpotent orbits can be described combinatorially, and induction can be totally understood in terms of partitions [CM93,§7].
The sheets of a Lie algebra are the irreducible components of the rank varieties which consist of points whose orbit has a fixed dimension. These were classified in terms of rigid induction data by Borho [Bo81]. For each induction datum pl, O 0 q define Dpl, O 0 q :" AdpGqpzplq reg`O0`r q Spl, O 0 q :" Dpl, O 0 q reg .
Theorem 6.1. [Bo81, Satz 4.3, Satz 5.6] The sheets of g are the sets Spl, O 0 q where pl, O 0 q varies over conjugacy classes of rigid data. Furthermore dim Spl, O 0 q " dim zplq`dim Ind g l pO 0 q. 6.2. Classical finite W -algebras and Slodowy slices. Resume the notation κ, te, h, f u, χ used in Section 2.2, so g " À iPZ gpiq is the adphq-grading. The Slodowy slice is the affine variety e`g f . Using κ we can identify g with g˚as G-modules, so that Spgq is identified with the coordinate ring Crgs. Then Spgqgpă -1q χ is identified with the defining ideal of e`gpă -1q K " tx P g | κpx´e, gpă -1qq " 0u " e`gpď 1q.
It follows from [GG02, Lemma 2.1] that the adjoint action map of G restricts to an isomorphism Gpă 0qˆe`g f " ÝÑ e`gpď 1q.
Via this isomorphism we identify Spg, eq " Cre`g f s, and thus we equip e`g f with a Poisson structure. Let γ : CˆÑ G be the cocharacter with differential d 1 γp1q " h. There is a Cˆ-action on g given by t Þ Ñ t 2 Adpγptq´1q which restricts to a contracting Cˆ-action on e`g f . The grading on Cre`g f s induced by this action coincides with the Kazhdan grading on Spg, eq.
Thanks to [GG02, §3.2] we know that the symplectic leaves of e`g f coincide with the irreducible components of the intersection of adjoint orbits with the slice. Since Spg, eq is Poisson graded in degree´2 this shows that the Cˆ-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 local geometry of g in a neighbourhood of the adjoint orbit of e. Let S 1 , ..., S l be the sheets of g containing e. Recall from the introduction that we define the Katsylo variety to be e`X :" pe`g f q X ď i"1 S i . (6.1) Let m 0 denote the maximal ideal of Cre`Xs corresponding to e. One of the main tools in this paper is the tangent cone TC e pe`Xq " Specpgr m 0 Cre`Xsq, which we equip with the reduced scheme structure.
(1) Spec Spg, eq ab red " e`X as reduced schemes.
(2) Let g be a classical Lie algebra. We have bijections ÝÑ Comp TC e pe`Xq.
The first bijection reduces dimension by dim AdpGqe whilst the second is dimension preserving.
Proof. By Lemma 5.5 we know that the reduced scheme associated to Spec Spg, eq ab is the union of zero 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 pe`g f q where O Ď g is an adjoint orbit. Since e`g f intersects orbits transversally it follows that the zero dimensional leaves are precisely the components of O X pe`g f q where dim O " dim AdpGqe. Since the contracting Cˆ-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 pe`g f q is irreducible. It was proven in [Im05, Theorem 6.2] that S i X pe`g f q can be expressed 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 pe`g f q the latter are also smooth, and so TC e pe`Xq " Now the second bijection will follow if we can show that T e pS i X e`g f q is never contained in T e pS j X e`g f q for i ‰ j which is a consequence of the fact that the sheets S 1 , ..., S l intersect transversally at e [Bu]. Finally the claims about dimensions follow from the fact that e`g f is transversal to adjoint orbits and the sheets are smooth. □ 7. Abelian quotients of finite W -algebras 7.1. 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 2 -triple pe, h, f q in g " LiepGq and consider the grading g " À iPZ gpiq given by adphq-eigenspaces. Write χ :" κpe,¨q P g˚. Recall the notation gpă -1q and Gpă 0q.
The generalised Gelfand-Graev module is defined to be where C χ is the one dimensional representation of gpă -1q afforded by χ. The finite W -algebra is defined to be U pg, eq :" Q Gpă0q .
It inherits a natural algebra structure from U pgq viaū 1ū2 " u 1 u 2 whereū denotes the projection of u P U pgq to Q, andū 1 ,ū 2 P Q Gpă0q .
Remark 7.1. The finite W -algebra is more commonly defined as a quantization of a Hamiltonian reduction with respect to a certain subgroup of G lying between Gpă´1q and Gpă 0q. In [GG02,Theorem 4.1] Gan-Ginzburg show that this definition is equivalent to the one given here: in their notation our definition corresponds to the isotropic space ℓ " 0.
The Kazdan filtration on U pgq is defined by placing gpiq in degree i`2. This descends to a non-negative filtration on Q and U pg, eq and we write gr U pg, eq for the associated graded algebra. The graded algebra gr U pg, eq naturally identifies with a subalgebra of gr Q " Spgq{Spgqgpă -1q χ , and we have the following fundamental fact (see [Pr02,Proposition 6.3], [GG02, Theorem 4.1]).
One of the main objects of study in this paper is the maximal abelian quotient U pg, eq ab , which is defined to be the quotient by the derived ideal which is generated by all commutators ru, vs with u, v P U pg, eq.
The Kazhdan filtration descends to U pg, eq ab and we write gr U pg, eq ab for the associated Kazhdan graded algebra. Our main results describe the structure of the following affine schemes Epg, eq :" Spec U pg, eq ab , CEpg, eq :" Spec grpU pg, eq ab q.
The closed points of the first of these parameterises the one dimensional representations of U pg, eq, whilst the second is the asymptotic cone of the first (cf. Section 5.1).

7.2.
Bounding the asymptotic cone. The next result is equivalent to (1.3), and we view it as a version of Premet's theorem [Pr10, Theorem 1.2] for the asymptotic cone of Epg, eq. Our methods are adapted from his work.
Theorem 7.3. If g " LiepGq is the Lie algebra of a complex reductive group then there is a surjective homomorphism Spg, eq ab ↠ gr U pg, eq ab which induces an isomorphism of reduced algebras.
First of all we prove the existence of the surjection Lemma 7.4. There is a surjective homomorphism Spg, eq ab ↠ gr U pg, eq ab .
Proof. Let θ : g e Ñ Spg, eq be a PBW map coming from Theorem 2.4 and let Θ : g e Ñ U pg, eq be a filtered lift of θ. Using the Leibniz rule for the biderivation r¨,¨s we see that the derived ideal D of U pg, eq is generated by the set trΘpuq, Θpvqs | u, v P g e u. Similarly the bracket ideal B in Spg, eq is generated by ttθpuq, θpvqu | u, v P g e u. If u, v are homogeneous for the adphq-grading on g then the image of rΘpuq, Θpvqs in the graded algebra gr U pg, eq " Spg, eq is tθpuq, θpvqu, using the identification of Lemma 7.2. This shows that B Ď gr D. Hence we have gr U pg, eq ab " Spg, eq{ gr D ↠ Spg, eq{B " Spg, eq ab . □ In order to show that this surjection gives an isomorphism of reduced algebras we use reduction modulo p. This process was pioneered by Premet in the theory of W -algebras and used systematically to great effect in subsequent work [Pr07b,Pr10,To17,PT21]. Rather than repeating the details in full we state the important properties of the modular reduction procedure which we need for our proof.
Let h denote the greatest Coxeter number as we vary over simple factors of the derived subgroup pG, Gq. Pick a Chevalley Z-form g Z Ď g and write g Z " h Z ' À αPΦ Zx 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 [PT21, §3.3] that for every nilpotent orbit O Ď g there is an element e P g Z X O and a cocharacter λ P X˚pT q such that for all algebraically closed fields k of characteristic p ą h: (1) The Bala-Carter labels of the adjoint orbits of e and e k :" e b 1 P g k coincide.
(2) The differential d 1 λpCq is a semisimple element of g lying in an sl 2 -triple containing e.
(3) After canonically identifying to cocharacter lattices X˚pT q " X˚pT k q the grading on g k induced by λ is good for e, meaning e k P g k p2q and g e k Ď g k pě 0q. We mention that slightly weaker assumptions are sufficient for (1), (2), (3). One may work under the standard hypotheses of [Ja04, §6.3], however we avoid these technicalities for the sake of simplicity.
Fix an orbit O Ď g and a choice of e, λ as above. When the characteristic of k is p ą h there is a G-equivariant trace form κ k , for example the Killing form is non-degenerate on rg, gs and g " rg, gs ' zpgq under out hypothesis. Write χ k for the element of gk corresponding to e k via κ k . Using the grading g k " À iPZ g k piq coming from λ we let v denote a homogeneous complement to rg k , e k s in g k . If S 1 , ..., S l denote the sheets of g k containing e k then we define In the following we write e`X C for the complex Katsylo variety (6.1), and write X _ k Ď gk for the image of X k under the G-equivariant isomorphism g k Ñ gk coming from the trace form.
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]. □ Since g k " LiepG k q is algebraic there is a G k -equivariant restricted structure on g k . The p-centre Z p pg k q is a central subalgebra of U pg k q which identifies with krpgkq p1q s, the coordinate ring on the Frobenius 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 pg k , e k q " Q G k pă0q k where Q k " U pg k q{U pg k qg k pă -1q χ and g k pă -1q χ " tx´χ k pxq | x P g k pă -1qu [Pr10,GT19a]. If ϕ : U pg k q Ñ Q k is the natural projection then the p-centre of U pg k , e k q is defined to be pϕZ p pg k qq G k pă0q ; see [GT19a,§8] for more detail. It follows from Lemma 8.2 of loc.
cit. that Z p pg k , e k q " krpχ k`v _ q p1q s where v _ " κ k pv,¨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 pg k , e k q to be the quotient of U pg k , e k q by the ideal generated by the corresponding maximal ideal of Z p pg k , e k q (identifying χ k`v _ with its Frobenius twist, as sets).
Lemma 7.6. There is a finite, dominant morphism kEpg k , e k q Ñ X k .
Proof. Consider the homomorphism Z p pg k , e k q Ñ U pg k , e k q ab and denote the kernel by K. Write V pKq Ď Max Z p pg k , e k q for the algebraic subvariety defined by K. By definition the points of V pKq correspond to the p-characters η P χ k`v _ such that U η pg k , e k q admits a one dimensional module. In [Pr10, §2.6] Premet demonstrated the following relationship between the reduced enveloping algebra and reduced finite W -algebra where d χ k :" 1 2 dim Ad˚pG k qχ k , under the assumption p " 0. In [GT19a,Remark 9.4] the isomorphism (7.3) was recovered under weaker hypotheses (p ą h suffices). It follows that the points of V pKq correspond to p-characters η P χ k`v _ Ď gk such that U η pg k q admits a module of dimension p dχ k .
In [Pr95] Premet provided his first proof of the second Kac-Weisfeiler conjecture (he later refined his argument using W -algebras, see Theorem 2.3(ii) and §2.6 of [Pr02]). From this, we know that every U η pg k q-module has dimension divisible by p dη . On the other hand, [PT21, Theorem 1.1] states that every U η pg k q has a module of dimension p dη . Therefore We claim that this set is equal to pχ k`X _ k q p1q . In the current setting, χ k`v _ admits a contracting kˆ-action defined in the same manner as the one parameter group of automorphisms appearing in Section 6.2, using the cocharacter determined by the grading in (3) above. This contracting action preserves the sheets of gk and this implies the claim.
Let R denote the reduced algebra corresponding to Z p pg, eq{K. Since Z p pg, eq " krpχ k`v _ q p1q s as Kazhdan filtered algebras we see that R " krpχ k`X _ k q p1q s is also an identification of Kazhdan filtered algebras. Since χ k`X _ k is stable under the contracting kˆ-action inducing the Kazhdan grading, it follows that Rgr R. We conclude that there is a finite injective algebra homomorphism R ãÑ gr U pg, eq ab , which completes the proof. □ Proof of Theorem 7.3. Recall that finite morphisms are closed, and so a finite dominant morphism is surjective. Now it follows from Lemma 7.6 that there is a surjective map Comp kEpg k , e k q Ñ Comp X k such that for every Z P Comp X k there exists an element of Comp kEpg k , e k q mapping to Z of the same dimension. Combining with Lemma 7.5 we deduce that there is surjection Comp CEpg, eq ↠ Comppe`X C q which respects dimensions in the same manner. In particular we have dpCEpg, eqq ě dpe`X C q, where d denotes the dimension vector defined in Section 5.1. Thanks to Lemma 5.1, Proposition 6.2(1) and Lemma 7.4 the embedding CEpg, eq ãÑ Spec Spg, eq ab " e`X C is an isomorphism on the underlying reduced schemes. □ The next result follows immediately from Theorem 7.3.
Corollary 7.7. If Spg, eq ab is reduced then so are gr U pg, eq ab and U pg, eq ab . □ 8. The abelian quotient of the classical W -algebra In this section we study the finite W -algebras associated to classical Lie algebras, and so we refresh the notation once again. Let N ą 0 is an integer and ε P t˘1u such that ε N " 1. We let G Ď GL N be a classical simple algebraic group and g " Lie G such that 8.1. Partitions for nilpotent orbits in classical Lie algebras. We refer to the G-orbits in g consisting of nilpotent elements as nilpotent orbits, and similarly for GL N -orbits in gl N . Nilpotent GL N -orbits in gl N are classified by partitions PpN q :" tλ $ N u, where the parts of a partition λ correspond to the sizes of the Jordan blocks of elements in the orbit.
For each nilpotent GL N -orbit O λ Ď g the intersection O λ X g is either empty or it is a union of either one or two nilpotent G-orbits. Therefore an approximate classification 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 pN q, and they are characterised as follows.
Lemma 8.1. Let λ " pλ 1 , ..., λ n q $ N . Then λ P P ε pN q if and only if there exists an involution i Þ Ñ i 1 on the set t1, ..., nu such that: (1) λ i " λ i 1 for all i " 1, ..., n; (2) i " i 1 if and only if εp´1q λ i " 1; (3) i 1 P ti´1, i, i`1u. □ Whenever we choose λ P P ε pN q we will assume that a choice of involution has been fixed in accordance with Lemma 8.1. We also adopt the convention that our partitions are non-decreasing λ 1 ď¨¨¨ď λ n . For completeness we mention that λ P P ε pN q corresponds to one orbit unless ε " 1 and all parts of λ are even, which case there are two G-orbits of nilpotent elements of g with Jordan blocks given by λ and these are permuted by the outer automorphism group of g, see [CM93, §5.1] for example. 8.2. Rigid, singular and distinguished partitions. We introduce three classes of partition which correspond to important families of nilpotent orbit in classical Lie algebras of type B, C, D.
Let λ " pλ 1 , ..., λ n q P P ε pN q and adopt the convention that λ 0 " 0 and λ n`1 " 8. A 2-step is a pair of indexes pi, i`1q such that: We note that λ 0 ă λ 1 and λ n ă λ n`1 always hold due to our conventions. We write ∆pλq for the set of 2-steps for λ. A 2-step pi, i`1q P ∆pλq is called bad if one of the following conditions holds: ‚ λ i´λi´1 ą 0 is even; ‚ λ i`2´λi`1 ą 0 is even. We note that the second condition never holds for i " n´1, whilst if the first condition holds for i " 1 then we necessarily have ε "´1. If the first of these two conditions holds then we refer to i´1 as the bad boundary of the 2-step whilst if the second condition holds then i`2 is the bad boundary. We say that a partition is 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: ‚ ∆pλq " H; ‚ λ i´λi´1 ă 2 for all i " 1, ..., n. We say that a partition is distinguished if the following two conditions hold: ‚ λ i ă λ i`1 for all i " 1, ..., n; ‚ i " i 1 for all i. Lemma 8.2. Let e be a nilpotent element with Jordan block sizes given by the partition λ P P ε pN q in a classical Lie algebra preserving a non-degenerate form Ψ on C N such that Ψpu, vq " εΨpv, uq for u, v P C N . The following hold: Keep fixed a choice of λ " pλ 1 , ..., λ n q P P ε pN q. We say that i is an admissible index for λ if Case 1 or Case 2 occurs: Case 1: λ i´λi´1 ą 1; Case 2: pi´1, iq P ∆pλq and λ i´1 " λ i .
When i is an admissible index we define λ piq P P ϵ pN´2pn´i`1qq as follows: Case 1: λ piq " pλ 1 , λ 2 , ..., λ i´1 , λ i´2 , λ i`1´2 , ..., λ n´2 q; Case 2: λ piq " pλ 1 , λ 2 , ..., λ i´2 , λ i´1´1 , λ i´1 , λ i`1´2 , ..., λ n´2 q Now we extend this definition inductively from indexes to sequences. We say that piq is an admissible sequence for λ if i is an admissible index. We inductively extend these definitions to sequences of indexes by saying that i " pi 1 , ..., i l q is an admissible sequence for λ and define λ i " pλ pi 1 ,...,i l´1 q q pi l q if i 1 " pi 1 , ..., i l´1 q is an admissible sequence for λ and i l is an admissible index for λ i 1 . A sequence i is called maximal admissible if λ i does not admit any admissible indexes. These definitions first appeared in [PT14], where the opposite ordering on the parts of λ was used. We define admissible multisets to be the multisets taking values in t1, ..., nu obtained from admissible sequences by forgetting the ordering. If i is an admissible sequence then write ris for the corresponding admissible multiset.
For ε "˘1 let g be a classical Lie algebra in accordance with Lemma 8.2, and let O Ď g be a nilpotent orbit with partition λ. According to Proposition 8 and Corollary 8 of [PT14], for each addmisible multiset ris, there is a procedure for choosing: (1) a Levi subalgebra l i -gl i 1ˆ¨¨¨g l i lˆg`, where g`is a classical Lie algebra of the same Dynkin type as g and natural representation of dimension N´2 ř i j .
(2) a nilpotent orbit O i " t0uˆ¨¨¨t0uˆO λ i where O λ i Ď g`has partition λ i . These choices satisfy O " Ind g l pO i q. Furthermore ris Þ Ñ pl i , O i q sets up a one-to-one correspondence between maximal admissible multisets and rigid induction data for O. Combining with Theorem 6.1 this proves.
Proposition 8.3. The sheets of g containing e are in bijection with maximal admissible multisets for λ. The dimension of sheet corresponding to i is |i|`dimpAdpGqeq. □ 8.4. Distinguished elements and induction. The proof of the first main theorem will be reduced to the distinguished case, and the following result is one of the crucial steps.
Lemma 8.4. Let O Ď g be a nilpotent orbit with partition λ P P ε pN q in a classical Lie algebra of rank r. There exists a sequence of indexes i 1 , ..., i l and a classical Lie algebrag of rank r`ř j pní j`1 q of the same Dynkin type as g such that: (1) gl n´i 1`1ˆ¨¨¨ˆg l n´i l`1ˆg embeds as a Levi subalgebra l Ďg; (2) Identifying O with the as a nilpotent orbit t0uˆ¨¨¨ˆt0uˆO Ď l we have that Indg l pOq is a distinguished orbit ing.
Furthermore we can assume that the first part of the partition of Indg g pOq is arbitrarily large.
Iterating this procedure we eventually replace λ with a distinguished partitionλ P P ε pN`ř iPi 2pní`1 qq where i Ď t1, ..., nu is some multiset consisting of indexes which were chosen in the above iterations. By applying the latter procedure at index i " 1 we can make the first part ofλ as large as we choose.
Letg be the classical Lie algebra of the same type as g with natural representation of dimension N`ř jPi 2pn´j`1q. By Lemma 8.2 every orbit with partitionλ is distinguished. Sinceλ i " λ the remarks of the previous section show that there is a unique orbit r O Ďg with partition λ and a unique induction datum pl, Oq such that l and O satisfy the properties described in the current proposition, and r O " Indg l pOq. □ 8.5. 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. In Theorem 8.8 we shall use this variety to relate the m 0 -adic graded algebra of Spg, eq ab with the tangent cone TC e pe`Xq of the Katsylo variety, where m 0 Ď Spg, eq denotes the maximal Kazhdan graded ideal and the orbit of e P g has partition λ. Let λ " pλ 1 , ..., λ n q P P ε pN q such that i " i 1 for all i, all parts of λ are distinct, and λ 1 ą 1. In particular λ is distinguished. This condition ensures that λ i´λi´1 ą 0 is even for all i " 2, ..., n and that pi, i`1q P ∆pλq for i " 1, ..., n´1. We set s 1 :" t λ 1 2 u; s i :" λ i´λi´1 2 for i " 2, ..., n. (8.1) Introduce a set of variables S λ :" tx i,r | 1 ď i ď n, 1 ď r ď s i u Y ty j | j " 1, ..., n´1u (8.2) and we consider the following collection of quadratic elements Q λ in the polynomial ring CrS λ s Q λ :" tx i,1 y i | i " 1, ..., n´1, λ i´λi´1 ą 0 is evenu Y tx i`2,1 y i , y i y i`1 | i " 1, ..., n´2u.
Finally we define X λ to be the algebraic variety with coordinate ring CrX λ s :" CrS λ s{pQ λ q. (8.4) The main goal of this section is to describe the irreducible components of X λ , and calculate their dimensions. For this purpose it suffices to consider a special partition: define α " pα 1 , ..., α n q by α " # p2, 4, ..., 2nq if ε "´1; p3, 5, ..., 2n`1q if ε " 1. (8.5) It follows directly from the definitions that CrX λ s -CrX α s b Crx i,r | 1 ď i ď n, 1 ă r ď s i s, which implies that It is now sufficient to study X α in order to understand the structure of X λ for any λ.
Let P n be the set of subsets S Ď t1, ..., n´1u such that for all i " 1, ..., n´2 either i P S or i`1 P S or both. For S P P n we let S c :" t1, ..., n´1uzS be the complementary set. We define a map from P n to the set of subsets of tx 1,1 , ..., x n,1 , y 1 , ..., y n´1 u Ď CrX α s by Lemma 8.5. S Þ Ñ pιSq is a bijection from P n to the set of minimal prime ideals of CrX α s and dim CrX α s{pιSq " |tx 1 , ..., x n , y 1 , ..., y n´1 uzιS| (8.8) Proof. The torus T :" pCˆq 2n´1 acts on Crx 1,1 , ..., x n,1 , y 1 , ..., y n´1 s by automorphisms rescaling generators, and this action descends to CrX α s. Since T is connected it preserves the irreducible components of X α , hence preserves the minimal primes of CrX α s. Since the T -weight spaces on Crx 1,1 , ..., x n,1 , y 1 , ..., y n´1 s are one dimensional, it follows that each minimal prime p is generated by p X tx 1,1 , ..., x n,1 , y 1 , ..., y n´1 u.
By inspection we see that for each S P P n the set ιS contains at least one of the factors of each quadratic relation (8.3) in CrX α s and is minimal with respect to this property. This implies that ι maps P n to minimal primes.
The injectivity of S Þ Ñ pιSq is clear and it remains to show that every minimal prime p is in the image. Suppose that p is generated by some subset S Ď tx 1,1 , ..., x n,1 , y 1 , ..., y n´1 u. Using the primality of p it is straightforward to check that S 1 :" t1 ď i ď n´1 | y i P Su P P n and pιS 1 q Ď p. Now by minimality we have p " pιS 1 q. □ Proposition 8.6. There is a bijection i Þ Ñ X i λ from maximal admissible multisets for λ to the set of irreducible components of X λ . Furthermore dim X i λ " |i|.
Proof. Applying Case 1 of the Kempken-Spaltenstein algorithm exactly pλ i´λi´1 q{2´1 times at each admissible index i we reduce the partition λ to α. This shows that the maximal admissible multisets for λ are of the form i Y j where i is a maximal admissible multiset for α and j is the multiset taking values in t1, ..., nu in which j occurs with multiplicity pλ j´λj´1 q{2´1. Combining this with (8.6) it suffices to prove the current proposition in the case λ " α. By Lemma 8.5 the irreducible components of X α are parameterised by P n . In order to prove the first claim of the proposition we construct a bijection from P n to the set of maximal admissible multisets.
The admissible multisets for α have multiplicities ď 2. Thanks to [PT14, Lemma 6] we know that a maximal admissible multiset is totally determined by the collection of indexes in t2, ..., nu which occur with multiplicity 2: this is the set of indexes such that Case 2 occurs at some point in the KS algorithm. Therefore the collection of indexes of multiplicity 2 is only constrained by the fact that if i has multiplicity 2 then neither i´1 nor i`1 have multiplicity 2. It follows that if S P P n then there is a unique maximal admissible sequence i for α such that the indexes in t2, ..., nuzpS`1q occur with multiplicity 2. This gives the desired bijection.
It remains to prove the claim regarding dimensions. Let S Ď P n and enumerate t1, ..., n´1uzS " pi 1 , ..., i l q. By the remarks of the previous paragraph the maximal admissible sequence associated to S P P n is determined as follows: first we let i 1 " pi 1`1 , i 1`1 , i 2`1 , i 2`1 , ..., i l`1 , i l`1 q and then we let i be the maximal admissible sequence obtained from i 1 by applying Case 1 of the KS algorithm to α i 1 as many times as possible. The set of indexes where Case 1 can be applied to α i 1 are precisely t1 ď i ď n | α i 1 i´α i 1 i´1 " 2u " t1 ď i ď n | i ‰ i t and i´2 ‰ i t for all tu (8.9) and it follows that the length of the maximal admissible sequence associated to S is n´1´|S|`#t1 ď i ď n | i, i´2 P Su. (8.10) According to (8.7) and Lemma 8.5 the dimension of the component of X α corresponding to S is 2n´1´|S|´#t1 ď i ď n | i P S c or i´2 P S c u (8.11) It is straightforward to see that (8.10) and (8.11) are equal. □ 8.6. The semiclassical abelianisation I: the distinguished case. Let λ " pλ 1 , ..., λ n q P P ε pN q be a distinguished partition satisfying λ 1 ą 1 and let e P g be a nilpotent element with partition λ. The indexes s 1 , ..., s n´1 defined in (8.1) allow us to construct an nˆn symmetric shift matrix σ " ps i,j q 1ďi,jďn via s i`1,i " s i,i`1 :" s i`1 for i " 1, ..., n´1. (8.12) By (3.2) these values determine the entire matrix, which coincides with the matrix constructed from λ in (4.1). Next we describe generators of gr m 0 Spg, eq ab and certain relations between them.
Proof. As explained in Section 4.5 there is a surjective Poisson homomorphism φ : Rpy n pσq, τ q ↠ Spg, eq. In the following proof we abuse notation and identify Spg, eq with the corresponding quotient of Rpy n pσq, τ q. This justifies the use of generators and relations appearing in Theorem 3.7.
We let θ :" η prq i`m 2 0 P gr m 0 Spg, eq. It follows from the proofs of Propositions 4.8 and 4.10 that gr m 0 Spg, eq is generated as a commutative algebra by elements Since natural map π : Spg, eq ↠ Spg, eq ab is a Kazhdan graded homomorphism the unique maximal graded ideal of Spg, eq maps to the unique maximal graded ideal of Spg, eq ab , and we indulge in another abuse of notation by writing m 0 for either ideal. Thus π induces a homomorphism π 0 : gr m 0 Spg, eq ↠ gr m 0 Spg, eq ab . We shall show that the kernel of π 0 contains the left-hand sides of (8.14)-(8.17), as well as the following elements Since (8.18) is the union of (8.13) and (8.19), this will complete the proof.
Let B Ď Spg, eq be the defining ideal of Spg, eq ab , which is generated by the Poisson brackets in Spg, eq. Then the kernel of π 0 is equal to gr m 0 B. Corollary 4.11 allows us to identify gr m 0 Spg, eq with Spg e q. Under this identification the vector space θpg e q Ď Spg, eq spanned by elements (8.18) identifies with the subspace g e Ď Spg e q. Furthermore the same Corollary also implies that tθpg e q, θpg e qu`m 2 0 identifies with rg e , g e s Ď g e Ď Spg e q. We can express this more informally by saying that the linear terms of Poisson brackets in Spg, eq are just the Lie brackets in g e , which is well-known in general (see [DSKV16,Theorem 2.11]). Now [PT14, Theorem 6] implies that the elements (8.19) identify with a spanning set for rg e , g e s, although we warn the reader that the notation there is different. It follows that every element of (8.19) can be expressed as b`m 2 0 for some b P B, and so these elements lie in gr m 0 B " Ker π 0 as required.
In order to complete the proof we show that the left-hand sides of (8.14)-(8.17) lie in gr m 0 B. Using (3.71) we see that θ ps i`1`1 q i θ ps i`2`1 q i`1 P B and it follows that the left-hand side of (8.14) lies in gr m 0 B. Now suppose that i " 1 and that ε "´1, so that λ 1 is even. Thanks to Proposition 4.10 we know that 9 θ pλ 1`s1`1 q 1 " 0 in Spg, eq. Using Example 4.9 and (3.69) we see that Note that λ 1 " 2s 1 in this case, since λ 1 is even. Projecting η pλ 1 q 1 θ ps 1`1 q 1 into gr m 0 Spg, eq we see that the left-hand side of (8.15) lies in gr m 0 B.
The relations (8.16), (8.17) can be deduced similarly by considering the quadratic terms appearing in (3.69), (3.74) and projecting into gr m 0 Spg, eq. Since the arguments for these two cases are almost identical we just provide the details for (8.17).
Before we proceed we record a single relation which is a special case of (3.69). For r ą s i`1`1 we have Fix i " 1, ..., n´2. Using (3.74) we examine the expression ␣ θ . There is a unique linear term θ p2s i`2`si`1`1 q i corresponding to m " m 1´1 and m 2 " 0. Thanks to (8.20) we can subtract an element of B to eliminate this linear term. Now consider the quadratic terms. If we choose any term such that m 2 ă m 1 then this term has a factor of θ p2pm 1´m2 q`s i`1`1 q i and so once again we can eliminate these particular quadratic terms using (8.20). The the only remaining quadratic terms are η which correspond to m 1 " m 2 " 0 and m 1 " m 2 " m´1 respectively. Finally one checks using (3.75) thatη This completes the proof of relation (8.17). The proof of (8.16) is almost identical, instead examining ␣ θ , and using (3.74) again. □ Theorem 8.8. Let g be a classical Lie algebra and e P g a distinguished nilpotent element. Let m 0 Ď Spg, eq ab be the maximal Kazhdan graded ideal. There is an isomorphism gr m 0 Spg, eq ab " ÝÑ CrTC e pe`Xqs.
In particular Spg, eq ab is reduced.
Proof. Let A denote the algebra with generators (8.13) and relations θ ps i`1`1 q i θ ps i`2`1 q i`1 " 0 for i " 1, ..., n´2; θ ps 2`1 q 1η p2s 1 q 1 " 0 for ε "´1; θ ps i`1`1 q iη p2s i q i " 0 for i " 2, ..., n´1; θ ps i`1`1 q iη p2s i`2 q i`2 " 0 for i " 1, ..., n´3. (8.21) Before we proceed, we outline the argument. First of all we will show that A -CrX λ s where X λ is the combinatorial Katsylo variety from Section 8.5. By Lemma 8.7 we have that A ↠ gr m 0 Spg, eq ab and since the reduced algebra of Spg, eq ab is Cre`Xs by Proposition 6.2(1), we see that CrX λ s ↠ CrTC e pe`Xqs. We then combine deductions of the previous sections to see that the components of X λ and TC e pe`Xq have the same dimensions. Since X λ is reduced we can apply Lemma 5.1 to deduce that we have isomorphisms CrX λ s -gr m 0 Spg, eq ab -CrTC e pe`Xqs, from which the current theorem follows.
Step (i): Comparing (8.21) with (8.3) we see there is an isomorphism A " ÝÑ CrX λ s defined bȳ Step (ii): Now we show that A ↠ gr m 0 Spg, eq ab . We adopt the conventionη 2r i " 0 when r ą s i or i " 0. The algebra endomorphism of A defined byη for all i " 1, ..., n and acting identically on the other generators, is a unimodular subsitution. Therefore it induces an automorphism. By slight abuse of notation we denote the images of the generators by the same symbols. With this new set of generators the relations are precisely (8.14)-(8.17). By Lemma 8.7 and part (i) we have a surjection A " CrX λ s ↠ gr m 0 Spg, eq ab .
Step (iii): Recall that we equipTC e pe`Xq with the reduced scheme structure. By Corollary 4.11 we have CrX λ s ↠ gr m 0 Spg, eq ab ↠ CrTC e pe`Xqs (8.22) Applying Proposition 6.2(2), Proposition 8.3 and Proposition 8.6 we have dpX λ q " dpe`Xq " dpTC e pe`Xqq. Now by Lemma 5.1 we see that the surjections (8.22) are isomorphisms on the underlying varieties of closed points. Since CrX λ s is reduced these maps are algebra isomorphisms and gr m 0 Spg, eq ab is reduced. Applying Lemma 5.4 we see that Spg, eq ab is reduced, as claimed. □ 8.7. The semiclassical abelianisation II: the general case. The following result uses the local geometry of Poisson manifolds to infer reducedness of Spg, eq ab from the case where e is distinguished, where we can apply Theorem 8.8.
Theorem 8.9. Let g be a classical simple Lie algebra and let e P g be any nilpotent element. Then Spg, eq ab is reduced.
Proof. Part (i): In type A the fact that Spg, eq ab is reduced can be deduced from [Pr10, Theorem 3.3], and so we let g be simple of type B, C or D throughout the proof. We let O Ď g be the adjoint orbit of e in g, and letg and r O be the Lie algebra and distinguished nilpotent orbit introduced in Lemma 8.4. Pickẽ Pg and an sl 2 -triple pẽ,h,f q. Write r G for the simply connected, connected complex algebraic group withg " Liep r Gq. Recall that l " gl n´i 1`1ˆ¨¨¨ˆg l n´i l`1ˆg embeds as a Levi subalgebra ofg. If we choose e P O Ď g Ď l then O identifies with the adjoint L-orbit and we have the following Poisson isomorphism Spl, eq -Spgl i 1 q b¨¨¨b Spgl i l q b Spg, eq Therefore Spg, eq ab is reduced if and only if Spl, eq ab is reduced. This holds if and only only if pSpl, eqm 0 q ab is reduced, where m 0 is the maximal ideal of e, thanks to Lemma 5.4 and Lemma 5.6.
Part (ii): Now pick a regular element s P zplq so thatg sl. Write x " s`e Pg. We will explain below that x can be viewed as a point ofẽ`gf " Spec Spg,ẽq, and by slight abuse of notation we will also write m 0 for the maximal ideal of x in Spg,ẽq. We claim that Spl, eqm 0 -Spg,ẽqm 0 as Poisson algebras. Note that pSpg,ẽqm 0 q ab is reduced by Lemma 5.4 and Theorem 8.8. Thanks to part (i) the current proof will follow from the claim.

Part (iii):
Pick an sl 2 -triple pe, h, f q for e inside l, then define y " s`f Pg. By sl 2 -theory the affine variety x`g y is a transverse slice to the adjoint orbit Adp r Gqx at the point x. Also pick an sl 2 -triple pẽ,h,f q ing. After replacing the latter triple by some conjugate we can actually assume that x Pẽ`gf . By construction we have Indg l pOq " Adp r Gqẽ. It follows that x lies in a sheet of g containingẽ, and so we have x Pẽ`r X, where the latter variety is defined in the same way as (6.1).
Now we choose small neighbourhoods U x Ďẽ`gf 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 carry transverse Poisson structures on the ring of analytic functions, see [LPV13, §5.3.3] for example. Write C an pM q for the ring of analytic functions on a complex manifold M . There is a natural restriction map Cre`l f s Ñ C ab pW e q which is a Poisson homomorphism. Thanks to [Ser55,Proposition 3] this induces an isomorphism Spl, eqm 0 -C an pW e qê of complete Poisson algebras. Similarly we have a Poisson isomorphism Spg,ẽqm 0 -C an pU x qx.
Now the claim will follow from the existence of an isomorphism C an pU x qx -C an pW e qê of complete Poisson algebras. First of all we observe that x Pẽ`r X implies that bothẽ`gf and x`g y are transverse slices to Adp r Gqx 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. □ 8.8. The abelianisation of the finite W -algebra via deformation theory. Here we prove the first main theorem (Theorem 1.1), which states that Premet's component map (1.2) is a bijection. As we explained after the statement of that theorem, the result is know in type A and so we must complete the proof for the other classical types.
First of all we apply Proposition 7.7 and Theorem 8.9 to see that both gr U pg, eq ab and U pg, eq ab are reduced. Now apply Lemma 5.2, Proposition 6.2(1) and Theorem 7.3 to see that # Comp Epg, eq ď # Comp CEpg, eq " # Comppe`Xq.
Since Premet's map (1.2) is surjective and restricts to a dimension preserving bijection from some subset, the theorem follows.
Remark 8.10. Premet asked whether the abelian quotient U pg, eq ab of a finite W -algebra is reduced [Pr10, Question 3.1]. Combining Proposition 7.7 with Theorem 8.9 we have given an affirmative answer in the case of classical Lie algebras.
The problem of understanding whether U pg, eq ab is reduced for exceptional Lie algebra is rather subtle. 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 correspond to the first four columns of [PT21, Table 1]). For these W -algebras it is not hard to see that the reduced algebra associated to U pg, eq ab is isomorphic to Crxs{px 2´1 q as filtered algebras, with the generator x in some positive degree. This ensures that gr U pg, eq ab is not reduced, and Proposition 7.7 implies that Spg, eq ab admits nilpotent elements. 9. The orbit method 9.1. Quantizations and deformations of symplectic singularities. We say that an affine Poisson algebra A is graded of degree d ą 0 if A " À iě0 A i is a graded commutative algebra such that A 0 " C and the Poisson bracket is in degree´d, meaning tA i , A j u Ď A i`j´d . In this case the variety X :" SpecpAq is conical since the grading induces a contacting Cˆ-action with unique fixed point À ią0 A i P X, and the contracting action rescales the Poisson bivector. The smooth locus of X will be denoted X reg . From now on we fix such a Poisson algebra A.
A filtered Poisson deformation of A is a pair pA, ιq consisting of a filtered Poisson algebra A " Ť iě0 A i such that the Poisson bracket satisfies tA i , A j u Ď A i`j´d and an isomorphism of graded Poisson algebras ι : gr A Ñ A.
A filtered quantization of A is pair pA, ιq where A is a not necessarily commutative, filtered algebra satisfying rA i , A j s Ď A i`j´d such that ι : gr A Ñ A is a Poisson isomorphism (see [ACET20, 2.2] for example).
Let B be a non-negatively graded connected algebra. By a graded Poisson B-algebra we shall mean a graded B-algebra A with Poisson structure such that the structure map B Ñ A is graded and factors through the Poisson centre. A graded Poisson deformation of A over base B is a pair pA, ιq consisting of a flat graded Poisson B-algebra A and an isomorphism ι : Now let B be a connected graded algebra concentrated in non-negative degrees. We can then equip B with the obvious filtration whose jth filtered piece is the sum of the first j graded pieces, and gr B -B as algebras. A filtered B-algebra is a filtered algebra A equipped with a filtered map B Ñ A which factors through the centre. We say that pA, ιq is a filtered quantization (of a Poisson deformation) of A over base B if A is a flat filtered B-algebra and pgr A, ιq is a graded Poisson deformation of A over gr B " B, and the map ι is an isomorphism ι : gr A b gr B C Ñ A of Poisson algebras.
For more detail on the above definitions, and a precise description of morphisms, we refer the reader to [Lo22, §2.2, 2.3, 3.1] or [ACET20, §2]. Whilst surveying the construction of graded Poisson deformations of Poisson algebras, and their quantizations, below we shall occasionally refer to deformations and quantizations of a Poisson scheme X (the distinction between a deformation of an algebra and a scheme is important when X is not affine). We refer the reader to [Lo22, §2.2] for a brief introduction to these notions.
Following Beauville [Be00], we say that X " SpecpAq has symplectic singularities if: ‚ X is normal; ‚ the Poisson bivector ω on X has full rank on X reg , which is therefore a smooth symplectic variety; ‚ there exists a projective resolution p X Ñ X such that the pullback of ω| X reg extends to a regular (not necessarily symplectic) bivector on p X.
From now on we also assume that A " CrXs has symplectic singularities and is graded in degree d ą 0. In this case the variety X is known as a conical symplectic singularity.
Example 9.2. Let G be a complex reductive group and g " LiepGq. If O Ď g is a nilpotent orbit then the affinisation Spec CrOs is a conical symplectic singularity. It is well-known that the affinisation is isomorphic to the normalisation of the closure (see [Ja04,Proposition 8.3]). The conical structure on O arises from the vector space structure on g (see [Ja04, Lemma 2.10]), whilst the presence of symplectic singularities follows from the work of Panyushev [Pa91]. Now suppose that r O Ñ O is a G-equivariant finite cover of a nilpotent orbit. Then the affinisation Spec Cr r Os is also a conical symplectic singularity (see [Lo18, Lemma 2.5]). We remark that the conical structure comes from lifting the Cˆ-action, and possibly replacing Cˆwith a cover.
A conical symplectic singularity X " SpecpAq admits a very nice deformation theory: there is a classification of filtered Poisson deformations and filtered quantizations of A, due to Namikawa [Na10,Na11] and Losev [Lo22], which we now recall.
Recall that a normal variety r X is said to be Q-factorial if every Weil divisor admits a non-zero integer multiple which is Cartier. We refer the reader to [Lo22, Proposition 2.3] and [BCMH00] for the defining properties of a Q-factorial terminalisation.
Let r X Ñ X be a Q-factorial terminalisation and define P :" H 2 p r X reg , Cq. This is known as the Namikawa-Cartan space of X. Then according to [Lo22, Proposition 2.6] there exists a universal graded Poisson deformation of r X over the base P, which should be understood as scheme-theoretic deformation, in the sense of [Lo22, §2.2]. This means the following: if r X B is a graded Poisson deformation of r X over SpecpBq there exists a unique Cˆ-equivariant morphism SpecpBq Ñ P and a (non-unique) isomorphism SpecpBqˆP r X P " ÝÑ r X B . In other words every graded Poisson deformation of r X can be obtained from r X P by base change. A universal graded Poisson deformation of A can be produced from r X P . First of all we describe an important finite group W X associated to X which controls isomorphisms of deformations. The following facts are explained in more detail in [Lo22,§2.3]. A result of Kaledin [Ka06] states that X has only finitely many symplectic leaves. Let L 1 , ..., L k be the collection of leaves of codimension 2. Another of the main results of op. cit. states that there exists a formal slice Σ i to each leaf L i . By [Be00, Proposition 1.3] we know that X has rational singularities, and so the formal neigbourhood of 0 P Σ i is a Du Val singularity, i.e. the formal completion of a rational double point of type ADE. Let Ă W i be the Weyl group associated to the Dynkin diagram of the singularity Σ i . The fundamental group of L i acts on Ă W i by diagram automorphisms and we let W i denote the fixed points. The Namikawa-Weyl group of X is W X :" ś k i"1 W i . We are now ready to describe a Poisson deformation of X with a remarkable property. Let X P :" Spec Cr r X P s and W " W X .
Proposition 9.3. [Lo22, Proposition 2.9, Proposition 2.12, Corollary 2.13] The algebras A P :" CrX P s W and B :" CrPs W satisfy the following properties: (1) A P is a finitely generated algebra and a free B-module.
(2) A P b B C " A where C denotes the unique one dimensional graded B-module.
(3) For every finitely generated positively graded, connected algebra B 1 and graded Poisson B 1 algebra A 1 such that A 1 b B 1 C -CrXs there exists a unique graded algebra homomorphism B Ñ B 1 and a (not necessarily unique) B 1 -linear graded Poisson isomorphism We call A P the universal graded Poisson deformation of A.
Similar to the situation for Poisson deformations, the smooth locus of the Q-factorial terminalisation r X of X admits a "universal quantization" (Cf. [Lo22, Proposition 3.1]). This is a sheaf of filtered, flat CrPs-algebras in the conical topology on r X. The pushforward of this sheaf of algebras via r X reg Ñ r X is denoted D P [Lo22, Corollary 3.2]. It follows that there is a map from CrPs to the global sections of D P . The algebra CrPs is graded whilst ΓpD P q is filtered. With respect to the natural filtration induced on CrPs the map CrPs Ñ ΓpD P q is strictly filtered in the sense of [ACET20, §2.2].
Lemma 9.4. [Lo22, Proposition 3.3] Write W " W X . We have the following: (1) ΓpD P q is a filtered quantization of CrX P s and the associated graded homomorphism of CrPs Ñ ΓpD P q is CrPs Ñ CrX P s, appearing in Proposition 9.3.
(2) W acts on ΓpD P q by filtered automorphisms preserving CrPs. The induced action on grpCrX P sq " CrX P s coincides with the action on CrX P s.
Crucially the filtered quantizations of Poisson deformations of A are classified by the same space as the graded Poisson deformations of A.
Proposition 9.5. [Lo22, Proposition 3.5] Let A P :" ΓpD P q W and B :" CrPs W . Let B 1 be any finitely generated, positively graded commutative algebra and let A 1 be a graded, flat Poisson B 1algebra such that A 1 b B 1 C " A. Also let A 1 be a B 1 -algebra which is filtered quantization of A 1 , such that the associated graded map of B 1 Ñ A 1 is B 1 Ñ A 1 . Then there is a unique filtered algebra homomorphism B Ñ B 1 and a (not necessarily unique) isomorphism Furthermore the associated graded homomorphism of (9.2) is (9.1).
We call A P the universal filtered quantization of Poisson deformations of A. For clarity we remark that A P is not a filtered quantization of A, rather it is a filtered quantization of CrX P s. However filtered quantizations of A can be obtained by specialising A P over points of P (Cf. Remark 9.1). 9.2. Birational induction and the orbit method. Let G be a connected complex reductive algebraic group and write g " LiepGq. In this section we recap some ideas from [Lo22, §4 & §5] and explain the construction of the orbit method map.
We identify g with g˚in what follows, and so Crgs is equipped with a natural Poisson structure. Furthermore the action of G on g is Hamiltonian.
Recall from Section 6.1 that an induction datum is a triple pl, O 0 , zq consisting of a Levi subalgebra, a nilpotent orbit therein and z P zplq. When z " 0 we usually omit this element, and say that pl, O 0 q is an induction datum. After fixing a datum we can choose a parabolic subalgebra p " LiepP q admitting l as a Levi factor, with r " Radppq, and construct a homogeneous bundle GˆP pz`O 0`r q over G{P . This bundle admits a map to g via the adjoint representation of G, which we denote π. The image of π is the closure of an adjoint orbit, and this orbit depends only on the induction datum, not on the choice of P [Lo22, Lemma 4.1]. This is the (Lusztig-Spaltenstein) induced orbit, denoted Ind g l,z pO 0 q. ‚ When π is generically finite onto its image we say that the orbit datum is birational. We say that Ind g l,z pO 0 q is birationally induced from pz, l, O 0 q ‚ Furthermore we say that O Ď g is birationally rigid if it cannot be birationally induced from a proper Levi subalgebra. ‚ If π is birational and O 0 is birationally rigid then we say that the datum is birationally minimal.
One of the key properties of birational induction is the following. For background we refer the reader to [Am20], where the theory of birational induction was studied in the setting of conjugacy classes in groups.
Now pick an induction datum such that O 0 Ď l is birationally rigid. Also choose a parabolic subgroup P such that p " LiepP q admits l as a Levi factor and nilradical r. Let X 0 :" Spec CrO 0 s be the affinisation of O 0 (this coincides with the normalisation of O, by [Ja04, Proposition 8.3]) and consider the bundle GˆP pX 0ˆr q. This contains a dense open orbit, say G{H for some subgroup H, and this is a finite G-equivariant cover of Ind g l pO 0 q. Thanks to Example 9.2 the affinisation X " Spec CrG{Hs is a conical symplectic singularity. In this setting one can describe the Namikawa-Cartan space, the Namikawa-Weyl group of X and the universal graded Poisson deformation of X.
Lemma 9.7. [Lo22, Proposition 2.9 & 4.7] If G is semisimple then: (1) P " zplq; (2) W X is a normal subgroup of W pl, O 0 q :" Z N pO 0 q{L where N " N G pLq is the normaliser of L, and Z N pO 0 q denotes the set of elements of N which stabilise the orbit O 0 . (3) The scheme r X P is isomorphic toX :" GˆP pzplqˆX 0ˆr q as a Poisson scheme over P with Cˆ-action.
We note that the Cˆ-action onX arises from the fact thatX is a graded deformation of GˆP pX 0ˆr q, in the sense of [Lo22, §2.2] (see the proof of [Lo22, Proposition 4.7]). Now pick an orbit O Ď g, and denote the corresponding birational induction datum by pl, O 0 , zq. Let G{H be the dense orbit in GˆP pX 0ˆr q, as above. It follows from [Lo22, Proposition 4.7(2)] that the affinisation of the dense orbit in GˆP ptzuˆX 0ˆr q is isomorphic to the fibre of the universal graded Poisson deformation of CrG{Hs corresponding to the parameter z P P (terminology of Proposition 9.3). We denote the regular functions on the fibre by CrX z s.
Thanks to Proposition 9.5 the above construction has a quantum analogue: let A z be the fibre of the universal filtered quantization of CrG{Hs corresponding to parameter z P P.
The action of G on G{H is Hamiltonian and the comoment map g Ñ CrX 0 s lifts to a quantum comoment map g Ñ A z . This gives rise to an algebra homomorphism U pgq Ñ A z and the kernel is denoted J pOq.
Lemma 9.8. The ideal J pOq is a completely prime, primitive ideal.
Proof. Since A z is a filtered quantization of CrG{Hs and the latter is an integral domain it follows that J pOq is completely prime.
Again let pl, O 0 , zq be the birationally minimal orbit datum inducing to O. The comoment map Crgs Ñ CrG{Hs factors through the comoment map Crgs Ñ CrInd g l pO 0 qs, and since the induced orbit is contained in the nilpotent cone, which is the vanishing locus of the non-constant homogeneous invariant polynomials Crgs G , it follows that the kernel of the map Crgs G Ñ CrG{Hs has codimension 1. The associated graded map of quantum comoment U pgq Ñ A z is Crgs Ñ CrG{Hs, and it follows that the kernel of the map Zpgq Ñ A z has codimension 1. This means that the prime ideal J pOq admits a central character. Now by [Di96, §8.5.7] it follows that J pOq is a primitive ideal. □

Hence this construction yields a map
J : g{G Ñ Prim U pgq, which is injective for classical Lie algebras [Lo22, Theorem 5.3].
Recall the notation Prim O U pgq from the introduction. The following refinement of Theorem 1.2 uses the notation of Section 7. It is the main result of this section.
Theorem 9.9. Let e P O Ď g be an element of a nilpotent orbit in a classical Lie algebra. The following sets coincide: (1) The image of J intersected with Prim O U pgq.
(2) The ideals Ann U pgq pQ b U pg,eq C η q where C η P U pg, eq -mod is one dimensional.
The proof of the theorem will occupy the rest of the paper. The first step is to describe the associated variety of the primitive ideal J pOq. This was stated in [Lo22, §5.4], but we provide a proof for completeness. Another proof has appeared recently in [LMM21, Proposition 6.1.2(1)].
Lemma 9.10. For O P g{G we let S be a sheet of g containing O. Then VApJ pOqq " S X N pgq.
Proof. Let pl, O 0 , zq be the birationally minimal orbit datum inducing to O. For t P C we can consider the orbit Optq which is dense in the image of the generalised Springer map GˆP ptzÒ 0`n q Ñ g. By definition we have Op1q " O and Op0q " Ind g l pO 0 q, whilst dim Optq is constant. It follows that O and Ind g l pO 0 q lie in a sheet. Since O 0 is nilpotent and every sheet contains a unique nilpotent orbit, S X N pgq " Ind g l pO 0 q for every sheet containing O. Now write G{H for the dense orbit in GˆP pX 0ˆn q. Since gr is exact on strictly filtered vector spaces, the graded ideal gr J pOq is the kernel of the comoment map Crgs Ñ CrG{Hs. This factors through the comoment map for Crgs Ñ CrInd g l pO 0 qs, which shows that a gr J pOq is the defining ideal of Ind g l pO 0 q. This completes the proof. □ 9.3. Harish-Chandra bimodules and Losev's dagger functors. Now we keep fixed e P N pgq, the adjoint orbit O :" AdpGqe, an sl 2 -triple te, h, f u for e and write G e p0q for the pointwise stabiliser of the triple. Note that the notation g e p0q " Lie G e p0q is consistent with that of Section 2.2. In [Lo10a] Losev constructed a map I Þ Ñ I : from two-sided ideals of U pg, eq to two-sided ideals of U pgq, with remarkable properties. In his subsequent work [Lo11] this construction was upgraded to a pair of adjoint functors between the categories of Harish-Chadra U pgq-bimodules and G e p0qequivariant Harish-Chandra U pg, eq-bimodules, as we now recall.
The category HC U pgq of Harish-Chandra U pgq-bimodules consists of U pgq-bimodules, finitely generated on both sides, which are locally finite for adpgq. This implies that there is a G-action which differentiates to adpgq. If O :" G¨e then we denote by HC O U pgq the full subcategory of bimodules supported on O, see [Lo11,§1.3] for the precise definition.
It follows from [GG02] that U pg, eq admits G e p0q-action by filtered automorphisms, and a quantum comoment map g e p0q ãÑ U pg, eq (9.3) This map is injective, thanks to [Pr07a, Lemma 2.4]. Therefore one can define the category HC G e p0q U pg, eq of G e p0q-equivariant Harish-Chandra U pg, eq-bimodules to be the category of U pg, eqbimodules, finitely generated on both sides, with a compatible G e p0q-action which differentiates to the ad g e p0q-action (see [Lo11,§1.3]). We denote the subcategory of finite dimensional objects by HC G e p0q fin U pg, eq. In [Lo11] Losev constructed a pair p‚ : , ‚ : q of adjoint functors ‚ : : HC O U pgq Ô HC G e p0q fin U pg, eq : ‚ : which behave nicely when applied to quantizations of nilpotent orbit covers (see [Lo22,§5]). The definition of these functors depends upon the decomposition theorem from [Lo10a] which we will not describe here. Instead we recall from [Lo11, §3.5] that there is an isomorphism M : -pM {M gpă -1q χ q adpgpă0qq (9.4) for any M P HCpgq, which is in the spirit of the definitions used throughout this paper. One immediate consequence which we use often is that U pgq : -U pg, eq.
Let pl 0 , O 0 q be an induction datum for O such that O 0 is birationally rigid in l 0 . This leads to to a finite cover G{H ↠ O discussed in Section 9.2. Let A denote the universal filtered quantization of CrG{Hs described in Proposition 9.5. We can assume without loss of generality that pG e q˝Ď H Ď G e . By Lemma 9.7 this is a Crzpl 0 qs W -algebra, where X denotes the affinisation of G{H and W is a subgroup of the relative Weyl group N G pLq{L.
The image of Crzpl 0 qs W Ñ A is central and so for every ideal I Ď Crzpl 0 qs W we see that A{IA P HC U pgq. Thus there is a natural G e p0q-action on pA{IAq : .
The following fact will be used a few times in the sequel.
For brevity we write B " Crzpl 0 qs W and for a closed point z P Spec B we write I z Ď B for the corresponding maximal ideal, and C z " B{I z for the one dimensional B-module, so that A z " A{I z A.
Since B Ď A is central, the image of the map B Ñ A{Agpă -1q χ lies in the space of adpgpă 0qqinvariants, giving an algebra homomorphism B Ñ A : . The multiplication on A : is induced from that on A it follows that the image of the map is central.
Proof. By (9.4) we have A : b B C z -pA{Agpă -1q χ qq ad gpă0q {pI z A{Agpă -1q χ qq ad gpă0q . Since ad gpă 0q is exact on the category of Whittaker modules [GG02, Theorem 6.1] it will suffice to show that pA{Agpă -1q χ q{pI z A{Agpă -1q χ q -pA{I z Aq{pAgpă -1q χ {I z Aq This is easy to see: both sides are isomorphic to A{Apgpă -1q χ`Iz q. □ By the lemma we can (and shall) identify pA z q : with pA : q{I z A : in what follows.
Corollary 9.13. The algebra A : is commutative.
Proof. Pick a P A : . Let adpaq : A : Ñ A : be the map b Þ Ñ ra, bs. For any closed point z P Spec B we let ad z paq : pA : q z Ñ pA : q z be the induced map on the quotient. Since A z is a filtered quantization of CrG{Hs it follows from Lemma 9.11 that pA z q : " pA : q z is a commutative algebra, and so ad z paq " 0. We conclude adpaqA : Ď I z A : for all maximal ideals z P Spec B. By Proposition 9.3(1) we see that gr A is a free gr B-module, and it follows by a standard filtration argument that A is a free B-module. We deduce that Ş z pI z Aq " 0, and this implies that Ş z pI z A : q " p Ş z I z Aq : " 0 by (9.4) and Lemma 9.12. Finally we have adpaqA : Ď Ş z pI z A : q " 0, and the proof is complete. □ Proposition 9.14. The G e p0q-action on A : factors through the component group Γ :" G e p0q{G e p0qa nd A Γ : :" pA : q G e p0q -Crzpl 0 qs W . (9.5) Proof. As B is central in A, it follows that B is adpgq-invariant, hence G-invariant. Therefore the image of B Ñ A : is G e p0q-invariant. Now consider the homomorphism φ : B Ñ A G e p0q : . We shall show that this is an isomorphism.
Since G e p0q is a reductive group the functor of G e p0q-invariants is exact, and so by Lemma 9.12 we have A . The latter is nonzero by Lemma 9.11. If 0 ‰ b P Ker φ then, since B is reduced, we can choose a maximal ideal I z Ď B with b R I z and we find A G e p0q : b B C z " 0. This contradiction shows that φ is injective. Now consider the cokernel C of φ in the category of B-modules. We claim that φ is surjective after specialisation at any point z P SpecpBq. By Lemma 9.11 we have pA{I z Aq G e p0q " C. Therefore the specialisation is just C " B{I z Ñ C. If this map is zero then we can choose b P B such that Cb`I z " B and φpbq Ď I z A G e p0q : . This would force φpBq Ď I z A G e p0q : Ĺ A G e p0q : which would contradict the fact that φp1q " 1. Therefore φ is surjective after specialisation, as claimed.
(2) The set of ideals Ann U pgq Q b U pg,eq C η Ď Prim U pgq where C η is a one dimensional U pg, eqmodule appearing in the image of the map of maximal spectra µ: : Spec A G e p0q : " Spec B Ñ Spec U pg, eq ab .
Proof. The first claim follows directly from Corollary 9.13. We now prove that (1) " (2). The set (1) is equal to the collection of kernels of the maps U pgq Ñ A b B C z as we vary over all closed points z P Spec B. For any fixed z (using Lemma 9.12) the kernel of the map µ : : U pg, eq Ñ A : b B C z is the annihilator of a one dimensional U pg, eqmodule, thanks to our previous observations. We denote it C η . Our chosen z lies in Spec B i for some i and we claim that the kernel K " KerpU pgq Ñ A b B C z q is equal to the annihilator of Q b U pg,eq C η . This will complete the proof.
As we mentioned above, the functor ‚ : on Harish-Chandra bimodules was preceded by Losev's construction of a map I Þ Ñ I : from ideals of U pg, eq to ideals of U pgq [Lo10a]. Also note that two sided ideals of U pgq can be regarded as Harish-Chadra U pgq-bimodules, and that the map J Þ Ñ J : sends ideals of U pgq to ideals of U pg, eq [Lo11, Theorem 1.3.1].
By loc. cit. we know that K : " Ann U pg,eq C η . By Lemmas 9.8 and 9.10, we know that K is a primitive ideal with associated variety G¨e, so by [Lo10a, Theorem 1.2.2(viii)] the ideals I Ď U pg, eq which satisfy I : " K are precisely minimal primes over K : . Since K : is primitive it is prime, and in particular pK : q : " K. Now [Lo10a, Theorem 1.2.2(ii)] implies that K " pAnn U pg,eq C η q : " Ann U pgq pQ b U pg,eq C η q.
This concludes the proof. □ Finally we prove the main theorem of this section.
Proof of Theorem 9.9. Thanks to Lemma 9.15 it suffices to show that µ: : Spec A G e p0q : Ñ Spec U pg, eq ab is surjective. We will show that the map is finite and dominant. Since finite morphisms are closed this will complete the proof. Since gr U pg, eq ab is reduced (Corollary 7.7 and Theorem 8.9), and since the property of being a finite module or an injective homomorphism can both be lifted through the associated graded construction, it is enough to show that the map gr µ : : Cre`g f s ab Ñ gr A G e p0q : is injective and gives a finite extension of algebras. The G e p0q-action on A : factors through a finite group (Proposition 9.14) and so it will suffice to show that µ : : Spg, eq ab Ñ gr A : is injective and finite.
By Lemma 9.7 we know that gr A is isomorphic to CrXs W whereX " ś s i"1 GˆP i pzpl i qX iˆri q and W " ś s i"1 W i . It will be convenient to work with a slightly larger algebra: letǍ denote the algebra ś s i"1 ΓpD i q where D i is the sheaf associated to the conic symplectic singularity Spec CrG{H i s in Proposition 9.5. Then we have grǍ " CrXs and it will suffice to show that Spg, eq ab Ñ grpǍ : q is injective and finite, sinceǍ W " A and W is a finite group.
The map Spg, eq Ñ grpǍ : q coincides with the pullback to e`g f of Crgs Ñ CrXs (see [Lo22, Lemma 5.1(4)] or [Lo11, Lemma 3.3.2(3)]). This means that if we take the preimage Y of e`g f insideX then grpǍ : q " CrY s. The algebra Spg, eq ab is reduced (Theorem 8.9) and it follows that the map is injective if and only if the corresponding morphism of spectra is dominant. Now it suffices to show that Y Ñ Spec Spg, eq ab is finite and dominant.
The morphismX Ñ g is proper, since it is the collapsing map of a bundle over a projective variety ś i G{P i . Properness is stable under base change and it follows that Y Ñ e`g f is proper as well.
We now show that Y Ñ e`g f is quasi-finite. The map GˆP i pzpl i qˆX iˆri q Ñ g factors through ξ : GˆP i pzpl i q`O i`ri q Ñ g and, since GˆP i pzpl i qˆX iˆri q is the normalisation of GˆP i pzpl i q`O i`ri q, it suffices to show that ξ map is quasi-finite, when restricted to the preimage of e`g f . The image of ξ is the closure of the decomposition class D i associated to pl i , O i q (see [Bo81, Lemma 2.5]). The variety Y inherits a Cˆ-action from the contracting Kazhdan action on e`g f described in Section 6.2. The map Y Ñ e`g f is Cˆ-equivariant, and so to show that it is quasi-finite it suffices to show that the fibre over e is finite. By Lemma 9.7(3) the fibre over e is contained in GˆP i pO i`ri q. The image of the map GˆP i pO i`ri q Ñ g is contained in Ind g l i pO i q, and so by dimension comparison we see that GˆP i pO i`ri q Ñ Ind g l i pO i q is generically finite-to-one. There is an open subset of Ind g l i pO i q over which the fibres are finite and, since the collapsing map of the fibre bundle is G-equivariant, we see that the fibre is finite over e P Ind g l i pO i q. Thus Y Ñ e`g f is quasi-finite.
We have shown that Y Ñ e`g f is quasi-finite and proper, hence finite. It remains to show that Y Ñ Spec Spg, eq ab is dominant. Let S 1 , ..., S k be the sheets of g containing O. By Proposition 6.2(1) we know that Spec Spg, eq ab " pe`g f q X Ť i S i . Since the sheets are classified by the rigid induction data (Theorem 6.1) and the tpl i , O i q | i " 1, ..., su denote all induction data for G¨e, we might as well assume that pl i , O i q is the induction datum corresponding to S i for i " 1, ..., k. By the remarks of the previous paragraph we know that the image of GˆP i pzpl i qˆX iˆri q Ñ g is dense in S i and so Y Ñ Spec Spg, eq ab is dominant. This completes the proof. □