W-algebras Associated to Truncated Current Lie Algebras

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W-algebras Associated to

Truncated Current Lie Algebras

Thèse Xiao He Doctorat en mathématiques Philosophiæ doctor (Ph.D.) Québec, Canada © Xiao He, 2018

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Résumé

Étant donné une algèbre de Lie g semi-simple de dimension finie et un élément nilpotent non nul e ∈ g, on peut construire plusieurs algèbres-W associées à (g, e). Parmi eux, l’algèbre-W affine est une algèbre vertex qui peut être réalisée comme une cohomologie semi-infinie d’une sous-algèbre nilpotente de ˜g, où ˜g est l’algèbre de Kac-Moody associée à g. L’algèbre-W finie est l’algèbre de Zhu de l’algèbre-W affine. Dans les constructions des algèbres-W, une forme bilinéaire non dégénérée invariante et une bonne Z-graduation de g jouent des rôles essentiels. Les algèbres de courants tron-qués associées à g sont des quotients de l’algèbre de courants g ⊗ C[t]. On peut montrer que: (1) des formes bilinéaires non dégénérées invariantes existent sur des algèbres de courants tronqués; (2) une bonne Z-graduation de g induit des bonnes Z-graduations des algèbres de courants tronqués. Alors, les constructions des algèbres-W fonctionnent bien dans le cas des algèbres de courants tronqués. Les résultats de cette thèse sont les suivants. Premièrement, nous introduisons les algèbres-W finies et affines associées aux algèbres de courants tronqués et nous généralisons certaines propriétés des algèbres-W associées aux algèbres de Lie semi-simples. Deuxièmement, nous developpons une ver-sion ajustée de la cohomologie semi-infinie, ce qui nous permet de définir les algèbres-W affines associées à des éléments nilpotents généraux d’une façon uniforme. À la fin, nous prouvons que les algèbres de Zhu de niveaux plus hauts d’une algèbre vertex conforme sont toutes isomorphes à des sous-quotients de son algèbre enveloppante universelle.

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Abstract

Given a finite-dimensional semi-simple Lie algebra g and a non-zero nilpotent element e ∈ g, one can construct various W-algebras associated to (g, e). Among them, the affine W-algebra is a vertex algebra which can be realized through semi-infinite cohomology, and the finite W-algebra is the Zhu algebra of the affine W-algebra. In the constructions of W-algebras, a non-degenerate invariant bilinear form and a good Z-grading of g play essential roles. Truncated current Lie algebras associated to g are quotients of the current Lie algebra g ⊗ C[t]. One can show that non-degenerate invariant bilinear forms exist on truncated current Lie algebras and a good Z-grading of g induces good Z-gradings of truncated current Lie algebras. The constructions of W-algebras can thus be adapted to the setting of truncated current Lie algebras.

The main results of this thesis are as follows. First, we introduce finite and affine W-algebras as-sociated to truncated current Lie algebras and generalize some properties of W-algebras asas-sociated to semi-simple Lie algebras. Second, we develop an adjusted version of semi-infinite cohomology, which helps us to define affine W-algebras associated to general nilpotent elements in a uniform way. Finally, we consider vertex operator algebras in general, and show that their higher level Zhu algebras are all isomorphic to subquotients of their universal enveloping algebras.

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Acknowledgements

I would like to express my deepest gratitude to my Ph.D. advisor Professor Michael Lau for his insightful advice, constant encouragement and inspiring discussions. I would also like to thank Pro-fessors Hugo Chapdelaine and Antonio Lei for their interesting lectures. Also, I would like to thank my master’s advisor Professor Kaiming Zhao, who opened the door to the math world for me, and Professor Yang Han, who introduced me to representation theory.

I am also very grateful for the financial support received from the China Scholarship Council (File No.201304910374), l’Institut des sciences mathématiques, the NSERC Discovery Grant of my advisor and the Département de mathématiques et de statistique of Université Laval.

I enjoyed and learned much from the discussions with my “mathematical brothers and sister”, Jean Auger, Rekha Biswal and Abdelkarim Chakhar, to whom I would like to express my sincere thanks. Also, I would like to thank the secretary of the math department, who created a comfortable environ-ment for us. Special thanks are due to Emmanuelle Reny-Nolin who encouraged me to work at the CDA, where I got unforgettable experience and improved my French a lot.

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Notation

Z≥0, Z, R, C Non-negative integers, integers, real numbers and complex numbers.

a, b, c, ... _{Finite-dimensional Lie algebras over C.}

U (a) The universal enveloping algebra of the Lie algebra a. Z(a) The center of U (a).

g, g∗ _{A finite-dimensional semi-simple Lie algebra over C and its dual.} (· | ·) A non-degenerate invariant symmetric bilinear form on g.

e A non-zero nilpotent element of g.

{e, f, h} An s`2-triple such that [e, f ] = h, [h, e] = 2e, [h, f ] = −2f .

gx The centralizer of x in g, i.e., gx = {y ∈ g | [x, y] = 0}. gp The level p truncated current Lie algebra associated to g.

Γ :L

i∈Zg(i) A Z-grading of g.

Γp:L_i∈Zgp(i) A Z-grading of gp.

(· | ·)p A non-degenerate invariant symmetric bilinear form on gp.

Hχp The finite W-algebra associated to (gp, e).

Wk(gp, e) The affine W-algebra associated to (gp, e).

L, L∗ A quasi-finite Z-graded Lie algebra and its restricted dual. Λ∞/2+•L∗ The space of semi-infinite forms on L.

H∞/2+•(L, M ) The semi-infinite cohomology of L with coefficients in M . Zhu(V ) or A0(V ) The Zhu algebra of a vertex algebra V .

An(V ) Higher level Zhu algebras of V .

U (V ) The universal enveloping algebra of V . Tensor products are taken over C except other declaration.

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Introduction

0.1. Given a complex finite-dimensional semi-simple Lie algebra g with a non-degenerate invariant symmetric bilinear form (· | ·), one can construct: (1) the universal enveloping algebra U (g); (2) the coordinate ring C[g∗] of the Poisson variety g∗, where g∗ is the dual of g; (3) the level k vacuum representation Vk(g) of ˜g, where ˜g is the Kac-Moody affinization of g; (4) the coordinate ring of the arc space J g∗ of g∗. Among them, U (g) is an associative algebra, C[g∗] is a Poisson algebra, Vk(g) is a vertex algebra and C[Jg∗] is a Poisson vertex algebra. It is well known that there is a filtration on U (g) whose associated graded gr U (g) ∼= S(g) ∼= C[g∗], so that U (g) can be considered as a quantization of g∗. Similarly, Vk(g) can be considered as a quantization of J g∗. The Zhu algebra functor sends Vk(g) to U (g) [FZ92] and C[Jg∗] to C[g∗] [DSKV16]. Therefore, we have the left side of the following diagram [Ara17,DSK05].

C[J g∗] Zhu Vk(g) gr oo Zhu “H-R" C[J S] Zhu Wk(g, e) gr oo Zhu C[g∗] oo _gr U (g) C[S] oo _gr Wf in(g, e)

Figure 0.1: Hamiltonian reductions

Given a non-zero nilpotent element e ∈ g, one can embed it into an s`2-triple [CM93] and perform

quantum or classical Hamiltonian reductions to get the right side of the above diagram [Ara17,DS14]. The Slodowy slice S obtained through Poisson reduction of g∗ inherits a Poisson structure from g∗. The arc space J S of S can be considered as an infinite-dimensional Poisson variety. The coordinate rings of S and J S are called classical finite and affine W-algebras, respectively. While C[S] is a Pois-son algebra, C[JS] is a PoisPois-son vertex algebra [Ara12]. The quantum finite and affine W-algebras Wf in(g, e) and Wk(g, e) are quantizations of S [GG02,Pre02] and J S [DSK06], respectively. Clas-sical and quantum finite W-algebras were proved to be the Zhu algebras of clasClas-sical and quantum affine W-algebras (see [Ara07] for the principal case, and [DSK06,DSKV16] for the general case). Convention: Whenever we refer to finite or affine W-algebras, we mean the quantum versions. If we want to refer to the classical versions, we will say classical finite and affine W-algebras.

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Finite W-algebras appeared first in B. Kostant’s work [Kos78], where he considered the case of a principal (i.e., regular) nilpotent element in a semi-simple Lie algebra, and proved that the resulting algebra is isomorphic to the center of the universal enveloping algebra. Then his student T. Lynch generalized the construction to even grading nilpotent elements [Lyn79]. It was A. Premet who gave the general definition of finite W-algebras associated to arbitrary nilpotent elements [Pre02].

Affine W-algebras, though more complicated, appeared a bit earlier in [BPZ84], where the authors introduced the affine W-algebra W3 associated to s`3 and its principal nilpotent element. The name

W-algebra, came partially from the notation W3 used there. After that, people started to study their

generalizations [FF90]. It was V. Kac, S. Roan and M. Wakimoto who gave the general definition of affine W-algebras around 2003 [KRW03], where they also realized most of the important supercon-formal algebras through affine W-algebras.

Classical affine W-algebras associated to principal nilpotent elements were introduced by V. Drin-feld and V. Sokolov [DS84] as algebras of local functions on infinite-dimensional Poisson manifolds, where they also constructed an integrable hierarchy of bi-Hamiltonian equations associated to each principal classical affine W-algebra. These constructions of integrable hierarchies were generalized recently to classical affine W-algebras associated to general nilpotent elements in the theory of Poisson vertex algebras [DSKV13].

Explicit generators and their products of classical finite and affine W-algebras were well-studied in [MR15,DSKV16]. However, even the explicit generators of quantum finite and affine W-algebras are not clear except some special cases [Bro11,AM17,DSKV18]. On the representation theory side, there are more interesting results. Skryabin equivalence (appendix of [Pre02]) established an equivalence between the category of Whittaker modules for Lie algebras and the category of modules for finite W-algebras. Finite-dimensional irreducible modules for finite W-algebras were also classified (see [BK06] for type A and [Los11,LO14] for general cases). A highest weight theory of finite W-algebras was also studied [BGK08]. As Zhu algebras of affine W-algebras, the representation theory of finite W-algebras are closely related to that of affine W-algebras [Ara05,Ara07].

In the literature, people usually assume that g is semi-simple or reductive in the constructions of W-algebras. In this thesis, we study W-algebras associated to truncated current Lie algebras.

Given a finite-dimensional semi-simple Lie algebra g, the current algebra associated to g is the Lie algebra g ⊗ C[t] with Lie bracket: [a ⊗ tm, b ⊗ tn] := [a, b] ⊗ tm+nfor a, b ∈ g, m, n ∈ Z. The level p truncated current Lie algebra gpis the quotient

g_{⊗ C[t]} g⊗ tp+1

C[t] with Lie bracket [a ⊗ ti, b ⊗ tj] = [a, b] ⊗ ti+j, where ti+j ≡ 0 when i + j > p.

In the language of jet schemes [Mus01], gpis the p-th jet scheme of g. In the constructions of various

W-algebras, a non-degenerate invariant bilinear form and a good Z-grading (see Definition1.2.1) play essential roles. Given a non-degenerate invariant bilinear form on g, one can construct a series of non-degenerate invariant bilinear forms on gp (see Lemma 2.1.3). Moreover, a good Z-grading of

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g naturally induces a good Z-grading of gp (see Lemma2.2.1). W-algebras associated to truncated

current algebras can be defined in a similar way to the semi-simple case [He].

0.2. The notion of semi-infinite cohomology is the mathematical counterpart of BRST reduction in physics. It was first introduced by B. Feigin [Fei84] around 1984 for Lie algebras. The affine W-algebra Wk(g, e) associated to a principal nilpotent element can be realized as the semi-infinite cohomology of a nilpotent subalgebra of the Kac-Moody affinization of g with coefficients in the vacuum representation Vk(g) [FKW92]. General affine W-algebras, however, differ a bit as there is an extra part in the definition of the cohomology complex [KRW03]. Computing semi-infinite cohomology requires that the Lie algebra in question admits a semi-infinite structure. Motivated by the construction of general affine W-algebras, we develop an adjusted version of semi-infinite cohomology to include the cases where the Lie algebra does not admit a semi-infinite structure but satisfies a mild condition. We also give a characterization of the differential in the (adjusted) semi-infinite cohomology [He17a]. Moreover, we show that general affine W-algebras can also be realized as semi-infinite cohomology with coefficients not in the vacuum module but in a different module. 0.3. The Zhu algebra of a vertex operator algebra was first introduced by Y. C. Zhu [FZ92]. It plays very important roles in the representation theory of vertex algebras. There is a one-to-one corre-spondence between the isomorphism classes of irreducible admissible modules of the vertex operator algebra and the isomorphism classes of irreducible modules of its Zhu algebra [FZ92]. Around 1998, C. Dong, H. Li and G. Mason [DLM98] generalized Zhu algebra to a series of associative algebras which we call higher level Zhu algebras. They proved similar results about the correspondence be-tween representations of higher level Zhu algebras and those of the vertex operator algebra. Frenkel and Zhu observed that the Zhu algebra is isomorphic to a subquotient of the universal enveloping alge-bra [FZ92]. We show that their observation can be generalized to higher level Zhu algebras [He17b]. 0.4. The organization of the thesis is as follows. In Chapter1, we recall some preliminaries on Poisson geometry and good Z-gradings of finite-dimensional Lie algebras. In Chapter2, we define finite W-algebras associated to truncated current Lie W-algebras and show that they are quantizations of Slodowy slices. We also show that Skryabin equivalence and Kostant’s theorem hold in the truncated current setting. In Chapter 3, we develop an adjusted version of semi-infinite cohomology. In Chapter 4, we define affine W-algebras associated to truncated current Lie algebras. In Chapter5, we show that higher level Zhu algebras of a vertex operator algebra are isomorphic to subquotients of its universal enveloping algebra.

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Chapter 1

Preliminaries

1.1 Poisson geometry

Let K be the field of real numbers R or complex numbers C. 1.1.1 Poisson manifold

Definition 1.1.1. A Poisson algebra over K is a commutative associative K-algebra (A, ·) with an additional bilinear binary operation {·, ·}, which is called a Poisson bracket, such that (A, {·, ·}) is a Lie algebra and the two operations {·, ·} and · satisfy Leibniz’s rule:

{a, b · c} = {a, b} · c + b · {a, c} for all a, b, c ∈ A.

Let (A1, ·1, {·, ·}1) and (A2, ·2, {·, ·}2) be two Poisson algebras over K. A Poisson algebra

homo-morphism_{is K-linear map ϕ : A}1 → A2which satisfies

ϕ(a ·1b) = ϕ(a) ·2ϕ(b) and ϕ({a, b}1) = {ϕ(a), ϕ(b)}2for all a, b ∈ A1.

Definition 1.1.2. A real (resp. complex) Poisson manifold is smooth (resp. complex) manifold M , such that the ring of smooth (resp. holomorphic) functions C∞(M ) (resp. O(M )) on M admits a Poisson algebra structure over R (resp. over C).

Let M be a smooth manifold. A smooth bi-vector field P on M is a smooth section of the bundle Λ2T M → M , where T M is the tangent bundle of M and Λ2T M its second exterior power. Here a smooth section means a smooth assignment of an element Pm ∈ Λ2TmM to each point m of M ,

where TmM is the tangent space of M at m. In local coordinates {U, x1, · · · , xn}, P can be expressed

as Px = n X i,j=1 Ki,j(x) ∂ ∂xi ∂ ∂xj for all x ∈ U,

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Let Ω1(M ) be the space of smooth 1-forms on M , which is the collection of smooth sections of the cotangent bundle T∗M → M . Let Vect(M ) be the space of smooth vector fields on M , which is the collection of smooth sections of the tangent bundle T M → M . Given a smooth bi-vector field P , we can define a map from Ω1(M ) to Vect(M ) by the natural pairing between the cotangent space T_x∗M and the tangent space TxM . In local coordinates {U, x1, · · · , xn}, a smooth 1-form and a

smooth vector field can be expressed asPn

i Fi(x)dxi and Pn i Gi(x) ∂ ∂xi respectively, where Fi(x)

and Gi(x) are smooth functions on U . The map induced by the bi-vector field P is given by

Px: n X i Fi(x)dxi 7→ n X i,j=1 Ki,j(x)Fj(x) ∂ ∂xi . (1.1)

There is a natural differential map diff : C∞(M ) → Ω1(M ), which sends a smooth function f to its differentialPn

i=1

∂f ∂xi

dxi. Composing with the map (1.1), a bi-vector field P also defines a map

P : C∞(M ) → Vect(M ), which in local coordinates reads Px : f 7→ n X i,j=1 Ki,j(x) ∂f ∂xj ∂ ∂xi .

The vector field defined by Xf(x) := Px(f ) is called the Hamiltonian vector field associated to f

with respect to P .

Recall that a smooth vector field X on M defines a linear map X : C∞(M ) → C∞(M ), which sends a smooth function g to X(g). Given a smooth bi-vector P , we have the following map

P : C∞(M ) × C∞(M ) → C∞(M ),

(f, g) 7→ {f, g} := Xf(g). (1.2)

Definition 1.1.3. A Poisson structure on a smooth manifold M is a Poisson bi-vector field P , i.e., a smooth bi-vector field P such that the map (1.2) gives C∞(M ) a Poisson algebra structure. A Poisson manifold is a smooth manifold with a Poisson structure.

In the above discussion, when M is a complex manifold, and if we replace P by a holomorphic section in Λ2ΘM, where ΘM is the holomorphic tangent bundle and Λ2ΘM its second exterior power, and

replace C∞(M ) by O(M ), then a complex Poisson manifold is a complex manifold with a Poisson structure, i.e., a holomorphic section in Λ2ΘM which induces a Poisson bracket on O(M ).

Example 1.1.4. Let a be a finite-dimensional Lie algebra over K with basis {xi}n_i=1, and a∗ be the

dual of a. Regarding {xi}ni=1as local coordinates on a∗, define a bi-vector field P on a∗by

Pε= n X i,j=1 ε([xi, xj]) ∂ ∂xi ∂ ∂xj for all ε ∈ a∗. (1.3)

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Then (1.3) defines a Poisson structure on a∗. Passing to the Poisson algebra structure on K[a∗], which we identify with the symmetric algebra S(a) of a, the Poisson bracket is

{f, g}(ε) =X i,j ε([xi, xj]) ∂f ∂xi ∂g ∂xj

for f, g ∈ S(a) and ε ∈ a∗. In particular, when f, g are elements of a, their Poisson bracket is the Lie bracket of a. 1.1.2 Symplectic foliation

A smooth 2-form ω on a smooth manifold M is a smooth assignment of a bilinear form ωm : TmM ×

TmM → R for each point m ∈ M . It is called skew-symmetric if each ωm is skew-symmetric. It is

called non-degenerate if each ωmis non-degenerate, and is called closed if dω = 0.

Definition 1.1.5. A symplectic manifold is a smooth manifold M with a symplectic form, i.e., a closed non-degenerate and skew-symmetric smooth 2-form ω.

Remark 1.1.6. The hypothesis of smooth should be replaced by holomorphic in the complex case.

Let G be an algebraic group with Lie algebra g. Conjugation g : G → G, g · x = gxg−1induces a G-action on the tangent space of the identity, which can be identified with g. This action on g is called the adjoint action. The adjoint G-action on g gives rise to the transposed coadjoint action on g∗. We denote by Ad and Ad∗ the adjoint and coadjoint actions on g and g∗, respectively. The orbits of the coadjoint (resp. adjoint) action of G on g∗(resp. on g) are called coadjoint (resp. adjoint) orbits. Example 1.1.7. Symplectic structure on a coadjoint orbit. Let α ∈ g∗ and O∗α = Ad∗G · α be the

coadjoint orbit through α. One can define a symplectic form ω on O∗α in the following way. For

ξ ∈ O∗α, let Gξ = {g ∈ G | Ad∗g(ξ) = ξ} be the isotropy group of ξ and gξ its Lie algebra. The

tangent space TξO∗αcan be identified with Tξ(G/Gξ) ∼= g/gξ∼= ad∗g(ξ). Let ωξ: TξO∗α×TξO∗α → C

be the skew-symmetric bilinear form defined by

ωξ(ad∗x(ξ), ad∗y(ξ)) := ξ([x, y]).

As the kernel of the bilinear form ξ([·, ·]) : g × g → C is exactly gξ, the bilinear form ωξ is

non-degenerate on TξO∗α. Moreover, the 2-form ω defined by the assignment ξ 7→ wξis closed, hence it

gives a symplectic structure on O∗α. Different proofs for the closure of ω are given in [Kir04],

Every Poisson manifold M has a symplectic foliation in the sense that: (1) M = tαSαdecomposes as

a disjoint union of submanifolds, which are called symplectic leaves; (2) the Poisson structure on M restricts to a symplectic structure on each Sα. If x ∈ Sα, then Sαis called the symplectic leaf through

x. It consists the points of M which can be connected to x by piecewise Hamiltonian paths, where a Hamiltonian path is an integral curve of the Hamiltonian vector field Xf associated to a smooth

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Example 1.1.8. The decomposition g∗ = tαO∗αas coadjoint orbits is the symplectic foliation of g∗.

Given a non-degenerate bilinear form (· | ·) on g, one can identify g with g∗ through (· | ·) and hence equip g itself with a Poisson structure. When the bilinear form is invariant, i.e., ([x, y] | z) = (x | [y, z]) for all x, y, z ∈ g, the symplectic foliation of g is given by the adjoint orbits. Let O be an adjoint orbit and x ∈ O. Then the tangent space TxO of O at x can be identified with [g, x], and the symplectic form on TxO becomes

ωx([a, x], [b, x]) = (x | [a, b]) for a, b ∈ g. (1.4)

Theorem 1.1.9 ([Vai94]). Let M be a Poisson manifold with the symplectic foliation given by tαSα.

LetN be a submanifold of M such that for all α,

(1) N is transversal to Sα, i.e.,TnN + TnSα= TnM for all n ∈ N ∩ Sα.

(2) For alln ∈ N ∩ Sα, the subspace TnN ∩ TnSα is a symplectic subspace ofTnSα, i.e., the

symplectic form onTnSαis non-degenerate when restricted toTnN ∩ TnSα.

Then there is an induced Poisson structure onN . The symplectic foliation of N is given by tα(N ∩Sα)

and the symplectic form onTn(N ∩ Sα) for all n ∈ N ∩ Sα is the restriction of the symplectic form

onTnSα.

1.1.3 Poisson reduction

Poisson reduction is a procedure of taking a subquotient of a Poisson algebra or of a Poisson manifold, such that the resulting object has a Poisson algebra or Poisson manifold structure. It allows us to construct new Poisson algebras or Poisson manifolds from old ones.

Definition 1.1.10. Let (A, ·, {·, ·}A) be a Poisson algebra, and I an ideal of (A, ·). Let B be a

sub-algebra of A/I. We say that the triple (A, I, B) is Poisson reducible if there is a Poisson sub-algebra structure {·, ·}Bon B, such that

{a, b}B={˜a, ˜b}_A for all a, b ∈ B,

where ˜a, ˜b ∈ A are arbitrary representatives of a, b in A, and ¯x is the image of x ∈ A in A/I. The Poisson bracket {·, ·}Bon B is called the reduced Poisson bracket.

We have the following diagram for a Poisson reducible triple, A

π

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Let (A, ·, {·, ·}A) be a Poisson algebra, and I an ideal of (A, ·). We denote by

N (I) := {a ∈ A | {a, b}A∈ I for all b ∈ I},

and call it the normalizer of I. One can show that N (I) is a Poisson subalgebra of A, i.e., N (I) is closed under both the product · and the bracket {·, ·}A. Denote by π the canonical projection

π : A A/I. Then (A, I, B) is a Poisson reducible triple if and only if π−1(B) is a Poisson subalgebra of N (I), i.e., π−1(B) ⊆ N (I) and is closed under {·, ·}A.

Remark 1.1.11. There is also a Poisson manifold version of Poisson reduction [Vai94].

1.1.4 Quantization

Definition 1.1.12. An associative algebra B is called Z-filtered if there is a filtration of subspaces {FiB}i∈Z, with FiB ⊆ Fi+1B and B =S_i∈ZFiB, such that FiB ·FjB ⊆ Fi+jB for all i, j ∈ Z. An

associative algebra B is called Z-graded if there is a Z-grading B =L

i∈ZBisuch that Bi·Bj ⊆ Bi+j

for all i, j ∈ Z.

Given a Z-filtered algebra B with filtration {FiB}i∈Z, one can associate it a Z-graded algebra grFB

by setting (grFB)i=

FiB

Fi−1B

and with multiplication defined by

(a + Fi−1B) · (b + Fj−1B) = a · b + Fi+j−1B for a ∈ FiB, b ∈ FjB.

The Z-filtered algebra B is called almost commutative if the associated graded algebra grFB is

com-mutative, i.e.,

a · b − b · a ∈ Fi+j−1B for all a ∈ FiB, b ∈ FjB.

Lemma 1.1.13. Let B be an almost commutative associative algebra with a Z-filtration {FiB}i∈Z.

Thengr_FB is a Poisson algebra with the Poisson bracket defined by

{a + Fi−1B, b + Fj−1B} := a · b − b · a + Fi+j−2B for a ∈ FiB, b ∈ FjB.

Proof. Since gr_FB is already commutative, we only need to prove that (gr_FB, {·, ·}) is a Lie algebra and it satisfies Leibniz’s rule. For the well-definedness, let a0 = a + s, b0 = b + t with s ∈ Fi−1B and

t ∈ Fj−1B be other representatives of a and b, respectively, in grFB. Then we have

a0· b0− b0· a0 = a · b − b · a + (s · b − b · s + a · t − t · a + s · t − t · s) ≡ a · b − b · a mod Fi+j−2B.

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Once {·, ·} is well-defined, it gives grFB a Lie algebra structure since the multiplication · is

associa-tive. For Leibniz’s rule, assume that c ∈ FkB. Then b · c ∈ Fj+kB, and

{a + Fi−1B, b · c + Fj+k−1B}

= a · (b · c) − (b · c) · a + Fi+j+k−2B

= (a · b) · c − (b · a) · c + (b · a) · c − (b · c) · a + Fi+j+k−2B

= {a + Fi−1B, b + Fj−1B} · c + b · {a + Fi−1B, c + Fk−1B} + Fi+j+k−2B.

Definition 1.1.14. Let A be a Poisson algebra and B a Z-filtered associative algebra with a Z-filtration {F_iB}_i∈Z. If there is an isomorphism between grFB and A as Poisson algebras, then we say that B

is a quantization of A.

Remark 1.1.15. When M is a Poisson manifold, a quantization of C∞(M ) or O(M ) is also called a quantization ofM .

Example 1.1.16. The PBW filtration on U (a) is given by U (a)n= span_C{x1· · · xm| m ≤ n, xi ∈

a}. It is well-known that its associated graded algebra gr U (a) ∼= S(a) ∼= C[a∗], where S(a) is the symmetric algebra of a and C[a∗] is the coordinate ring of the Poisson variety a∗. Indeed, the isomorphism is a Poisson algebra isomorphism. Therefore, the universal enveloping algebra U (a) is a quantization of the Poisson variety a∗.

Example 1.1.17. Let a =L

i∈Za(i) be a Z-grading of a, i.e., [a(i), a(j)] ⊆ a(i + j) for all i, j ∈ Z.

For x ∈ a(i), define its degree to be deg x = 2 + i. The Kazhdan filtration on U (a) associated to the Z-grading of a is given by setting KnU (a) = span_C{x1· · · xm | P_ideg xi ≤ n}. Assume

that x ∈ a(i), y ∈ a(j). Then deg x = 2 + i, deg y = 2 + j and deg [x, y] = 2 + i + j as [x, y] ∈ a(i + j). By an induction on the number of factors in a monomial, one can show that if u ∈ K2+iU (a), v ∈ K2+jU (a), then [u, v] ∈ K2+i+jU (a), i.e., U (a) is almost commutative with

respect to the Kazhdan filtration. The Poisson algebra isomorphism gr U (a) ∼= S(a) preserves the Z-gradation, while the Z-grading on gr U (a) comes from the Kazhdan filtration, and that on S(a) from by setting the degree of x ∈ a(i) to be 2 + i.

Remark 1.1.18. Note that the PBW filtration on U (a) is a Z≥0-filtration, while the Kazhdan filtration

is a Z-filtration.

1.2 _{Good Z-grading of finite-dimensional Lie algebras}

Let a be a finite-dimensional Lie algebra over C. A Z-grading of a is a Z-gradation a =L

i∈Za(i),

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Definition 1.2.1. Let Γ : a = L

i∈Za(i) be a Z-grading of a. An element e ∈ a(2) is called a good

elementwith respect to Γ if

ad e : a(i) → a(i + 2) is injective for i ≤ −1 and surjective for i ≥ −1. A Z-grading of a is called good if it admits a good element.

Given a good Z-grading Γ and a good element e, the following properties are immediate: (1) the element e is nilpotent;

(2) the centralizer aeof e in a lies inL

i≥0a(i);

(3) ad e : a(−1) → a(1) is bijective.

A standard s`2-triplein a Lie algebra a is a triple {e, f, h} ⊆ a with [e, f ] = h, [h, e] = 2e and

[h, f ] = −2f . The subalgebra spanned by a standard s`2-triple {e, f, h} is isomorphic to s`2.

Many important examples of good Z-gradings of a finite-dimensional Lie algebra a come from a standard s`2-triple {e, h, f }. It follows from the representation theory of s`2 that the eigenspace

decomposition of a with respect to ad h is a good Z-grading of a with a good element e. Good Z-gradings thus obtained are called Dynkin Z-gradings.

Theorem 1.2.2 (Jacobson-Morozov). Let g be a finite-dimensional semi-simple Lie algebra over C and e ∈ g be a non-zero nilpotent element. Then e can be embedded into a standard s`2-triple

{e, f, h} of g. If h0 _{∈ [e, g] satisfies that [h}0_{, e] = 2e, then {e, h}0_{} can be embedded into a standard}

s`2-triple{e, f0, h0} of g.

The proof of Theorem1.2.2is not constructive but by an induction on the dimension of g. Lemma 1.2.3. Let Γ : g = L

i∈Zg(i) be a Z-grading of a complex semi-simple Lie algebra g and

e ∈ g(2). Then there exists h ∈ g(0) and f ∈ g(−2), such that {e, h, f } form a standard s`2-triple.

Proof. By Theorem 1.2.2, we can embed e in an s`2-triple, say {e, h, f }. Write h = P_i∈Zhi and

f =P

i∈Zfiwith hi, fi ∈ g(i). Then [hi, e] = δi,02e as [hi, e] ∈ g(i + 2) and [h, e] = 2e. We also

have [e, fi] = hi+2as [e, f ] = h. In particular, we have [e, f−2] = h0. Therefore, by Theorem1.2.2,

there exists f0, such that {e, h0, f0} form a standard s`2-triple. Write f0 =P_i∈Zf_i0 with f_i0 ∈ g(i),

then {e, h0, f−20 } is a standard s`2-triple that we are looking for.

Definition 1.2.4. Given an associative algebra A (resp. a Lie algebra L), a linear map D : A → A (resp. D : L → L) is called a derivation of A (resp. of L) if D(ab) = D(a)b + aD(b) (resp. D([a, b]) = [D(a), b] + [a, D(b)]) for all a, b ∈ A (resp. ∈ L). The derivations are denoted by Der A or Der L.

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For a Lie algebra g, we have a notion of inner derivation. Let a ∈ g, and ad a : g → g be the map defined by ad a(x) = [a, x]. Then the Jacobi identity in the definition of a Lie algebra implies that ad a is a derivation of g. Such derivations are called inner derivations. We denote by Inn g the collection of inner derivations of g. It is well-known that Der g has a Lie algebra structure and Inn g is an ideal of Der g. When g is semi-simple, we have Inn g = Der g.

Lemma 1.2.5. Let g be a semi-simple Lie algebra and Γ : g =L

i∈Zg(i) be a Z-grading of g. Then

there exists an elementhΓ∈ g, such that [hΓ, x] = ix for all x ∈ g(i).

Proof. It is clear that the linear operator δ : g → g defined by δ(x) = ix for x ∈ g(i) is a derivation of g. Since all derivations of a semi-simple Lie algebra are inner, there exists an element hΓ∈ g such

that [hΓ, x] = δ(x) = ix for x ∈ g(i).

Remark 1.2.6. A complete classification of good Z-gradings of finite-dimensional simple Lie algebras over C was given in [EK05].

1.3 Vector superspace

A vector superspace is a Z2-graded vector space V = V¯₀ ⊕ V¯₁. An element a ∈ V is said to be

homogeneous if a ∈ V¯_i for some ¯i ∈ Z2, and p(a) = ¯i is called the parity or degree of a. Given a

vector superspace V , its endomorphism space End V is naturally Z2-graded by setting

(End V )¯_j := {f ∈ End V | f (V¯_i) ⊆ V¯_i+¯_j for ¯i ∈ Z2}.

Elements of (End V )¯0 are called even endomorphisms and those of (End V )¯1 odd endomorphisms.

A homomorphism f between two vector superspaces V and W is said to be parity-preserving if f (V¯_i) ⊆ W¯_ifor ¯i ∈ Z2.

Degree Convention: Whenever we use the notation p(a), we assume that a is homogeneous.

Definition 1.3.1. Let V = V¯₀⊕ V¯₁be a vector superspace. A bilinear form (· | ·) : V × V → C is

called supersymmetric if (V¯₀| V¯₁) = (V¯₁| V¯₀) = 0 and it is symmetric on V¯₀and skew-symmetric on

V¯1. It is called skew-supersymmetric if (V¯0| V¯1) = (V¯1| V¯0) = 0 and it is skew-symmetric on A¯0and

symmetric on A¯₁.

A product in a superspace V is called supercommutative if a · b = (−1)p(a)p(b)b · a and is called anti-supercommutative if a · b = −(−1)p(a)p(b)b · a for all a, b ∈ V .

Sign Convention: For a superspace V , when we need to change the positions of two adjacent elements a, b in a product, we usually need to add a sign ±(−1)p(a)p(b).

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Chapter 2

Finite W-algebras associated to truncated

current Lie algebras

In this chapter, we define finite W-algebras associated to truncated current Lie algebras and study some of their properties.

2.1 Truncated current Lie algebras

Given a finite-dimensional Lie algebra a, the current algebra associated to a is the Lie algebra a ⊗ C[t] with Lie bracket defined by [a ⊗ tm, b ⊗ tn] := [a, b] ⊗ tm+nfor a, b ∈ a, m, n ∈ Z≥0. One can show

that the subspace a ⊗ tpC[t] is an ideal of a ⊗ C[t] for any nonnegative integer p.

Definition 2.1.1. The level p truncated current Lie algebra associated to a is the quotient Lie algebra a_p:= a⊗ C[t]

a⊗ tp+1_C[t] ∼= a ⊗

C[t] tp+1_C[t].

The Lie bracket of apis

[a ⊗ ti, b ⊗ tj] = [a, b] ⊗ ti+j, where ti+j ≡ 0 when i + j > p.

Remark 2.1.2. In the language of jet schemes [Mus01], ap is the p-th jet scheme of a. Truncated

current Lie algebras are also called generalized Takiff algebras or polynomial Lie algebras.

For convenience, we write xti for x ⊗ ti. An element of ap can be uniquely expressed as a sum

Pp

i=0xiti with xi ∈ a. When q ≥ p, the canonical surjective map πq,p : aq apsending a ⊗ tk to

zero for k ≥ p + 1 is a Lie algebra homomorphism. For a subspace b ⊆ a, we let bp = b ⊗ C[t]

tp+1_C[t],

which is a subspace of ap. If b is a subalgebra of a, then bp is a subalgebra of ap. For a nonnegative

integer k ≤ p, we denote by a(k) = a ⊗ tk. By a(≥1) we mean L

k≥1a(k). Then a(0) ∼= a is a

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Let (· | ·) be a symmetric bilinear form on a. Let ¯c := (c0, · · · , cp) with ci ∈ C. Define a symmetric

bilinear form on apby the formula

(x | y)p:= p X k=0 ck X i+j=k (xi| yj), (2.1) where x =Pp i=0xiti and y = Pp i=0yitiwith xi, yi ∈ a.

Lemma 2.1.3 ([Cas11]). Assume that (· | ·) is non-degenerate and invariant on a. Then the bilinear form(· | ·)pdefined by (2.1) is invariant and symmetric. It is non-degenerate if and only ifcp 6= 0.

Proof. Let x =P

ixiti, y =

P

iyitiand z =

P

izitiwith xi, yi, zi ∈ a. For the invariance, we have

If cp = 0, it is clear that a(p) lies in the kernel of the form (· | ·)p, so it is degenerate. When cp 6= 0,

assume that a =P

i≥i0ait

i_{, with a}

i0 6= 0. By the non-degenerancy of (· | ·), there exists an element

b ∈ a, such that (ai0 | b) 6= 0. Then (a | bt

p−i0₎ p= cp(ai0 | b) 6= 0, i.e., (· | ·)pis non-degenerate. Lemma 2.1.4. Der C[t] htp+1_i ∼= tC[t] htp+1_i d dt.

Proof. _{Given a polynomial f (t) ∈ tC[t]/ht}p+1i, setting g(t) 7→ f (t)d

dtg(t) defines a derivation of C[t]/htp+1i. Conversely, let D be a derivation of C[t]/htp+1i. As C[t]/htp+1i is generated by {1, t} and D(1) = 0, D is determined by D(t). Assume that D(t) = g(t) for some g(t) ∈ C[t]/htp+1i. Then Leibniz’s rule implies that D(tk) = ktk−1g(t), i.e., D = g(t)d

dt. But (p + 1)t

p_{g(t) =}

D(tp+1) = 0 implies that g(0) = 0, so g(t) ∈ tC[t]/htp+1i and D ∈ tC[t]/htp+1_id

dt. Let M be a g-module. A derivation from g to M is a linear map f : g → M satisfying

f ([a, b]) = a · f (b) − b · f (a) for all a, b ∈ g.

The derivations from g to M is denoted by Der(g, M ). Given an element m ∈ M , define ad m(x) = x · m for all x ∈ g. Then the Lie algebra action of g on M implies that ad m ∈ Der(g, M ). Such derivations are called inner derivations and are denoted by Inn(g, M ). We have Der g = Der(g, g) and Inn g = Inn(g, g), where g is considered as the adjoint module of g.

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In the language of Lie algebra cohomology (see Section3.1), a derivation from g to M is a 1-cocycle with coefficients in M and an inner derivation from g to M is a 1-coboundary with coefficients in M , so H1(g, M ) = Der(g, M )/Inn(g, M ).

Lemma 2.1.5 (Whitehead). Let g be dimensional semi-simple Lie algebra and M a finite-dimensional non-trivial simple g-module. ThenHi(g, M ) = 0 for all i > 0, in particular, we have Der(g, M ) = Inn(g, M ).

Let ϕ ∈ Homg(g, g) and d ∈ Der C[t]

htp+1_i. Consider the map D = ϕ ⊗ d : gp → gpdefined by sending

a ⊗ f (t) to ϕ(a) ⊗ df (t). We have

D([a ⊗ f (t), b ⊗ g(t)]) = D([a, b] ⊗ f (t)g(t)) = ϕ([a, b]) ⊗ d(f (t)g(t)).

Since ϕ ∈ Homg(g, g), we have ϕ([a, b]) = [a, ϕ(b)] = −ϕ([b, a]) = −[b, ϕ(a)] = [ϕ(a), b]. Since

d ∈ Der C[t]

htp+1_i, we have d(f (t)g(t)) = d(f (t))g(t) + f (t)d(g(t)). Therefore, we have

D([a ⊗ f (t), b ⊗ g(t)]) = ϕ([a, b]) ⊗ (d(f (t))g(t) + f (t)d(g(t)))

= [ϕ(a), b] ⊗ d(f (t))g(t) + [a, ϕ(b)] ⊗ f (t)d(g(t)) = [D(a ⊗ f (t)), b ⊗ g(t)] + [a ⊗ f (t), D(b ⊗ g(t))], i.e., ϕ ⊗ d ∈ Der gp.

Proposition 2.1.6. Let g be a finite-dimensional semi-simple Lie algebra. Then Der gp ∼= Homg(g, g) ⊗ Der C[t] htp+1_i n Inn gp.

Proof. Given ϕ ∈ Homg(g, g) and d ∈ Der_htC[t]_p+1_i, we have (ϕ ⊗ d)(g(0)) = 0, so every element of

Homg(g, g) ⊗ Der_htC[t]_p+1_i kills g(0). But we have ad x(g(0)) 6= 0 for all x ∈ gpwhich is non-zero, so

Inn gp∩

Homg(g, g) ⊗ Der_htC[t]_p+1_i

= 0. We know that Inn gpis an ideal of Der gp, so we only need to prove that

Der gp= Homg(g, g) ⊗ Der C[t]

htp+1_i + Inn gp.

For 0 ≤ i ≤ p, let πibe the projection of gpto the subspace g(i), i.e., πi(Pp_k=0xktk) = xiti.

Note that gp is generated by g(0) ⊕ g(1), so a derivation D ∈ Der gp is determined by its value on

g(0)⊕ g(1)_{. Let D}

i = πi◦ D. Then we have D =Ppi=0Di. Composing πi with Leibniz’s rule, we

get

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That means, when restricted to g(0), Di ∈ Der(g(0), g(i)). Since g(0) ∼= g is semi-simple, we have

Der(g(0), g(i)) = Inn(g(0), g(i)) by Lemma2.1.5. Therefore, there exists xi⊗ ti ∈ g ⊗ ti for each

0 ≤ i ≤ p, such that ad (xi⊗ ti) = Diwhen restricted to g(0). Let D0= D −Pp_i=0ad (xi⊗ ti). Then

D0|_g(0) = 0. Let D0_i = πi ◦ D0. Applying D0 to [a ⊗ 1, b ⊗ t] and composing with πi, by Leibniz’s

rule, we get

D_i0([a ⊗ 1, b ⊗ t]) = [a ⊗ 1, D_i0(b ⊗ t)]. (2.2) When restricted to g(1), (2.2) implies that D0_i: g(1)→ g(i)_{is a g}(0)_{-module homomorphism. As g}(1) ∼₌ g(i) ∼= g as g-modules, there exist g-module homomorphisms ϕi : g → g such that D0i = ϕi ⊗ ti−1

when restricted to g(1). Note that for i ≥ 1, D0_i = ϕi⊗ ti

d dt ∈ Homg(g, g) ⊗ Der C[t] htp+1_i, when D 0 i is restricted to g(1). Let D00= D0−Pp i≥1ϕi⊗ ti d dt. Then D 00_| g(0) = 0 and D00(g(1)) ⊆ g(0). We show

that D00= 0. Note that we have D00= D₀0 = ϕ0⊗ t−1when restricted to g(1), where ϕ0 : g(1)→ g(0)

is a g(0)-module homomorphism. By Leibniz’s rule, we have

D00([a ⊗ t, b ⊗ t]) = [D00(a ⊗ t), b ⊗ t] + [a ⊗ t, D00(b ⊗ t)] = [ϕ0(a), b] ⊗ t + [a, ϕ0(b)] ⊗ t

= ϕ0[a, b] ⊗ 2t.

Since [g, g] = g, we have D00(a ⊗ t2) = ϕ0(a) ⊗ 2t for all a ∈ g. Inductively, we have D00(a ⊗ tk) =

ϕ0(a) ⊗ ktk−1. In particular, D00(a ⊗ tp+1) = ϕ0(a) ⊗ ptp for all a ∈ g. Since a ⊗ tp+1= 0 in gp,

we have ϕ0(a) = 0 for all a ∈ g, i.e., D00= 0, and

D = p X i=1 ad (xi⊗ ti) + p X i≥1 ϕi⊗ ti d dt ∈ Homg(g, g) ⊗ Der C[t] htp+1_i + Inn gp.

2.2 Finite W-algebras via Whittaker model definition

Let g be a finite-dimensional semi-simple Lie algebra over C with a non-degenerate invariant symmet-ric bilinear form (· | ·). By Lemma2.1.3, there exists a non-degenerate invariant symmetric bilinear form (· | ·)pon gp, which we fix from now on.

Let Γ : gad hΓ

== L

i∈Zg(i) be a good Z-grading of g with a good element e ∈ g(2), and {e, f, h} an

s`2-triple containing e with h ∈ g(0) and f ∈ g(−2). Let gp(i) := {x ∈ gp | [hΓ, x] = ix }. Then

Γp: gp=L_i∈Zgp(i) is a Z-grading of gp.

Lemma 2.2.1. The Z-grading Γp of gpis good with good elemente.

Proof. Note that gp(i) = g(i)p. For the map ad e : gp(i) → gp(i + 2), we have ker ad e = (g(i)e)p

and im ad e = ([g(i), e])p, so it is injective for i ≤ −1 and surjective for i ≥ −1 as e is a good

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Remark 2.2.2. We call Γp a good Z-grading of gpinduced from a good Z-grading of g.

Example 2.2.3. In this example, we show that not every good Z-grading of gpis induced from a good

Z-grading of g as in Lemma 2.2.1. Let g = s`2 with canonical basis {e, f, h} such that [e, f ] =

h, [h, e] = 2e, [h, f ] = −2f . Consider g2, which has a basis {e, f, h, e ⊗ t, f ⊗ t, h ⊗ t}. Let

x = h + 2e ⊗ t + 2f ⊗ t. Then with respect to ad x, we have the Z-grading on g2

g2 = g2(−2) ⊕ g2(0) ⊕ g2(2) (2.3)

with g2(−2) = span_C{f ⊗ t, f − h ⊗ t}, g2(0) = span_C{h ⊗ t, h + 2e ⊗ t + 2f ⊗ t}, and g2(2) =

span_C{e ⊗ t, e − h ⊗ t}. It is easy to check that e − h ⊗ t is a good element with respect to (2.3). Moreover, Jacobson-Morozov’s lemma does not work in truncated current Lie algebras. Indeed, when p ≥ 1, x ⊗ t is nilpotent in gp for any x ∈ g and it cannot be embedded into any s`2-triple.

Lemma 2.2.4. Let Γp : L_i∈Zgp(i) be a Z-grading of gp induced from a good Z-grading of g. We

have(gp(i) | gp(j))p = 0 if i + j 6= 0.

Proof. Let hΓbe the semi-simple element defining Γp. Let x ∈ gp(i), y ∈ gp(j) and i + j 6= 0. Then

([hΓ, x] | y)p= −(x | [hΓ, y])p, i.e., (i+j)(x | y)p= 0. Since i+j 6= 0, that implies (x | y)p = 0.

Let χp = (e | ·)p ∈ g∗p. Define a skew-symmetric bilinear form on gp(−1) by

h·, ·ip :gp(−1) × gp(−1) → C, (x, y) 7→ hx, yip := χp([x, y]). (2.4)

Lemma 2.2.5. The bilinear form on gp(−1) defined by (2.4) is non-degenerate.

Proof. This follows from the surjectivity of ad e : gp(−1) → gp(1), the invariance of the bilinear

form (· | ·)pand the pairing property (gp(i) | gp(j))p= 0 if i + j 6= 0.

Let lpbe an isotropic subspace of gp(−1) with respect to the bilinear form (2.4), i.e., (e | [lp, lp])p = 0.

Let l⊥_p := {x ∈ gp(−1) | (e | [x, y])p = 0 for all y ∈ lp}, and let

m_p := M i≤−2 g_p(i), m_l,p:= mp⊕ lp, nl,p:= mp⊕ l⊥p, np := M i≤−1 g_p(i). (2.5) Obviously, mp ⊆ ml,p⊆ nl,p⊆ np are all nilpotent subalgebras of gp.

One can easily show that (e | [ml,p, nl,p])p = 0, thanks to the property (e | gp(i))p = 0 for i ≤ −3

and the definition of lp and l⊥p. In particular, χp = (e | ·)p is a character of ml,p hence defines a

one-dimensional representation of ml,p, which we denote by Cχp. Let

Qχp:= U (gp) ⊗U (ml,p)Cχp ∼= U (gp)/Iχp,

where Iχp is the left ideal of U (gp) generated by {a − χp(a) | a ∈ ml,p}. We denote by ¯u := u + Iχp

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Lemma 2.2.6. The adjoint action of nl,ponU (gp) leaves the subspace Iχpinvariant.

Proof. Let x ∈ nl,pand y =Piui(ai− χp(ai)) ∈ Iχp, with ui ∈ U (gp) and ai ∈ ml,p. Then

[x, y] =X i [x, ui(ai− χp(ai))] =X i ([x, ui](ai− χp(ai)) + ui[x, ai− χp(ai)]) .

Since χp([nl,p, ml,p]) = 0, we have [x, ai− χp(ai)] = [x, ai] ∈ Iχp, hence [x, y] ∈ Iχp.

Since ad nl,ppreserves Iχp, it induces a well-defined adjoint action on Qχp, such that

[x, ¯u] = [x, u] for x ∈ nl,p, u ∈ U (gp).

Let

Hχp := Q

ad nl,p

χp = {¯u ∈ Qχp| [x, u] ∈ Iχpfor all x ∈ nl,p}.

Lemma 2.2.7. There is a well-defined multiplication on Hχp by

¯

u · ¯v := uv for ¯u, ¯v ∈ Hχp.

Proof. First, we show that the multiplication ¯u · ¯v does not depend on the representatives. It is obvious that it does not depend on the representatives of v. For that of u, we need to show that yv ∈ Iχpfor

all y ∈ Iχp, ¯v ∈ Hχp. Assume that y =

P iui(ai− χp(ai)) with ai ∈ ml,p, then yv = [y, v] + vy =X i ui[ai− χp(ai), v] + X i [ui, v](ai− χp(ai)) + vy. (2.6)

By the definition of Hχp, we have [ai + χp(ai), v] = [ai, v] ∈ Iχp since ai ∈ ml,p ⊆ nl,p, hence

yv ∈ Iχp.

Next we show that Hχp is closed under the multiplication. Let ¯u1, ¯u2 ∈ Hχp, we need show that

u1u2 ∈ Hχp, i.e., [x, u1u2] ∈ Iχp for all x ∈ nl,p. By Leibniz’s rule, we have

[x, u1u2] = [x, u1]u2+ u1[x, u2].

By the definition of Hχp, we have [x, u1], [x, u2] ∈ Iχp. Therefore, [x, u1]u2 ∈ Iχp by (2.6).

Once the multiplication is well-defined, Hχpinherits an associative algebra structure from U (gp).

Definition 2.2.8. The finite W-algebra Wf in(gp, e) associated to the pair (gp, e) is defined to be Hχp.

Remark 2.2.9. When p = 0, we get the definition of the finite W-algebra associated to the semi-simple Lie algebra g and the nilpotent elemente given by A. Premet in [Pre02].

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When lpis a Lagrangian subspace, i.e., lp = l⊥p hence ml,p= nl,p, we can realize Hχpas the opposite

endomorphism algebra (EndU (gp)Qχp)

op_{in the following way. As Q}

χp = U (gp)/Iχpis a cyclic gp

-module, an endomorphism ϕ is determined by its value on the generator ¯1. Since ¯1 is killed by Iχp,

ϕ(¯1) must be annihilated by Iχp. On the other hand, given an element ¯y ∈ Qχp, which is killed by

Iχp, ¯1 7→ ¯y defines an endomorphism of Qχp. We thus have

(EndU (gp)Qχp)

op∼_{= {¯}_{y ∈ Q}

χp| (a − χp(a))y ∈ Iχp for all a ∈ ml,p}

= {¯y ∈ Qχp| [a, y] ∈ Iχp for all a ∈ nl,p}

= Hχp.

Remark 2.2.10. When p = 0, it was proved that the finite W-algebras Hχ0 with respect to different

good gradingsΓ0[BG07] and different isotropic subspaces l0[GG02] are all isomorphic. Forp ≥ 1,

we will show the independence of isotropic subspace lpin the sequel following [GG02].

Remark 2.2.11. As in the semi-simple case [BGK08], there are other definitions of finite W-algebras in the truncated current setting.

2.3 Quantization of Slodowy slices

We keep the notation of Section2.1and Section2.2.

2.3.1 Poisson structure on Slodowy slices

The non-degenerate invariant symmetric bilinear form (· | ·)pon gp defines a bijection κp : gp → g∗p

through x 7→ (x | ·)p. Let gfp be the centralizer of f in gp. Set

S_e_p := e + gf_p and S_χ_p := χp+ ker ad∗f = κp(Sep).

When p = 0, Se:= Se0 is called the Slodowy slice through e [Slo80]. In the language of jet schemes

[Mus01], Sepis the p-th jet scheme of Se. We also call Septhe Slodowy slice through e in gpand Sχp

the Slodowy slice through χpin g∗p.

By the representation theory of s`2, we have gp = gep ⊕ [gp, f ] = gfp ⊕ [gp, e], which implies that

ad e : [f, gp] 1:1

−−→ [e, gp] and ad f : [e, gp] 1:1

−−→ [f, gp] are both bijective.

Lemma 2.3.1. Let r ∈L

i≤1gp(i). Then

(a) [e + r, [f, gp]] ∩ gfp = 0.

(b) The mapad (e + r) : [f, gp] → [e + r, [f, gp]] is bijective.

(c) Ifa ∈ gpis such that[e + r, a] ∈ gfp and(a | [e + r, gp] ∩ gfp)p= 0, then [e + r, a] = 0.

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Proof. Let a = P

iai with ai ∈ gp(i) such that [f, a] 6= 0. Let i0 be such that [f, ai0] 6= 0 but

[f, ai] = 0 for all i > i0. Then the i0-th component (which belongs to gp(i0)) of [e + r, [f, a]] is

[e, [f, ai0]] as r ∈

L

i≤1gp(i) and e ∈ gp(2). Since [f, ai0] 6= 0 and ad e : [f, gp] → [e, gp] is

bijective, we have [e, [f, ai0]] 6= 0.

(a) Assume a ∈ gp satisfies that 0 6= [e + r, [f, a]] ∈ gfp. Then [f, a] 6= 0. Let i0 be as above,

then 0 6= [e, [f, ai0]] ∈ g

f

p(i0) i.e., [f, [e, [f, ai0]]] = 0. This contradicts to the bijectivity of

ad f : [e, gp] → [f, gp].

(b) We just need to show that ad (e + r) is injective on [f, gp]. Suppose that [e + r, [f, a]] = 0 with

[f, a] 6= 0. Let i0be as above. Then its i0-th component [e, [f, ai0]] 6= 0, a contradiction.

(c) For a subspace V of gp, we denote by V⊥ its orthogonal complement with respect to (· | ·)p.

Then ([e + r, gp] ∩ gfp)⊥= [e + r, gp]⊥+ (gfp)⊥. Note that (gfp)⊥= [f, gp] and [e + r, gp]⊥=

ker ad (e + r) as (· | ·)pis non-degenerate and invariant. Therefore, (c) is equivalent to saying

that if a = u+v with u ∈ (gfp)⊥= [f, gp], v ∈ [e+r, gp]⊥and [e+r, a] ∈ gfp, then [e+r, a] = 0.

Since u ∈ [f, gp] and v ∈ ker ad (e + r), we have [e + r, a] = [e + r, u] ∈ gfp∩ [e + r, [f, gp]],

which must be zero by (a).

(d) It is enough to prove [e + r, [f, gp]] ⊕ gfp = gp. It is a direct sum because of (a). Let us count

dimensions. We have dim[e + r, [f, gp]] = dim[f, gp] by (b). Note that dim gfp = dim gep and

dim[f, gp] = dim gp− dim gepas we have gp = [gp, f ] ⊕ gep, so dim gp= dim gfp+ dim[f, gp],

and (d) is proved.

Remark 2.3.2. Lemma2.3.1was proved in [DSKV16] forr ∈L

i≤0g(i) and g semi-simple, where

Γ : g =L

i∈Zg(i) is a good Z-grading of g with a good element e ∈ g(2). We have used the same

argument to prove the truncated current version above.

Combining Theorem1.1.9and Lemma2.3.1, we have the following lemma. Lemma 2.3.3. The slice Sep has a Poisson structure.

Proof. We show that the two conditions in Theorem 1.1.9are satisfied for the submanifold Sep of

gp. Let x = e + r ∈ Sep∩ Ox, where Oxis the adjoint orbit of gp through x. As r ∈

L

i≤0gp(i),

Lemma2.3.1applies. Note that TxSep = g

f

p and TxOx= [gp, x]. Part (d) of Lemma2.3.1shows that

Sep is transversal to Oxat x. Next we show that the restriction of the symplectic form ωx defined by

(1.4) on the subspace TxOx∩ TxSep = [gp, x] ∩ g

f

p is non-degenerate. Assume that there exists an

element [a, x] ∈ [gp, x] ∩ gfp such that [a, x] ∈ ker ωx|_[g_p_,x]∩gf p, i.e.,

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for all [b, x] ∈ [gp, x] ∩ gfp. Part (c) of Lemma 2.3.1shows that [a, x] = 0. Therefore, ωx is

non-degenerate when restricted to [gp, x] ∩ gfp and Sepinherits a Poisson structure from that of gp.

Corollary 2.3.4. The Slodowy slice Sχphas a Poisson structure.

Definition 2.3.5. The classical finite W-algebra associated to (gp, e) is defined to be the Poisson

algebra C[Sχp].

Remark 2.3.6. Explicit formulas for the Poisson bracket of C[Sχp] were calculated in [DSKV16] for

p = 0.

2.3.2 An isomorphism of affine varieties

Let Gpbe the adjoint group of gpand Nl,pthe unipotent subgroup of Gpwith Lie algebra nl,p. Let

m⊥_l,p:= {x ∈ gp| (x | y)p = 0 for all y ∈ ml,p}

be the orthogonal complement of ml,pwith respect to the bilinear form (· | ·)p. One can show that

m⊥_l,p=L

i≤0gp(i)

⊕ [l⊥

p, e]. As nl,pis nilpotent, the subgroup Nl,pis generated by exp(ad x) with

x running through nl,p. Restrict the adjoint action of Nl,pto e + m⊥_l,p. Assume that y ∈ m⊥_l,p. Note that

exp(ad x)(e + y) = (1 + ad x + · · · + ad

n_x

n! + · · · )(e + y) ∈ e + m

⊥ l,p.

Therefore, the image of the action map Nl,p× (e + m⊥l,p) is contained in e + m ⊥

l,p. Since Sep ⊆ e + m

⊥ l,p,

we can moreover restrict the adjoint action map to Nl,p× Sep. There is an Nl,p-action on Nl,p× Sep

defined by u · (v, x) = (uv, x) for u, v ∈ Nl,pand x ∈ Sep. Note that

u · (v, x) = (uv, x) = (uv) · x = u · (v · x), so the adjoint action map Nl,p× Sep → e + m

⊥

l,pis Nl,p-equivariant, where Nl,pacts on e + m⊥l,pby

adjoint action.

Lemma 2.3.7. The adjoint action map β : Nl,p×Sep→ e+m

⊥

l,pis an isomorphism of affine varieties.

Proof. The adjoint action map is obviously a morphism of varieties, so we only need to show that it is bijective. Since gp has trivial center, we can identify gp with a subalgebra of End gp through

the map ad : gp → End gp. Since ad is injective, we have nl,p ∼= ad nl,p. The adjoint group of

g_p is the subgroup of Aut(gp) generated by exp(ad u) with u running through gp, and Nl,p is the

subgroup generated by exp(ad v) with v running through nl,p. As nl,p is nilpotent, the exponential

map exp : ad nl,p → Nl,pis surjective, i.e., every element of Nl,pcan be expressed as exp(ad v) for

some v ∈ nl,p. Now we show that given an element e + z ∈ e + m⊥l,p, there exists a unique element

e + y ∈ Sepand a unique element x ∈ nl,p, such that exp(ad x)(e + y) = e + z. Note that

m⊥_l,p=   M i≤0 gp(i)  ⊕ [l ⊥ p, e], nl,p=   M i≤−2 gp(i)  ⊕ l ⊥ p and gfp ⊆ M i≤0 gp(i).

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For an element u ∈ gp, we write u =Piuiwith ui∈ gp(i). Let x ∈ nl,p, y ∈ gfp, and z ∈ m⊥l,p. Then x =P i≤−1xi, y = P j≤0yjand z = P

k≤1zkwith x−1 ∈ l⊥p and z1∈ [l⊥p, e]. Note that

exp(ad x)(e + y) = e + y + [x, e] + [x, y] +X

n≥2

(ad x)n

n! (e + y). The equation exp(ad x)(e + y) = e + z means that

X k zk = X j yj+ X i [xi, e] + X i,j [xi, yj] + X n≥2 (P iad xi)n n! (e + X j yj), (2.7)

which is equivalent to a series of equations, i.e., for k ≤ 1, zk− yk− [xk−2, e] = X i+j=k ad xi(yj) + X n≥2 P i1+···+in=k−2ad xi1· · · ad xin(e) n! +X n≥2 P i1+···+in+j=kad xi1· · · ad xin(yj) n! . (2.8)

We use a decreasing induction on k to show that given z, there is a unique solution (x, y) for (2.7). We remark that

• Given k, ad xi, yjappear on the right side of (2.8) only wehn i > k − 2 and j > k. Moreover, if

we have already found values for {xi, yj}i≥k0−2,j≥k0 such that (2.8) is satisfied for all k ≥ k0,

and if we only change the values of {xi, yj}i<k0−2,j<k0, then (2.8) is still valid for k ≥ k0.

• We have the decomposition gp = gfp ⊕ [gp, e], i.e., gp(i) = gfp(i) ⊕ [gp(i − 2), e], where

gf_p(i) = gfp ∩ gp(i) for all i.

• ad e : gp(i) → gp(i + 2) is injective for i ≤ −1.

When k = 1, (2.8) reads [x−1, e] = z1, which has a unique solution for x−1 when given z1, as

x−1 ∈ l⊥p, z1 ∈ [lp⊥, e] and ad e : l⊥p → [l⊥p, e] is injective. For k = k0 ≤ 0, we assume that we

have uniquely determined {xi, yj}i≥k0−1,j≥k0+1such that (2.8) is satisfied for k ≥ k0+ 1. We show

that we can uniquely determine (xk0−2, yk0) (while {xi, yj}i≥k0−1,j≥k0+1 will not change), such that

(2.8) is satisfied for k ≥ k0. Set k = k0in (2.8), since the values of {xi, yj}i≥k0−1,j≥k0+1are already

determined, the right side of (2.8) is determined, which is an element of gp(k0). Denote it by wk0.

Then (2.8) becomes [xk0−2, e] = wk0+yk0−zk0. This equation has a unique solution for (xk0−2, yk0)

when zk0 and wk0 are given, as gp(k0) = g

f

p(k0) ⊕ [gp(k0− 2), e] and ad e is injective on gp(k0− 2).

By induction, we can find a unique solution (x, y) for (2.7) when z is given.

Remark 2.3.8. The above isomorphism of affine varieties was proved in [Kos78] whene is a principal nilpotent element, and then generalized by W. Gan and V. Ginzburg in [GG02_{] for Dynkin good}

Z-grading. Their proof involves a C∗-action on both varieties and then applies a general theorem in algebraic geometry. Our proof here is purely algebraic and works for all good Z-gradings.

Corollary 2.3.9. The coadjoint action map α : Nl,p× Sχp→ χp+ m

⊥,∗

l,p is an isomorphism of affine

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2.3.3 Quantization of Slodowy slices

Recall the Kazhdan filtration on U (gp) induced by the Z-grading Γpin Example1.1.17. Let {Un(gp)}

be the PBW-filtration on U (gp) and

Un(gp)(i) := {x ∈ Un(gp) | [hΓ, x] = ix}.

Then KnU (gp) =Pi+2j≤nUj(gp)(i). The Kazhdan filtration is separated and exhaustive, i.e.,

\ n∈Z KnU (gp) = {0} and U (gp) = [ n∈Z KnU (gp).

The Kazhdan filtration on U (gp) induces filtrations on Iχp, Qχp and Hχp, which we also denote

by Kn. Moreover, grKIχp is just the ideal of C[g

∗

p] defining the affine subvariety χp + m⊥,∗l,p , i.e.,

gr_KQχp ∼= C[χp+ m

⊥,∗

l,p ]. Note that KnQχp = 0 for n < 0 as {a − χp(a) | a ∈ ml,p} contains all the

negative-degree generators of U (gp) with respect to the Kazhdan filtration.

Since Hχp ⊆ Qχp, we have a natural inclusion map

ν1: grKHχp → grKQχp.

On the other hand, as Sχp ⊆ χp+ m

⊥,∗

l,p , we have a restriction map

ν2 : C[χp+ m⊥,∗_l,p ] → C[Sχp].

Composing these two maps, we get a homomorphism, as grKQχp ∼= C[χp+ m

⊥,∗ l,p ],

ν = ν2◦ ν1 : grKHχp → C[Sχp].

We are going to show that ν is an isomorphism.

The module Qχpis a filtered U (nl,p)-module, where the filtration on U (nl,p) is the Kazhdan filtration

induced from that of U (gp). This filtration induces filtrations on the cohomologies Hi(nl,p, Qχp), and

there are canonical homomorphisms

hi : grKHi(nl,p, Qχp) → H

i_(n

l,p, grKQχp). (2.9)

Theorem 2.3.10. The homomorphism ν : gr_KHχp → C[Sχp] is an isomorphism.

Proof. First, we show that Hi(nl,p, grKQχp) = δi,0C[Sχp]. Recall the isomorphism of affine varieties

in Lemma2.3.7, which is Nlp-equivariant. Thus we have an nl,p-module isomorphism C[χp+m

⊥,∗ l,p ] ∼= C[Nlp] ⊗ C[Sχp]. Hence Hi(nl,p, grKQχp) = H i_(n l,p, C[χp+ m⊥,∗_l,p ]) = Hi(nl,p, C[Nlp]) ⊗ C[Sχp].

The cohomology Hi(nl,p, C[Nlp]) is equal to the algebraic de Rham cohomology of Nlp [CE48],

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Next we show that the homomorphisms hi in (2.9) are all isomorphisms. The standard cochain

com-plex for computing the cohomology of nl,pwith coefficients in Qχp is

0 → Qχp→ n

∗

l,p⊗ Qχp→ · · · → Λ

n_n∗

l,p⊗ Qχp → · · · . (2.10)

Recall that there is a grading on g∗p hence a grading on n∗l,p, which is positively graded as nl,p is

negatively graded in gp. We write the gradation as n∗l,p=

L

i≥1n ∗

l,p(i). Define a filtration of Λnn ∗ l,p⊗

Qχp by setting Fs(Λ

n_n∗

l,p⊗ Qχp) to be the subspace spanned by (x1∧ · · · ∧ xn) ⊗ v for all xi ∈

n∗_l,p(ni), v ∈ KjQχpsuch that j +P ni≤ s, where Kjis the Kazhdan filtration on Qχp. This defines

a filtered complex on (2.10) whose associated graded complex gives us the standard cochain complex for computing the cohomology of nl,pwith coefficients in grKQχp.

Consider the spectral sequence with

E₀s,t= Fs(Λ s+t_n∗ l,p⊗ Qχp) Fs−1(Λs+tn∗_l,p⊗ Qχp) . Then E₁s,t = Hs+t(nl,p, KsQχp Ks−1Qχp

) and the spectral sequence converges to

E_∞s,t= FsH

s+t_(n

l,p, Qχp)

Fs−1Hs+t(nl,p, Qχp)

,

i.e., the maps hi: grKHi(nl,p, Qχp) → H

i_(n

l,p, grKQχp) are isomorphisms hence

gr_KHχp = grKH

0_(n

l,p, Qχp) ∼= H

0_(n

l,p, grKQχp) ∼= C[Sχp].

Remark 2.3.11. For p = 0, the isomorphism in Theorem2.3.10was proved by A. Premet [Pre02] when l is a Lagrangian subspace of g(−1) and then generalized by W. Gan and V. Ginzburg [GG02] for general isotropic subspaces l. Our method here follows [GG02].

Remark 2.3.12. Theorem2.3.10shows that_(C[g∗_p], gr_KIχp, C[Sχp]) is a Poisson reducible triple and

the Poisson structure onS_χ_pcan be considered as a Poisson reduction of g∗_p. Corollary 2.3.13. The algebra Hχp does not depend on the isotropic subspace lp.

Proof. Let lp ⊆ l0pbe two isotropic subspaces of gp(−1), and Hχp, H

0

χp the corresponding finite

W-algebras. Then we have a natural map π : Hχp ,→ H

0

χp hence a natural map gr π : grKHχp ,→

gr_KH_χ0_p. By Theorem2.3.10, we know that gr π is an isomorphism as they are both isomorphic to C[Sχp], so π is itself an isomorphism.

Since Hχp does not depend on the isotropic subspace lp, we choose it to be a Lagrangian subspace of

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2.4 Kostant’s theorem and Skryabin equivalence

2.4.1 Kostant’s theorem

Given a finite-dimensional Lie algebra a and a linear functional ϕ ∈ a∗, define aϕ:= {x ∈ a | ϕ([x, y]) = 0 for all y ∈ a}.

The index of a is defined to be χ(a) = Inf{dim aϕ | ϕ ∈ a∗}. We say that ϕ ∈ a∗ is regular if dim aϕ= χ(a).

Given x ∈ a, let ax= {y ∈ a | [x, y] = 0} be the centralizer of x in a. Then x is called regular if its centralizer ax has minimal dimension, i.e., dim ax ≤ dim ax0

for all x0 ∈ a. When a admits a non-degenerate invariant symmetric bilinear form which identifies a and a∗, the regularity of an element is the same thing as the regularity of the corresponding linear function. It is well known that the subset of regular elements in a is a dense open subset under the Zariski topology.

Let e be a regular nilpotent element in g, which we also call principal nilpotent. We show that the finite W-algebra Hχp associated to (gp, e) is isomorphic to Z(gp), the center of the universal enveloping

algebra U (gp).

Let S(gp) be the symmetric algebra of gp. It is well known that there is a canonical isomorphism of

g_p-modules ϕ : S(gp) → grU (gp), where gr is the associated graded of the PBW filtration of U (gp).

Let I(gp) := {g ∈ S(gp) | [x, g] = 0 for all x ∈ gp} be the gp-invariants in S(gp) and Z(gp) be the

center of U (gp). Then the restriction of ϕ to I(gp) yields an isomorphism of vector spaces

ϕ : I(gp) → grZ(gp).

Recall that Sep = e + g

f

p and Sχp = κp(Sep). Since Sχp ⊆ g

∗

p, we have a canonical restriction

ιp : C[g∗p] → C[Sχp]. Identifying C[g

∗

p] with S(gp) and restricting ιpto I(gp), we get a natural map

from I(gp) to C[Sχp], which we still denote by ιp.

Lemma 2.4.1 ([RT92, MS16]). Let g be a finite-dimensional semi-simple Lie algebra and x = P

ixiti∈ gpwithxi ∈ g. Let e be a regular nilpotent element of g. Then

(1) x is regular in gpif and only ifx0is regular in g.

(2) Every element ofSepis regular. Moreover, the adjoint orbit of every regular element intersects

Sep in a unique point.

(3) The mapιp : I(gp) → C[Sχp] is an isomorphism of vector spaces.

Theorem 2.4.2. Let e be a regular nilpotent element of g. Then the finite W-algebra Hχpassociated

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Proof. Since Z(gp) ⊆ U (gp) is obviously invariant under the adjoint action of nl,p, we have a natural

map jp : Z(gp) → Hχp, which preserves the Kazhdan filtrations on Z(gp) and Hχp. Passing to their

associated graded, we have gr jp : grZ(gp) → grHχp, which is the isomorphism ι : I(gp) → C[Sχp].

Since the associated graded of jp is an isomorphism, jpitself is an isomorphism of algebras.

Z(gp) jp _// gr Hχp gr I(gp) gr jp ∼ = //C[Sχp]

Remark 2.4.3. When p = 0, i.e., in semi-simple cases, Lemma2.4.1and Theorem2.4.2were proved by B. Kostant [Kos78]. T. Macedo and A. Savage [MS16] generalized Lemma 2.4.1 to truncated multicurrent Lie algebras, on which non-degenerate invariant bilinear forms exist. Therefore, all the lemmas and theorems in this section can be generalized to those algebras, i.e., finite W-algebras associated to truncated multicurrent Lie algebras can be defined and Kostant’s theorem holds. Remark 2.4.4. Explicit generators of I(gp) were constructed in [RT92], but corresponding

genera-tors ofZ(gp) are not known in general. When g = s`n, A. Molev [Mol97] has given a description of

generators ofZ(gp).

2.4.2 Skryabin equivalence

Definition 2.4.5. A gp-module M is called a Whittaker module if a − χp(a) acts locally nilpotently

on M for all a ∈ ml,p. Given a Whittaker module M , an element m ∈ M is called a Whittaker vector

if (a − χp(a)) · m = 0 for all a ∈ ml,p. Let Wh(M ) be the collection of the Whittaker vectors of M .

Lemma 2.4.6. The gp-moduleQχpis a Whittaker module, withWh(Qχp) = Hχp.

Proof. Remember that Qχp = U (gp)/Iχp, where Iχp is the left ideal of U (gp) generated by {a −

χp(a) | a ∈ ml,p}. Since ml,p is negatively graded in the good grading Γp of gp, it acts nilpotently

on gp hence locally nilpotently on U (gp). Note that ad a = ad (a − χp(a)) for all a ∈ ml,p, so

ad (a − χp(a)) acts locally nilpotently on U (gp), and also on its quotient Qχp, i.e., Qχpis a Whittaker

module. Since we choose lp to be a Lagrangian subspace of gp(−1), we have nl,p = ml,p. Then by

the definition of Hχp, we have Wh(Qχp) = H

0_(m

l,p, Qχp) = Hχp.

Let gp-Wmodχp be the category of finitely generated Whittaker gp-modules and Hχp-Mod be the

category of finitely generated left Hχp-modules.

Since Hχp ∼= (EndgpQχp)

op_{, Q}

χp admits a right Hχp-module structure. Given N ∈ Hχp-Mod, we

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Lemma 2.4.7. (1) LetM ∈ gp-Wmodχp. ThenWh(M ) = 0 implies that M = 0.

(2) LetM ∈ gp-Wmodχp. ThenWh(M ) admits an Hχp-module structure, with(y + Iχp) · v = y · v

fory + Iχp ∈ Hχp, v ∈ M .

(3) LetN ∈ Hχp-Mod. ThenQχp⊗Hχp N ∈ gp-Wmod

χp_.

Proof. By definition, a Whittaker gp-module M is locally U (ml,p)-finite as U (ml,p) is generated by 1

and {a − χp(a) | a ∈ ml,p}. Given a nonzero vector v ∈ M , we have dim U (ml,p) · v < ∞. Since

a − χp(a) are nilpotent operators on U (ml,p) · v, by Engel’s theorem, we can find a nonzero common

eigenvector for them, which is a Whittaker vector, so Wh(M ) 6= 0 if M 6= 0.

For (2), we only need to show that y · v ∈ Wh(M ) for all y + Iχp ∈ Hχpand v ∈ Wh(M ), because

the module structure comes from the U (gp)-module structure on M . We have

(a − χp(a))y · v = [a − χp(a), y] · v + y(a − χp(a)) · v = [a − χp(a), y] · v.

By the proof of Lemma2.2.6, we have [a, y] ∈ Iχp, so (a − χp(a))y · v = 0, i.e., y · v ∈ Wh(M ).

For (3), note that Qχp is a Whittaker gp-module, so a − χp(a) acts locally nilpotently on it. But

the U (gp)-action on the tensor product is from the left side, so a − χp(a) acts automatically locally

nilpotently on the tensor product Qχp⊗HχpN for all a ∈ ml,p.

By Lemma2.4.7, we have two functors,

Wh : g_p-Wmodχp _{−→ H}

χp-Mod, M 7−→ Wh(M ),

Qχp⊗Hχp − : Hχp-Mod −→ gp-Wmod

χp_, _{N 7−→ Q}

χp⊗Hχp N.

The functor Wh(−) is left exact and the functor Qχp⊗Hχp − is right exact.

Theorem 2.4.8. The two functors Wh(−) and Qχp⊗Hχp− give an equivalence of categories between

g_p-Wmodχp_and_H

χp-Mod.

Proof. Since Hχp does not depend on the isotropic subspace lp, we choose it to be a Lagrangian

subspace of gp(−1), so we have ml,p = nl,p. First, we show that Wh(Qχp ⊗Hχp N ) ∼= N for all

N ∈ Hχp-Mod. Assume that N is generated by a finite-dimensional subspace N0. Setting KnN :=

(KnHχp)N0 gives a filtration on N and it becomes a filtered Hχp-module. We twist the ml,p-action

on Qχp⊗HχpN by −χp, i.e., we define a new action by

a · (u ⊗ v) = (a − χp(a))u ⊗ v = ad(a − χp(a))(u) ⊗ v for a ∈ ml,p, u ∈ Qχp, v ∈ N.

Then Wh(Qχp ⊗Hχp N ) = H

0_(m

l,p, Qχp ⊗Hχp N ) with respect to this new action. The Kazhdan

filtrations on Qχp and N induce a Kazhdan filtration on Qχp⊗HχpN , with

Kn(Qχp⊗Hχp N ) =

X

i+j=n

W-algebras Associated to Truncated Current Lie Algebras