Slodowy Varieties, Parabolic W-Algebras, and an Introduction to Shifted Yangians

SLODOWY VARIETIES, PARABOLIC W-ALGEBRAS, AND AN INTRODUCTION TO SHIFTED YANGIANS

SURYA RAGHAVENDRAN

Abstract. Slodowy varieties are certain symplectic varieties related to Slodowy slices in reductive lie algebras. I’ll introduce these gadgets and their quantizations which are certain generalizations of ﬁnite W- algebras. I’ll end with an introduction to shifted Yangians via a result of Brundan-Kleschev that uses them to give presentations for ﬁnite W-algebras.

1. Slodowy Varieties Slodowy varieties were brieﬂy mentioned in Vasya’s talk on the Mirkovic-Vybornov isomorphism. These are certain generalizations of slices to nilpotent orbits in reductive Lie algebras. Let us begin by recalling the latter construction. 1.1. Warm-up: Slodowy slices.

Deﬁnition 1.1. Let e ∈ g be a nilpotent element. Extend this to a choice of sl2 triple he, h, fi ⊂ g. The Slodowy slice associated to (g, e) is the variety S = e + ker[f, −] ⊂ g. A large class of familiar examples are given by Kleinian singularities. Theorem 1.2 (Grothendieck-Brieskorn). Let g be simply laced, N ⊂ g denote the nilpotent cone, and S be a Slodowy slice to a subregular nilpotent. The intersection S ∩ N is a Kleinian surface singularity with the same Dynkin diagram as g.

Example 1.3. Let g = sl3 and let e be subregular nilpotent and complete this to an sl2 triple. Explicitly, we have 0 1 0 1 0 0 0 0 0 e = 0 0 0 , h = 0 −1 0 , f = 1 0 0 . 0 0 0 0 0 0 0 0 0 n a 1 o As a set, S = b a c . The intersection S ∩ N is the zero-fibre of the Chevalley restriction map d −2a 2 2 S → C /S3 given by taking coefficients of the characteristic polynomial. This yields the equations a + b = 0, 2a3 − 2ab + cd = 0, which imply 4a3 + cd = 0. Changing coordinates w = 41/3a, c = x + iy, d = x − iy, we 3 find an A2 surface singularity in C . Remark 1.4. We can give the algebra C[S] the so-called Kazhdan grading, which is defined as follows. Identifying g =∼ g∗ by means of a symmetric nondegenerate invariant form (−, −), let us view S ⊂ g∗. Let γ : C× → G be the one-parameter subgroup associated to the semisimple element h ∈ g. That is, if ξ ∈ g is such that [h, ξ] = iξ, then γ(t)ξ = tiξ. Consider the C× action on g∗ given by tα = t−2γ(t)α. Under this action, S has only negative weight spaces, so C[S] is in fact nonnegatively graded. Remark 1.5. We may also construct S as a Hamiltonian reduction. Consider the grading g = L g(i) i∈Z given by the decomposition of g into ad(h)-eigenspaces. Let χ = (e, −) ∈ g∗, and consider the skew-symmetric form on g given by (ξ, η) 7→ hχ, [ξ, η]i. The restriction of this skew-symmetric form to g(−1) is nondegenerate L so defines a symplectic structure. Let l ⊂ g(−1) be a lagrangian subspace and let m = l ⊕ i≤−2 g(i). The subalgebra m exponentiates to a unipotent subgroup M ⊂ G. Consider the moment map for the coadjoint action of M on g∗; this is a map µ : g∗ → m∗ simply given −1 by restriction of functions. It turns out that χ|m is a regular value for µ and that µ (χ|m) is M-stable. −1 ∗ Further, µ (χ|m)/M is isomorphic to S as a variety: the map M × S → g given by left multiplication −1 ∗ is readily seen to be an isomorphism onto µ (χ|m). Thus the Poisson structure on g reduces to give a Poisson structure on S.

Date: April 20, 2020. 1 1.2. Slodowy Varieties. Definition 1.6. Let e ∈ g be nilpotent and let S be the corresponding Slodowy slice. The equivariant Slodowy slice is the G-variety X = G × S equipped with the left G-action. Trivializing T ∗G =∼ G × g∗ note that there is a natural map X → T ∗G. It is a fact that the pullback of the canonical symplectic form ω along this map is still nondegenerate. Thus, X is symplectic. Now let P ⊂ G be a parabolic subgroup. The Hamiltonian action of P on T ∗G restricts to give a Hamiltonian action of P on X. Definition 1.7. Let e ∈ g be a nilpotent with corresponding slodowy slice S and let P ⊂ G be a parabolic subgroup. The Slodowy variety S(e, P ) is given by the symplectic reduction X///0P . ∗ ∗ Note that S(e, P ) has an obvious embedding into T G///0P = T (G/P ). Example 1.8. (1) Let e = 0. Then the corresponding Slodowy slice is all of g∗, so X = T ∗G. The equivariant Slodowy slice S(0,P ) is then given by T ∗(G/P ). (2) Let P = B a Borel subgroup. Then the image of S(e, B) in T ∗(G/B) is precisely the preimage of S ∩ N under the Springer resolution T ∗(G/B) → N . In particular, combining with theorem 1.2, we see that for g simply-laced and e subregular, S(e, B) is a resolution of a Kleinian singularity. ∗ ∗ ¯⊥ (3) Let G = SLn. Then the moment map T (G/P ) → g is generically injective with image Gp . This is the closure of a nilpotent orbit, the so-called Richardson orbit of p. Therefore, we see that S(e, P ) is a resolution of S ∩ G¯p⊥, which is a transverse slice to the nilpotent orbit through e inside the closure of the Richardson orbit of p. 1.3. Relation to Quiver Varieties in Type A. A result of Maffei [Ma] establishes an isomorphism between quiver varieties and Slodowy varieties in type A. Let g = slN and fix the following pieces of data: Pn • n nonnegative integers r1, ··· , rn ∈ Z≥0 with i=1 ri = N. These numbers determine a partial flag Pj (Fj) with dim Fj = i=1 ri, and in turn a parabolic P ⊂ G = SLn as the stabilizer of (Fj). Pn−1 • d = (d1, ··· dn−1) with i=1 idi = N. This determines a partition of N in which i occurs di times and hences a nilpotent element e ∈ g with Jordan type given by the partition. These two pieces of data determine a Slodowy variety S(e, P ) as well as a quiver variety M(v, d) attached to the An−1 quiver, with n n−1 X X vi = rj − (j − i)dj j=i+1 j=i+1 Theorem 1.9 (Maffei [Ma]). There is an isomorphism M(v, d) → S(e, P ). Remark 1.10. In [L] it is further proved that the above isomorphism is in fact a C×-equivariant symplec- tomorphism. The proofs of these statements are quite involved, so I will instead try to illustrate the isomorphism in some examples. Example 1.11.

(1) Let g = slN . We first consider the case where the ri are arbitrary and d1 = N all other di = 0. This determines the Slodowy variety S(0,P ), which as we saw in example 1.8, is isomorphic to T ∗(G/P ). The above formula for the dimension vector gives i X vi = N − rj. j=1 ∗ N Thinking of T (G/P ) as consisting of pairs (x, (Fj)) where (Fj) is a partial flag in C , and x satisfies x(Fj) ⊂ Fj−1, the above isomorphism is given by

(x, x,¯ p, q) 7→ (q1p1, {0} ⊂ ker p ⊂ ker x1p1 ⊂ · · · ⊂ ker xn−1 ··· x1p1). 2 (2) Let g = sl3. We consider the case where n = 3, ri = 1, di = 1. The parabolic determined by the ri is a Borel and the nilpotent determined by the di is subregular. The Slodowy variety is thus S(e, B) which by example 1.8 is the minimal resolution of the A2 surface singularity. The above prescription for the dimension vector yields v = (1, 1). We have an isomorphism M0(v, d) → S ∩ N given by

(x, x,¯ (p1, p2), (q1, q2)) 7→ (w1, w2, w3) = (q2xp1, q1xp¯ 2, p1q1).

3 The image of this map indeed satisﬁes w3 − w1w2 = 0.

2. Finite and Parabolic W-algebras In the previous section we constructed a Poisson variety S and symplectic varieties X,S(e, P ). We now turn to their quantizations.

2.1. Finite W-Algebras are Quantizations of Slodowy Slices. The ﬁnite W-algebra is constructed in a way that parallels the construction of S via Hamiltonian reduction in remark 1.5. Let χ, m,M be as in remark 1.5, and let mχ = span{a − χ(a): a ∈ m}.

Definition 2.1. The finite W-algebra W(e) is defind as the quantum Hamiltonian reduction U((g))///χM. Explicitly m W (e) = (U(g)/U(g)mχ) . The algebra W (e) has a natural filtration called the Kazhdan filtration induced from one on U(g), which we define as follows. Note that the adjoint action of h on g extends to a derivation on U(g); for i ∈ Z, denote j by Ui(g) the i-eigenspace of this action. Further, let U (g) denote the j-th piece of the PBW filtration. With this in hand, the Kazhdan filtration F•U(g) has

X j FkU(g) = Ui(g) ∩ U (g). i+2j≤k

One can show that the ideal generated by mχ is in fact a ﬁltered ideal, and the ﬁltration descends to one on W (e).

Theorem 2.2 (Gan-Ginzburg [GG]). gr W (e) =∼ C[S]. For e regular nilpotent, W (e) is well understood. Theorem 2.3 (Kostant [K]). For e a regular nilpotent, W (e) congZ(U(g).

1 2 1 Example 2.4. Let g = sl2. We claim that W (e) = C[e+ 4 h − 2 h]. Indeed, here, m = Cf and mχ = C(f −1). Note that we can write the casimir of g as 1 1 h2 − 2h + 4ef = h2 − 2h + 4ef + 4e − 4e = 4(e + h2 − h + 4e(f − 1) 4 2 . Furthermore the casimir is clearly m invariant. Recall the equivariant Slodowy slice X = G×S. This is symplectic and affine, so we can quantize this using ˜ the techniques of [BK] that we learned about in Dylan’s talk. Let W~ denote the canonical quantization ˜ G of X. This is a sheaf of graded algebras flat over C[~]. Consider the algebra Γ(X, W~) ; we have that ˜ G ˜ G Γ(X, W~) /~Γ(X, W~) = C[X]. × ˜ G × Let Wh denote the subalgebra of C finite vectors in Γ(X, W~) . Since the C -action on S is contracting, it follows that W~/~W~ = C[S].

Remark 2.5. In fact, in [L] the author shows that the filtered algebra W~/(~ − 1)W~ is isomorphic to the finite W-algebra as constructed via quantum Hamiltonian reduction above, with the Kazhdan filtration. 3 2.2. Parabolic W-Algebras are Quantizations of Slodowy Varieties. Let h ⊂ g be a Cartan subalgebra and let W denote the Weyl group. Also choose a Borel subgroup B ⊂ G. The usual Harish-Chandra G ∼ ∗ W G ∼ ∗ W isomorphism U(g) = C[h ] admits a homogenized version U~(g) = C[h , ~] where the W -action on C[h∗, ~] is given by w.f(λ) = f(w−1(λ + ρ~) − ρ~)) where ρ denotes the half-sum of positive roots as usual. ∗ With this in hand, we set W ,h = W ⊗U (g)G [h , ]. ~ ~ ~ C ~ Let P ⊂ G be a parabolic subgroup and let P0 denote its solvable radical. Let p = Lie P and let a be the × P/P0 quotient of p by its solvable radical. Let A~ denote the algebra of C -finite elements in Γ(G/P, D~(G/P0) ). ∗ The Hamiltonian G-action on T (G/P ) induces a quantum comoment map U~(g) → A~

Definition 2.6. The parabolic W-algebra is defined as the quantum Hamiltonian reduction A~///χM. Explicitly WP = (A /A m )M ~,a ~ ~ χ Proposition 2.7. WP / WP =∼ [S˜(e, P )] ~,a ~ ~,a C The proof in [L] appeals to a lemma that gives conditions on the M-action under which quantization ∗ commutes with reduction. Letting µ denote the moment map Spec(A~/~A~) → m , the conditions are −1 satisfied if µ (χ|m) is free. ∗ In the sequel, we’ll need a ρ-shifted version of the C[a , ~]-algebra structure on A~. Assume that the parabolic P contains the Borel B we fixed above and let L stand for the Levi subgroup of P containing the ∼ maximal torus exponentiating h. We may then identify a = z(l) where l = Lie(L). Letting Φ : a → A~ denote ∗ the map inducing the C[a ]-algebra structure, consider the map ι : a → A~ defined by ι(ξ) = Φ(ξ) − ~hρ, ξi. This induces the desired C[a∗, ~]-algebra structure. Note that the identification of a with the center of l gives a decomposition h = a ⊕ (h ∩ [l, l]), so the ∗ ∗ ∗ ∗ projection onto h makes C[a ] into a C[h ]-module. In what follows, let Wa,~ = C[a , ~] ⊗C[h ,~] Wh,~. Theorem 2.8. There is a natural graded algebra homomorphism W → WP . ~,a ~,a ∗ Sketch of proof. The idea is to construct a homomorphism [a , ] ⊗U (g)G U (g) → A and then apply C ~ ~ ~ ~ quantum Hamiltonian reduction by M. Note that we have an algebra homomorphism U~(g) → A~ given ∗ by the quantum comoment map. We also have the map ι : C[a , ~] → A~ constructed above. We need only G ∗ show that the two maps agree on U~(g) which injects in U~(g) and maps to C[a , ~] via the composition G ∼ ∗ W ∗ ∗ U~(g) = C[h , ~] → C[h , ~] → C[a , ~]. G Further since both of the morphisms U~(g) → A~ are graded, it suffices to check this after specializing ~ = 1. ( ∗ Now the algebra A = Γ(G/P0, D(G/P0) P/P0)) acts on C[G/P0] by differentiation. The subalgebra C[a ] G acts faithfully. Thus, the claim amounts to checking that the natural action of U(g) on C[G/P0] factors through the map U(g)G → C[a∗]. This supposedly follows from the construction of the Harish-Chandra homomorphism.

Remark 2.9. When P = B the above map is an isomorphism. Since Wa,~ is an extension of scalars of Wh,~, this establishes the sense in which the parabolic W-algebra generalizes the ﬁnite W-algebra. Remark 2.10. In type A, the above map is surjective. The kernel has an explicit description worked out in [WWY].

3. Presentations of Finite W-algebras from Shifted Yangians We now give an overview of some work of Brundan-Kleschev [BrKl] that exhibits isomorphisms between finite W-algebras and certain subalgebras of the Yangian called shifted Yangians. We begin by introducing the latter objects. Throughout this section we will restrict to g = glN . 3.1. Shifted Yangians. Shifted Yangians are defined to be certain subalgebras of the ordinary Yangian Y (gln) which is a certain deformation of U(gln[z]). r Definition 3.1. The Yangian of gln is the algebra generated by elements 1, ti,j where r > 0, i, j = 1, ··· , n subject to the relations r+1 s r s+1 r s s r [ti,j , tk,l] − [ti,j, tk,l ] = tk,jti,l − tk,jti,l. 4 A convenient way to package these relations is via the so-called RTT relation. We define a generating series X r −r ti,j(z) = ti,ju . r≥0 Definition 3.2. The transfer matrix is the matrix n X −1 T (z) = Ei,j ⊗ ti,j(z) ∈ gln ⊗ gln[z ]. i,j The R-matrix is the matrix n 1 X R(z) = 1 − E ⊗ E . z i,j j,i i,j

−1 −1 −1 The RTT relation is then the following equality in gln ⊗ gln ⊗ gln[z , w , (z − w) ]:

R12(z − w)T13(z)T23(w) = T23(w)T12(z)R1,2(z − w) where the subscripts correspond to factors of the tensor product. A gauss decomposition of the transfer matrix will lend itself to another presentation of the Yangian. Write T (z) = F (z)D(z)E(z), where       1 0 ··· 0 D1(z) 1 E1(z) · · · ∗ F1(z) 1 ··· 0  D2(z)  0 1 ···  F (z) =   ,D(z) =   ,E(z) =    ···   ···   ··· En−1(z)  Fn−1(z) 1 Dn(z) 0 0 ··· 1

−1 −1 Let D˜ i(z) = −Di(z) . All of the above matrix elements are power series in z . Our new generators of Y (glN ) will be the coeﬃcients of these power series. Namely, writing X r −r X r −r Ei(z) = Ei z ,Fi(z) = Fi z r≥1 r≥1 X r −r ˜ X ˜ r −r Di(z) = Di z , Di(z) = Di z r≥1 r≥1 the relations are r s • [Di ,Dj ] = 0 r s Pr+s−1 ˜ t r+s−1−t • [Ei ,Fj ] = δij t=0 Di Di+1 r s Pr−1 r+s−1−t t • [Di ,Ej ] = (δij − δi,j+1) t=0 Fj Di r s Pr−1 r+s−1−t t • [Di ,Fj ] = (δij − δi,j+1) t=0 Ej Di r s Ps−1 t r+s−1−t Ps−1 r+s−1−t t • [Ei ,Ei ] = t=1 Ei Ei − t=1 Ei Ei r s Ps−1 t r+s−1−t Ps−1 r+s−1−t t • [Fi ,Fi ] = t=1 Fi Fi − t=1 Fi Fi r s+1 r+1 s r s+1 • [Ei ,Ei+1 ] − [Ei ,Ei+1] = −Ei Ei r s+1 r+1 s r s+1 • [Fi ,Fi+1 ] − [Fi ,Fi+1] = −Fi Fi r s • [Ei ,Ej ] = 0 for |i − j| > 1 r s • [Fi ,Fj ] = 0 for |i − j| > 1 r s t s r t • [Ei , [Ei ,Ei ]] + [Ei , [Ei ,Ei ]] = 0 for |i − j| > 1 r s t s r t • [Fi , [Fi ,Fi ]] + [Fi , [Fi ,Fi ]] = 0 for |i − j| > 1

The shifted Yangian will be a certian subalgebra of Y (gln) that is generated by some subset of the above generators, but with the same relations. The subset of generators is picked out by imposing certain bounds on the poles of matrix elements in the factors of the above Gauss decomposition. These bounds are stipulated by a choice of shift matrix.

Deﬁnition 3.3. A shift matrix is an n × n matrix σ = (sij) where

sij + sjk = sik if |i − j| + |j − k| = |i − j| 5 Deﬁnition 3.4. Let σ be a shift matrix. The shifted Yangian Yn(σ) is a subalgebra of Y (gln) generated by r • Di , r > 0, 1 ≤ i ≤ n r • Ei , r > si,i+1, 1 ≤ i < n r • Fi , r > si+1,i, 1 ≤ i < n subject to the same relations as above. P Now, an n-step partition of N, λ = (λ1, ··· , λn), i λi = N determines both a nilpotent eλ ∈ glN with n Jordan blocks, and an n × n-shift matrix σ, given by ( λn+1−j − λn+1−i for i < j si,j = 0 for i ≥ j

The top step λn is typically denoted l and called a level.

Definition 3.5. The shifted Yangian of level l Yn,l(σ) is the quotient of Ynσ by the two-sided ideal r generated by D1 for r > λ1. ∼ Theorem 3.6 (Brundan-Kleschev [BrKl]). There is an isomorphism W (eλ) = Yn,l(σ). This result is proven under the assumption that the grading on g is even. Then the premet subalgebra m is a summand of g; denote the complement by p. The key step in the construction of the above isomorphism is an explicit construction of certain m-invariant elements of U(p) that are in bijection with the generators of Yn,l(σ) above. Example 3.7. Let N = 2 and consider a length 1 partition of 2. This determines the regular nilpotent in gl2 and an identically zero shift matrix. By example 2.4, we have that W (eλ) is just a polynomial algebra 1 2 r r in two variables. The above definition tells us that Y1,2(σ) is generated by D ,D ,E ,F . The relations kill off the E,F and stipulate that D1,D2 commute.

References [BK] R.Bezrukavnikov and D.Kaledin Fedosov Quantization in Algebraic Context arXiv:math/0309290v4 [BrKl] J. Brundan and A. Kleschev Shifted Yangians and finite W-algebras Advances Math. 200 (2006), 136–195. [DA] D. V. Artmanov Introduction to finite W-algebras arXiv:1607.01697v1 [GG] W.L. Gan and V. Ginzburg Quantization of Slodowy Slices Int. Math. Res. Not. 2002, no. 5, 243-255 [K] B. Kostant On Whittaker vectors and representation theory Invent. Math. 48 (1978), 101–184. [L] I. Losev Isomorphisms of quantizations via quantization of resolutions. arXiv:1010.3182v3 [Ma] A. Maffei Quiver varieties of type A Comment. Math. Helv. 80(2005), 1-27. [M] A. Moreau Nilpotent Orbits and Finite W-Algebras http://www-math.sp2mi.univ-poitiers.fr/ amoreau/Kent2014-Walg.pdf [WWY] B. Webster, A. Weekes, and O. Yacobi A quantum Mirkovic-Vybornov isomorphism arXiv:1706.03841v2 Email address: [email protected]