Symplectic implosion and the Grothendieck-Springer resolution

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to reduce a Hamiltonian $G$-space to a Hamiltonian $H$-space where $H$ is the maximal torus of $G$. We show that this procedure coincides with an algebraic version of symplectic implosion of Guillemin, Jeffrey and Sjamaar. We explain how to obtain generalizations of this picture to quasi-Hamiltonian spaces and their elliptic version.


Introduction
The goal of this paper is to introduce symplectic implosion in the realm of derived symplectic geometry. 0.1. Derived symplectic geometry. Pantev, Toën, Vaquié and Vezzosi introduced the notions of closed differential forms on derived stacks and defined shifted symplectic structures on such stacks. As in the classical context, a symplectic structure is a closed nondegenerate two-form on the stack, but now the form can have a nontrivial degree. Moreover, the form is not strictly closed, but closed only up to homotopy.
One can also introduce Lagrangians in a shifted symplectic stack X. These are morphisms f : L → X together with a trivialization of the pullback of the symplectic form from X to L; moreover, we require the trivialization to be nondegenerate in a certain sense. Note that Lagrangians L → X are not necessarily embeddings: for instance, if L → X is a Lagrangian in an n-shifted symplectic stack X and Y is an (n − 1)-shifted symplectic stack Y , then L × Y → X is also Lagrangian.
The key result about derived Lagrangians is the fact that their intersection carries a shifted symplectic structure. More precisely, if we have two Lagrangians L 1 , L 2 in an nshifted symplectic stack X, the intersection L 1 × X L 2 is (n − 1)-shifted symplectic. More generally, if X ← L → Y is a Lagrangian correspondence and N → Y is another Lagrangian, L × Y N → X also carries a Lagrangian structure. 0.2. Hamiltonian reduction. Let us now explain how shifted symplectic structures give a new point of view on Hamiltonian reduction (this perspective can be found in [Cal13] and [Saf13]).
We This picture generalizes to quasi-Hamiltonian reduction which is concerned with groupvalued moment maps M → G. In that case G G = BG × BG×BG BG is the self-intersection of the diagonal in BG, a 2-shifted symplectic stack, so it carries a 1-shifted symplectic structure. Asking the same question for G-equivariant maps M → G we get that Lagrangians in G G are the same as G-quasi-Hamiltonian spaces. One can perform Hamiltonian reduction given any two Lagrangians in a 1-shifted symplectic stack. Another example of a 1-shifted symplectic stack is Bun G (E), the moduli stack of G-bundles on an elliptic curve. Such "elliptic" Hamiltonian reduction is useful to construct symplectic structures on the moduli stacks of G-bundles on del Pezzo surfaces (see [Cal13]). 0.3. Symplectic implosion. Symplectic implosion was introduced by Guillemin, Jeffrey and Sjamaar as a way to produce H-Hamiltonian spaces out of G-Hamiltonian spaces, where H ⊂ G is the maximal torus. It is defined in a rather ad hoc way as a symplectic completion of the cross-section, which is roughly the preimage of the fundamental Weyl chamber under the moment map. It can also be interpreted as a G-Hamiltonian reduction with respect to (T * G) impl , the universal implosion. In [GJS01] it was shown that (T * K) impl ∼ = [G/N] af f , the affinization of the base affine space, where K ⊂ G is the maximal compact subgroup.
Symplectic implosion was generalized to the hyperKähler setting for K = SU(n) by Dancer, Kirwan and Swann [DKS12]. As one expects hyperKähler implosion for K to coincide with the holomorphic symplectic implosion for G, we will use their definition to compare our results. Let us mention that they obtain the universal implosion to be (T * K) HKimpl = [G × N b] af f in one of the complex structures.
Since Hamiltonian spaces are interpreted as Lagrangians in the adjoint quotient, to implode a G-Hamiltonian space to an H-Hamiltonian space one has to compose the Lagrangian with a Lagrangian correspondence between g/G and h/H. One famous such correspondence is the so-called Grothendieck-Springer simultaneous resolution Here g parametrizes Borel subgroups B ⊂ G together with elements in their Lie algebras. We show that this correspondence is Lagrangian and, moreover, that the composition of a Lagrangian in [g/G], a G-Hamiltonian space, gives the holomorphic symplectic implosion. More precisely, we compute the symplectic implosion of the universal space, T * G, and show that (T * G) impl = G × N b which we regard as a stack instead of an affine scheme.
The Grothendieck-Springer correspondence has generalizations to the group and elliptic cases. These are correspondences M , where M is the Levi factor of P .
We show that all these correspondences are Lagrangian, which allows one to perform symplectic implosion in the quasi-Hamiltonian and elliptic setting. The generalization of symplectic implosion to parabolics provides an interpolation between the original unreduced space in the case P = G and the usual symplectic implosion in the case P = B.
Having established the definition of symplectic implosion in our setting, we are able to compute H-Hamiltonian reduction of the imploded space in terms of the G-Hamiltonian reduction of the original space. Let us cite the result in the Hamiltonian case.
Theorem. Let X be a G-Hamiltonian space.
The H-Hamiltonian reduction of the symplectic implosion X impl at the zero moment map value is isomorphic to the G-Hamiltonian reduction of X with respect to the G-Hamiltonian space T * (G/B), the Springer resolution of the nilpotent cone.
The H-Hamiltonian reduction of X impl at a regular semisimple moment map value λ ∈ h is isomorphic to the G-Hamiltonian reduction of X along the adjoint orbit of λ.
The discrepancy between the Hamiltonian reduction of the implosion and the original space can be explained by noting that the procedure of implosion is not invertible. However, every Lagrangian correspondence has a dual and we discuss the procedure dual to symplectic implosion (which may perhaps be named "symplectic explosion"). It is an operation that takes H-Hamiltonian spaces to G-Hamiltonian spaces. We show that the composition of the symplectic implosion and its dual is not the identity. 0.4. Structure of the paper. The paper is organized as follows. In section 1 we recall the necessary material on derived symplectic geometry from [PTVV11]. In section 2 we explain how Hamiltonian and quasi-Hamiltonian reductions can be presented as Lagrangian intersections which explains the symplectic structure. Finally, section 3 is devoted to symplectic implosion. First, we show that various versions of the Grothendieck-Springer correspondence are Lagrangian. Second, we show that the universal implosion in the Hamiltonian case coincides with the one obtained by [DKS12] in the context of hyperKähler implosion if one interprets quotients as stacky quotients instead of affine quotients. We end with a discussion of an operation dual to symplectic implosion. 0.5. Acknowledgements. The author would like to thank Frances Kirwan for useful comments and a seminar talk which prompted the writing of this paper. This research was supported by the EPSRC grant EP/I033343/1.

Derived symplectic geometry
In this section we briefly recall the necessary basics of derived differential geometry. The reader is invited to consult [PTVV11] for details and precise statements.
1.1. Symplectic structures. Let X be a derived Artin stack. Such a stack has a cotangent complex L X with the differential d. If X is, moreover, assumed to be locally of finite presentation, then the cotangent complex is perfect and we can define its dual T X = L * X , the tangent complex.
We refer to n as the degree of the differential form and p as weight. There is a de Rham differential d dR : Ω p (X, n) → Ω p+1 (X, n) and it allows one to define the complex of closed differential forms Ω p,cl (X, n). An element ω 0 + uω 1 + ... of the complex of closed differential forms is represented by a sequence of differential forms ω 0 , ω 1 , ..., where ω 0 is a p-form of degree n, ω 1 is a (p + 1)-form of degree n − 1 and so on. The differential on the complex of closed differential forms is d + ud dR and a collection (ω 0 , ω 1 , ...) of differential forms is closed if the equations dω 0 = 0 d dR ω i + dω i+1 = 0 are satisfied. A way to think of these equations is as saying that ω i is not strictly d dR -closed, but only closed up to homotopy given by ω i+1 .
Definition. An n-shifted symplectic structure on a derived stack X is a closed nondegenerate two-form ω of degree n.
We will encounter the following two basic examples of symplectic stacks.
(1) Suppose X is an derived Artin stack. Then we can define the shifted cotangent stack . In this setting one can define a Liouville one-form λ of degree n and a closed twoform ω = d dR λ of the same degree n. A local calculation ([PTVV11, Proposition 1.21]) shows that ω is an n-shifted symplectic structure if X is a derived Deligne-Mumford stack.
(2) Let G be a reductive group. Then the cotangent complex of the classifying stack BG is . It can also be checked to be closed, so BG is a 2-shifted symplectic stack.
1.2. Lagrangians. In ordinary symplectic geometry, we say that a submanifold L ⊂ X of a symplectic manifold is isotropic if the symplectic form restricts to zero on L. Since we are working in the homotopical context, the form might restrict to zero only up to homotopy. Let X be an n-shifted symplectic stack.
Definition. An isotropic structure on a morphism f : L → X is a homotopy from f * ω X to 0 in Ω 2,cl (L, n).
Thus, an isotropic structure on a morphism f : L → X (not necessarily an embedding) is a collection of differential forms h = h 0 + uh 1 + ... satisfying Unpacking this definition, we see that we must have Since h 0 is not d-closed, it does not define a morphism of complexes T L → L L [n − 1]. However, one can check that the map In other words, h and ω together define a morphism of complexes We have the following important theorem about Lagrangian intersections [PTVV11, Theorem 2.9].
Theorem 1.1. Suppose L 1 , L 2 → X are two Lagrangians into an n-shifted symplectic stack. Then their intersection L 1 × X L 2 carries an (n − 1)-shifted symplectic structure.
The symplectic structure is constructed from the following observation: both Lagrangians carry a trivialization of the pullback of the symplectic structure on X. Therefore, their intersection carries two such trivializations and their difference defines an actual closed twoform.
We will need a slight generalization of this theorem [Saf13, Theorem 1.2]. Given a symplectic stack X, we denote by X the same stack with the opposite symplectic structure.
The previous theorem can be recovered if we let X = pt with its canonical n-shifted symplectic structure. In this case a Lagrangian structure is the same as an (n − 1)-shifted symplectic structure.
1.3. Examples of Lagrangians. Let us provide some further tools to construct Lagrangians.
Let Z → X be an embedding of smooth manifolds. Then it is well-known that the correspondence The map on the left is the obvious projection and the map on the right is given by the pullback of differential forms. Let us prove an immediate generalization of the theorem to shifted cotangent stacks.
is Lagrangian.
Proof. Let us recall the construction of the symplectic structure on the shifted cotangent stack T * [n]Z. Consider the map where p : T * [n]Z → Z is the projection. By adjunction it gives a map i.e. a degree n one-form on T * [n]Z. The symplectic structure on T * [n]Z is defined to be ω = d dR λ. The proof of [PTVV11, Proposition 1.21] in fact gives the following characterization of ω : Let λ Z and λ X be the Liouville one-forms on T * [n]Z and T * [n]X respectively. The pullbacks g * λ Z and g * λ X are adjoint to and respectively, which are clearly homotopic. Indeed, for any two dualizable objects V, W of a symmetric monoidal 1-category C with a morphism V → W the diagram ) and denote the map L → T * [n]X×T * [n]Z by g. Then we need to show that the sequence is a fiber sequence. It is enough to show this factétale-locally on X and Z in which case we have a characterization of the symplectic structures on the cotangent stacks.
We denote the projection L → Z by π. Then we have fiber sequences We have to show that the first, second and fourth terms in the middle column in where Spec denotes the relative spectrum of O Z -algebras.
We have a sequence of isomorphisms Therefore, N * [n](Z/X) → T * [n]X can be obtained as a composition of the zero section Z → T * [n]Z and the Lagrangian correspondence By Theorem 1.2 this implies that the morphism itself is Lagrangian.
Remark. Both the theorem and the corollary remain true for derived Artin stacks if one replaces Lagrangian structures with isotropic structures.

Hamiltonian reduction
Let us present Hamiltonian and quasi-Hamiltonian reductions from the point of view of Lagrangian intersections. The details can be found in [Cal13] and [Saf13].
2.1. Ordinary Hamiltonian reduction. The space X = [g * /G] ∼ = T * [1](BG) has a 1shifted symplectic structure. Let us write it down explicitly. The cotangent complex of X is L X ∼ = L g * → g * ⊗ O g * viewed as a complex of G-equivariant sheaves on g * with the differential given by the coadjoint action.
The Liouville one-form λ [g * /G] on [g * /G] is given by the identity function g * → g * viewed as an element of (g * ⊗ O g * ) G . We It is a closed two-form by construction. The element ω ∈ (L g * ⊗ g * ) G can be described as follows. Given a tangent vector to g * at some point, ω regards it as an element of g * using the vector space structure on g * . The tangent complex to [g * /G] can be represented by the complex Proof. Consider the projection morphism p : g * → [g * /G]. The pullback p * λ is zero and we define the isotropic structure on p to be the zero two-form. To prove that it is Lagrangian, consider the sequence It is clearly a fiber sequence since the nontrivial maps involved are identities.
Note that the same morphism can be written as The composition of the two correspondences on the left gives the zero section Y → T * [1]Y and so the Hamiltonian reduction is the self-intersection of the zero section, hence contractible. An easy check shows that the zero isotropic structure is in fact Lagrangian. The isotropic structure gives a two-form on O which is nothing else but the Kirillov-Kostant-Souriau symplectic structure on a coadjoint orbit.
We define the Hamiltonian reduction of M with respect to G along a coadjoint orbit O to be It is again a Lagrangian intersection, so it carries a symplectic structure.
2.2. Quasi-Hamiltonian reduction. Choose a G-invariant nondegenerate bilinear form on g. It gives a 2-shifted symplectic structure on the classifying stack BG. Therefore, a self-intersection of the diagonal BG, is a Lagrangian intersection and hence it carries a natural 1-shifted symplectic structure. Here and in the future the horizontal line denotes the adjoint quotient.
In [Saf13] we showed that this 1-shifted symplectic structure on G G has the following description. The cotangent complex of viewed as a complex of G-equivariant sheaves on G.
We have a two-form of degree 1 and a three-form of degree 0 where θ and θ are the Maurer-Cartan forms. The symplectic structure on G G is given by ω 0 + uω 1 .
Theorem 2.2. The data of a G-quasi-Hamiltonian space µ : M → G is equivalent to a Lagrangian structure on The reduced space is defined to be It is again a Lagrangian intersection, so it carries a symplectic structure. More generally, a conjugacy class O ⊂ G has a Lagrangian structure and we define the quasi-Hamiltonian reduction of M with respect to G along O to be the Lagrangian intersection

Symplectic implosion
In this section G denotes a split connected reductive group over a characteristic zero field k.
3.1. Grothendieck-Springer resolution. Let B ⊂ G be a Borel subgroup and p : B ։ H the abelianization map; we denote by b and h the corresponding Lie algebras. The constructions we are about to describe can be written in a way independent of the choice of the Borel, but choose it for the sake of exposition.
We define the Grothendieck-Springer simultaneous resolution g to be the vector bundle over the flag variety G/B. We have a map g → g given by and g can be described as the space of elements x of g together with a choice of a Borel containing x.
There is a G-action on g given by the left action on G. This makes g → g into a Gequivariant map.
We also have a map g → h given by the composition using the fact that B acts trivially on h.
Combining all these maps we get a correspondence Similarly, there is a group version of the Grothendieck-Springer resolution given by where B acts on itself by conjugation. This gives a correspondence where again 3.2. Lagrangian structure. Pick a nondegenerate bilinear pairing c G ∈ Sym 2 (g * ) G . By restriction we get a bilinear pairing in Sym 2 (b * ) B . Similarly, we have a restriction map which is an isomorphism. Indeed, pick a splitting H ⊂ B and consider the H-action on b.
where V α is a one-dimensional vector space for every positive root α. To show that the map is an isomorphism, we just need to show that (e α , x) = 0 for every x ∈ b. Indeed, we have for any element h ∈ h and e β ∈ V β . But α + β is never zero, so (e α , e β ) = 0. The same argument shows that (e α , h) = 0 for any h ∈ h. The choice of c G determines a symplectic structure on [g/G] and one on [h/H]. Let us prove the following statement.
Theorem 3.1. The Grothendieck-Springer correspondence To simplify the notation, let's denote the trivial vector bundle with fiber V by V when the base space is clear.
Then the tangent complex T L is with the differential given by the adjoint action. We let h be zero, so we want to prove that the vertical map For this it is enough to prove that is an exact sequence of vector spaces. First, we have to check that the composite of the two maps is zero. This means that (x, y) g = (p(x), p(y)) h for every elements x, y ∈ b. But this is equivalent to saying that is an isomorphism, which we already know. Clearly, the sequence is exact at the first term and the third term. Moreover, h → b * is injective. The Euler characteristic of the sequence is which coincides with the dimension of the cohomology of the middle term, which is, therefore, also zero.
Similarly, the group version of the Grothendieck-Springer correspondence is also Lagrangian. To show this, we need a lemma.
is Lagrangian. Moreover, the space of Lagrangian structures is contractible.
Proof. A Lagrangian structure is given by a degree 1 two-form h on BB which is necessarily zero. Let us now check that the zero h defines a Lagrangian structure.
We have to check that In other words, we have to show that the sequence of B-representations is exact which we have already checked in the course of the proof of the previous theorem.  The proof of this theorem is identical to the proof of the AKSZ theorem [PTVV11, Theorem 2.5], so we omit it. Let us present two corollaries.
Here Bun G (E) is the moduli stack of G-bundles on an elliptic curve E equipped with a trivialization of the canonical bundle.
Both statements are obtained by applying Map(S 1 B , −) and Map(E, −) to the correspondence in Lemma 3.2.
3.3. General parabolics. Let P ⊂ G be a parabolic subgroup with p : P ։ M the Levi factor. We denote by p and m the corresponding Lie algebras. Moreover, let u be the kernel of the projection p → m.
Let us recall the following basic fact about parabolic subalgebras.
Proposition 3.5. The orthogonal complement p ⊥ coincides with the radical u.
We have a generalization of the basic correspondence to the parabolic case.
Lemma 3.6. The correspondence is Lagrangian. Moreover, the space of Lagrangian structures is contractible.
Proof. On the level of tangent complexes, we have to show that the sequence The composite of the two maps is zero if (x, y) g = (p(x), p(y)) m for every x, y ∈ p. This is true if (n, y) g = 0 for every n ∈ u and y ∈ p. But this immediately follows from Proposition 3.5.
As before, the sequence is obviously exact in the first and the third terms, so we just need to show that its Euler characteristic is zero. That is, we have to show that dim g + dim m = 2 dim p.
Since dim m = dim p − dim u, this is equivalent to Corollary 3.7. The correspondences are Lagrangian.
Finally, we have a Lie algebra version of the correspondence.
Theorem 3.8. The correspondence (1) If pt ֒→ h is the inclusion of the origin, then Theorem 3.9. The H-Hamiltonian reduction of the symplectic implosion X impl at the zero moment map value is isomorphic to the G-Hamiltonian reduction of X with respect to the G-Hamiltonian space T * (G/B).
The H-Hamiltonian reduction of X impl at a regular semisimple moment map value λ ∈ h is isomorphic to the G-Hamiltonian reduction of X along the adjoint orbit of λ.
3.5. Some generalizations. The picture of symplectic implosion admits an immediate generalization to the quasi-Hamiltonian case since we have a similar Lagrangian correspondence there as well.
Definition. The group-valued symplectic implosion of a G-quasi-Hamiltonian space X is an H-quasi-Hamiltonian space The relation between H-quasi-Hamiltonian reduction of implosion and G-quasi-Hamiltonian reduction of the original space is similar to the Lie algebra case, so let us just state the result.
Theorem 3.10. The H-quasi-Hamiltonian reduction of the symplectic implosion X qimpl at the unit moment map value is isomorphic to the G-quasi-Hamiltonian reduction of X with respect to the G-quasi-Hamiltonian space N G .
The H-quasi-Hamiltonian reduction of X qimpl at a regular semisimple moment map value λ ∈ H is isomorphic to the G-quasi-Hamiltonian reduction of X along the conjugacy class of λ.
In the theorem N G is the Springer resolution of the unipotent variety N G ⊂ G. It can be explicitly written as N G ∼ = G × B N with the B-action on N given by conjugation.
Let us also give a version of implosion with more general parabolic subgroups. We will only give it in the group-valued case, the Lie algebra case is identical. Let P ⊂ G be a parabolic subgroup with U ⊂ P the unipotent radical and M = P/U the Levi factor.
Definition. The partial group-valued symplectic implosion of a G-quasi-Hamiltonian space X is an M-quasi-Hamiltonian space For instance, if P = G, the implosion is isomorphic to X again, so partial symplectic implosions interpolate between the original quasi-Hamiltonian space X in the case P = G and the imploded space X qimpl in the case P = B, a Borel subgroup.
3.6. Universal implosion. The group G has two commuting G actions given by the left and right action, so T * G is a (G × G)-Hamiltonian space. For any G-Hamiltonian space we have (X × T * G)//G ∼ = X, so T * G acts as a kind of identity.
As both symplectic implosion and Hamiltonian reduction are fiber products, the operations commute. Therefore, we have Pulling back this morphism along g → [g/G] we get Thus G × N b is the (G × H)-Hamiltonian space satisfying In particular, it is the universal implosion There is the following modular interpretation of (T * G) impl : it parametrizes Borel subgroups B ⊂ G together with an element in Lie(B) and an element in B/ [B, B].
is also a |W | : 1 cover. But [O/G] ∼ = BH, so the H-quasi-Hamiltonian space corresponding to [ O/G] is identified with the finite set W with the moment map given by sending the whole set to λ ∈ H. As we assume G (and hence H) is connected, the action of H on W is necessarily trivial.
Theorem 3.11. Let X be an H-quasi-Hamiltonian space.
The G-quasi-Hamiltonian reduction of X dimpl along the unit coincides with the H-quasi-Hamiltonian reduction of X along the H-quasi-Hamiltonian space BN.
The G-quasi-Hamiltonian reduction of X dimpl along the conjugacy class of a regular semisimple element λ ∈ H is a |W | : 1 cover of the H-quasi-Hamiltonian reduction of X along λ.
If a Lagrangian correspondence has an inverse, it must be given by the dual correspondence. So, non-invertibility of a Lagrangian correspondence can be analyzed by computing the composition of the correspondence and its dual.
For instance, the composition of the implosion and its dual in one order gives a correspondence