SYMPLECTIC IMPLOSION AND
THE GROTHENDIECK–SPRINGER RESOLUTION
Abstract. 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.
The goal of this paper is to introduce symplectic implosion in the realm of
derived symplectic geometry.
0.1. Derived symplectic geometry
Pantev, To¨en, Vaqui´e and Vezzosi [PTVV11] introduced the notions of closed dif-
ferential forms on derived stacks and deﬁned shifted symplectic structures on such
stacks. As in the classical context, a symplectic structure is a closed non-degenerate
two-form on the stack, but now the form can have a nontrivial cohomological de-
gree. 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 non-
degenerate 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, then L × Y → X is also
Lagrangian. Moreover, in contrast to the classical setting, being a Lagrangian is
not a property but an extra structure.
The key result about derived Lagrangians is the fact that their derived in-
tersection carries a shifted symplectic structure. More precisely, if we have two
in an n-shifted symplectic stack X, the derived intersection
is (n − 1)-shifted symplectic. More generally, if X ← L → Y is a La-
grangian correspondence and N → Y is another Lagrangian, L ×
N → X also
carries a Lagrangian structure. This should be contrasted to the case of ordinary
Received August 17, 2015. Accepted April 27, 2016.
Corresponding Author: P. Safronov, e-mail: email@example.com.
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Vol. 22, No.
, 2017, pp.