Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. Call a relation $U \to V$ a (linear) Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus V$, where $\overline U$ is the conjugate symplectic vector space $(U,-\omega)$.


These linear Lagrangian relations have the property that the composite (as relations) of two Lagrangian relations is again a Lagrangian relation. As any Lagrangian subspace of some space $U$ can be considered a Lagrangian relation $0 \to V$, this implies that Lagrangian relations map Lagrangian subspaces to Lagrangian subspaces. Does the converse hold?


*That is, if a relation maps Lagrangian subspaces to Lagrangian subspaces, is it a linear Lagrangian relation?*


EDIT: Nate Bottman provides an easy counterexample below. A refinement of this question to get closer to the intuition guiding it is to ask whether a relation that both preserves and reflects Lagrangian subspaces is Lagrangian. Any tips on this would be appreciated.


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Less precisely, in the category of symplectic vector spaces and linear Lagrangian relations, Lagrangian subspaces are thus the (monoidal unit-valued) points of each object. What can be said about Lagrangian relations as functions on this set of points, and is this a useful perspective? Is there any literature on this?


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(Cross-posted from math.stackexchange: <https://math.stackexchange.com/questions/917808/if-a-linear-relation-maps-lagrangian-subspaces-to-lagrangian-subspaces-is-it>.)



If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation?
Counterexample: set $U$ to be anything, $V := \text{pt}$, and $\Lambda \subset \overline{U} \oplus \text{pt}$ to be a non-Lagrangian subspace.


(But maybe true with some hypotheses, e.g. $\Lambda$ induces an injection on Lagrangian subspaces? Hmm, a linear symplectic analogue of Orlov's theorem...)


