I have recently been trying to learn some basic symplectic geometry, and I have come across two sesquilinear forms which are closely related to the symplectic form.
Fix $\mathbb K$ to be a field, and $V$ to be a $\mathbb{K}$-vector space equipped with a symplectic form $\omega(-,=):V^*\oplus V\to \mathbb{K}$.
When $\mathbb{K}=\mathbb{C}$ there is a Hermitian form on $V$ given by $(v,w)\mapsto i\omega\left(\overline{ v}, w\right) $. I have heard this referred to as a Kähler form, or Kähler structure, when used in geometrical quantization.
On the other hand, in the field of quantum error correction: fixing $\mathbb{K}=\mathbb{F}_p((x))$ to be the field of formal Laurent series over a finite field, from what I gather, a quantum convolutional code is isotropic with respect to the sesquilinear form on $V$ given by $(f(x),g(x))\mapsto \omega (f(x), g(1/x))$.
To me, the similarity is quite striking at a superficial level. Given a symplectic vector space and a distinguished automorphism, you can conjugate the symplectic form by the automorphism to obtain a sesquilinear form.
Therefore, my question is the following: first, is there a general theory of sesquilinear forms which arise in this way from symplectic forms? Second, is the second sesquilinear induced by the automorphism $x \mapsto 1/x$ in $\mathbb{K}((x))$ known more broadly outside of quantum error correction by a particular name?