Consider the ring of integer Laurent polynomials $\Lambda = \mathbb Z[t^\pm]$ equipped with the involutive automorpshim $\overline{(-)} :\Lambda \to \Lambda$ given by $\overline{p(t)} = p(1/t)$. I have a free module $M$ over $\Lambda$ and an antisymmetric sesquilinear form $\hat{\iota}:\overline{M} \otimes M \to \Lambda$. (This means that $\hat{\iota}$ is additive separately in both arguments and that $\hat{\iota}(p(t)x,q(t)y)=p(t)q(1/t)\cdot \hat{\iota}(x,y)$ for all $x,y \in M$.) The form is also non-degenerate.
I'm wondering if there is any reasonable way to define the "angle" between two elements of $M$ with respect to the form $\hat{\iota}$. You can certainly have "orthogonal" vectors when $\hat{\iota}(x,y)=0$ and maybe you can define the "norm" of a vector (and so tell when two vectors are parallel). Ultimately, I'm really curious about trying to define a "distance" between submodules of $M$ (specifically rank 2 submodules) and so I thought I could use some analog of "principal angles" as one uses to define the Grassmann distance between subspaces of $\mathbb{R}^n$.
Feel free to suggest whatever modifications would make this easier to work with. E.g., we can multiply $\hat{\iota}$ by $(t-1/t)$ so obtain a symmetric sesquilinear form (then this is more analogous to an inner product on $\mathbb{R}^n$ or a Hermitian form on $\mathbb{C}^n$). Or we can change the coefficients to a field of rational functions $\mathbb{C}(t)$. Etc.