Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal basis whose sections lift to theta functions on the complex plane.
My question is, what if we try to compare the Hilbert spaces obtained for different $\tau$? One method would be to look at the Hitchin connection. However, if we instead tried to use the Blattner-Kostant-Souriau (BKS) pairing between polarizations, would this just amount to doing a Fourier transform of sorts? In particular will this just give us something proportional to the Hermite-Jacobi action?