Mukai transform gives a derived equivalence between the (bounded) derived category of coherent sheaves $D^b_{\mathrm{coh}}(A)$ of abelian variety $A$ and that of dual $A^\vee$, $D_{\mathrm{coh}}^{b}(A^\vee)$. This is called as Fourier-Mukai tranform because of its resemblance with the classical Fourier transform. For example, it is defined via Poincare bundle on $A \times A^\vee$, which is somewhat analogous object to the kernel function $e^{2\pi i x y}$ of the Fourier transform. Also, it sends a skyscraper sheaf $\mathcal{S}_{\mathcal{L}}$ of $A^\vee$ supported at $\mathcal{L}$ to $\mathcal{L}$ on $A$, and this corresponds to the fact that delta function $\delta_a(x)$ maps to the exponential function $y\mapsto \exp(2\pi i ay)$ via Fourier transform. At last, classical Fourier transform converts convolution as a product ($\mathcal{F}(f*g) = \mathcal{F}(f)\cdot\mathcal{F}(g)$, and (maybe) this corresponds to the fact that the Mukai transform preserves exterior tensor product (here).
I wonder if there are more analogous properties that correspond to some of the properties of classical Fourier transform. For example, here is a list of properties that I want to seek the correspondents:
- Differentiation and multiplication: $\mathcal{F}(f^{(n)})(y) = (2\pi i y)^{n}\mathcal{F}(f)(y)$. (what kind of operation corresponds to differentiation / power of the identity function $f(x) = x$?)
- Plancherel formula: $\mathcal{F}$ is an isometry, i.e. it preserves $L^2$ norm. (what would be the correct analogue of $L^2$-norm for line bundles?)
- Poisson summation formula: $\sum_{n\in \mathbb{Z}}f(n) = \sum_{n\in\mathbb{Z}}\mathcal{F}(f)(n)$. (summation over integer may correspond to doing something over discrete subset of $A$ like torsion points...?)
- $\mathcal{F}^{2}(f)(x) = f(-x)$ and $\mathcal{F}^{4} = \mathrm{id}$.
And if $A$ is self-dual (e.g. elliptic curve),
- Eigenfunctions: the eigenfunctions of Fourier transform can be expressed in terms of Hermite polynomials.
I was not able to find appropriate analogues for these properties. Please let me know if there are any related results to these statements for Mukai transform (any one of these - I think I'm asking multiple questions at once). Thanks in advance.
