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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.

Seewoo Lee
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    Poisson summation is about functoriality for short exact sequence Z-->R-->S^1. It says pushforward from R to S^1 (:= averaging) agrees with restriction to Z and then transforming. As such I'd guess Poisson summation should make sense when u have an exact sequence B--->A--->B* of abelian varieties, and should express compatibility of push and restrict and transform. The L2 isometry property should probably correspond just to hom spaces matching up, so that L2 norm means "cohomology". For the differentiation question i think you'd need a version with D modules to make sense of it –  Feb 25 '23 at 16:19
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    @EBz Could you elaborate? It would be great if you can add as an answer. – Seewoo Lee Feb 26 '23 at 02:53
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    Your last point about F^2 and F^4 is also due to Mukai ( and you can find a good explanation in Huybrechts' book ). If you have a polarized abelian variety there is an action of SL(2,Z), corresponding to shifting and tensoring by the polarization. Mukai proves the analogous properties you mention ( Prop 9.30 in Huybrechts , for example ). – AT0 Feb 26 '23 at 10:03
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    Related: https://mathoverflow.net/questions/97361/explaining-mukai-fourier-transforms-physically – Tom Copeland Feb 26 '23 at 11:38
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    The exchange of convolution and product of Fourier transform becomes exchange of Pontrjagin product and tensor product of Fourier-Mukai transform, see (3.7) of Mukai's "Duality between $D(X)$ and $D(\hat{X})$ with its applications to Picard sheaves". – Doug Liu Jun 02 '23 at 12:45

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