General Idea
Let $X$ and $Y$ be smooth projective varieties, and $\mathcal{E}$ a suitable object in the derived category $\mathcal{D}^b(X\times Y)$ of sheaves over $X\times Y$. There is a transformation $$ \mathcal{D}^b(X)\xrightarrow{(Rp_X)^\ast} \mathcal{D}^b(X\times Y)\xrightarrow{\cdot\otimes^L\mathcal{E}}\mathcal{D}^b(X\times Y)\xrightarrow{(Rp_Y)_\ast} \mathcal{D}^b(Y) $$ between derived categories of sheaves. $(Rp_X)_\ast$ is the derived inverse image functor and $(Rp_Y)_\ast$ ist the derived direct image functor for the coordinate projections $p_X$ and $p_Y$.
Mukai proved that if $T$ is an abelian variety and $\hat{T}$ its dual, with $\mathcal{E}$ the structure sheaf of sections of the Poincare bundle $\mathcal{P}$, then the transform $$ \mathcal{D}^b(T)\to \mathcal{D}^b(\hat{T}) $$ is an isomorphism, with inverse the map induced by the dual Fourier-Mukai transform obtained by flipping $T$ and $\hat{T}$. The isomorphism is an instance of T-duality, and is called Fourier-Mukai duality.
Baum-Connes map for free abelian groups
Let $[D]$ be the class of the $\mathbb{Z}^d$-equivariant Dirac operator on $\mathbb{R}^d$. Apply Kasparov’s descent and Fourier transform $$ j:\operatorname{KK}^{\mathbb{Z}^d}_\ast(C_0(\mathbb{R}^d),\mathbb{C}) \to \operatorname{KK}_{-d}(C(\mathbb{T}^d),C^\ast(\mathbb{Z}^d)) \cong \operatorname{KK}_{-d}(C(\mathbb{T}^d),C(\widehat{\mathbb{Z}}^d)) $$ then compose with the class of the standard Morita equivalence bimodule in $\operatorname{KK}_0(C(\mathbb{T}^d),C_0(\mathbb{R}^d)\rtimes \mathbb{Z}^d)$.
The composition of these maps sends $[D]\in \operatorname{KK}_{-d}(C_0(\mathbb{R}^d),\mathbb{C})$ to the class of the Fourier-Mukai correspondence $[\mathcal{F}_d]\in \operatorname{KK}_{-d}(\mathbb{T}^d,\widehat{\mathbb{Z}}^d)$. The Poincare bundle’s appearance is due to the Morita equivalence bimodule, which has a twisting effect.
Deformation quantization
Fourier-Mukai equivalence is an instance of derived equivalences of algebraic varieties.
Deformations of a scheme in a particular direction may correspond to deformations to a rather on the other side of a derived equivalence, typically twisted or noncommutative spaces. The deformed derived equivalences often give an interpretation of the deformations of the moduli spaces as moduli spaces in their own right…
See
- Oren Ben-Bassat, Jonathan Block, Tony Pantev. Non-commutative tori and Fourier-Mukai duality https://arxiv.org/abs/math/0509161
D.Arinkin, J.Block, T.Pantev. $\ast$-Quantizations of Fourier-Mukai transforms. https://arxiv.org/abs/1101.0626