Original Laumon's local Fourier transform for l-adic sheaves:
Laumon, G.,
Transformation de Fourier, constantes d'equations fonctionnelles et conjecture de Weil,
Publ. Math. IHES, 65 (1987), pp. 131-210.
pdf
Exposition of the previous paper:
N. Katz: Travaux de Laumon
pdf
Basic definitions of local Fourier transform and stationary phase for D-modules:
B. Malgrange, Sur les points singuliers des équations différentielles
(1971-1972), Exp. No 20-22.
1
(pdf),
2
(pdf),
3
(pdf)
B. Malgrange, Modules microdiff´erentielles et classes de Gevrey
Advances in Mathematics supplementary Studies, vol.7B (1981), 513-530.
B. Malgrange, Equations diffeerentielles `a coefficients polynomiaux
Progress in Mathematics, vol. 96. Birkh¨auser, 1991.
R. Garcia Lopez. Microlocalization and stationary phase
Asian J. Math., 8(4):747-768, 2004.
arxiv
pdf
S. Bloch, H. Esnault: Local Fourier transforms and rigidity for D-modules
arxiv
pdf
D. Arinkin: Fourier transform and middle convolution for irregular D-modules
arxiv
pdf
More advanced topics on lFT
For l-adic sheaves
L. Fu: Calculation of l-adic local Fourier transformations
arxiv
pdf
T. Saito and A. Abbes: Local Fourier transform and epsilon factors
arxiv
pdf
For D-modules
C. Sabbah: An explicit stationary phase formula
for the local formal Fourier-Laplace transform (formal)
arxiv
pdf
T. Mochizuki: Notes on the Stokes structure of Fourier transform
pdf
Also I'd like to hear on Gross-Frenkel's work 0901.2163
About the twist:
B. Dubrovin: On almost duality for Frobenius manifolds (chapter 4)
arxiv
pdf
A. Givental