Fourier transform as a triangular matrix

Let V be a vector space of dimension 2n over the field F_2 with two elements. Assume that we are given a nondegenerate symplectic form on V with values in F_2. Let X be the vector space of complex valued functions on V. Fourier transform is an involution of X.  We show that there exists an interesting basis of X in which Fourier transform acts by a triangular matrix. This follows the tradition of Hermite who showed that Fourier transform on the real line can be diagonalized.

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