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Let F be a non-archimedean field, let g = gl ( n , F ) and let k be the lattice of matrices whose coefficients are integers. Consider a function f on g whose orbital integrals are supported on elements with integral eigenvalues and such that the Fourier transform of f has the same property. Then the orbital integrals of f are equal to the orbital integrals of the characteristic function of k , up to a constant. [ABSTRACT FROM AUTHOR]