A stability result for balanced dictatorships in Sn

We prove that a Boolean function on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn, is close to dictator, a function which is determined by the image or pre-image of a single element. As a corollary we get a stability result concerning extremal isoperimetric sets in the Cayley graph of transpositions on Sn. Our proof works in the case where the the expectation of the function is bounded away from 0 and 1. In contrast, [6] works in the case where the Boolean function is approximated by the indicator of the union of a constant number of cosets, and hence has expectation c/n.