SCHWARTZ FUNCTIONS ON THE HEISENBERG GROUP, SPECTRAL MULTIPLIERS AND GELFAND PAIRS

Abstract. We review recent results proved jointly with B. Di Blasio and F. Astengo. On the Heisenberg group Hn, consider the two commuting self-adjoint operators L and i−1T, where L is the sublaplacian and T is the central derivative. Their joint L2-spectrum is the so-called Heisenberg fan, contained in R2. To any bounded Borel function m on the fan, we associate the operator m(L, i−1T). The main result that we describe says that the convolution kernel of m(L, i−1T) is a Schwartz function if and only if m is the restriction of a Schwartz function on R2. We point out that this result can be interpreted in terms of the spherical transform for the convolution algebra of U(n)-invariant functions on Hn. We also describe extensions to more general situations. 1