Abstract
It is well known that ( H n ⋊ U ( n ) , U ( n ) ) $(\mathbb {H}^n\rtimes U(n),U(n))$ is a Gelfand pair, an exact analogue of the Heisenberg group result due to Narayanan and Ratnakumar is not possible for the Heisenberg motion group. We prove that if an integrable function on the Heisenberg motion group is supported on a set of finite measure, and its Weyl transform is non-zero only for finitely many Fourier-Wigner pieces and have finite rank, then the function must be zero. In the end, a quantitative interpretation of this result is described through strong annihilating pair for the Weyl transform.
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