Super Howe duality and rigidity of hybrid Fourier transforms

We characterize the hybrid Fourier transform on the Schwartz--Grassmann space (m) Λ 2n by operator intertwining relations. Using the bosonic oscillator Howe decomposition and the fermionic Lefschetz decomposition, we describe the action of the hybrid Fourier transform on both tensor factors. We show that first-order covariance with respect to coordinate multiplication and differentiation determines the transform up to a scalar. For m,n 1, we prove that covariance under all even quadratic operators, together with any one nonzero mixed quadratic operator, determines the linear span of the hybrid Fourier transform and its inverse. This rigidity result is obtained by computing the ordinary commutants of the even and full quadratic oscillator actions of (4n2m). After a degree-scaling change of normalization, we make the hybrid Fourier transform unitary with respect to the occupation number inner product, and factor it into a bosonic oscillator rotation and (up to a scalar phase) an SU(2) rotation in the fermionic Lefschetz representation. For m=1 and n 2, we show that full even quadratic covariance together with the parity-preserving cubic relation for qQ f determines the unitary transform up to a global phase and a fermion-parity phase. Finally we calculate the unitary conjugation formulas for Hubbard--Holstein Hamiltonian for a hybrid continuous-variable and discrete-variable quantum system.