Weyl quantization of exponential lie groups for square integrable representations

We construct a general quantization procedure for square integrable functions on well-behaved connected exponential Lie groups. The Lie groups in question will act as the phase space, and should admit at least one open co-adjoint orbit in the topology of $$\mathfrak {g}^*$$ . The construction is based on composing the Fourier-Wigner transform with another Fourier transform we call the Fourier-Kirillov transform. This quantization has many desirable properties including respecting function translations, inducing an isomorphism from square integrable functions to Hilbert-Schmidt operators, and inducing a well-behaved Wigner distribution. Moreover, we investigate the connection to the operator convolutions of quantum harmonic analysis. This is intricately connected to Weyl quantization in the Weyl-Heisenberg setting. We find that convolution relations in quantum harmonic analysis can be written as group convolutions of Weyl quantizations. This implies that the squared modulus of the wavelet transform of the representation can be written as a convolution between two Wigner distributions. Lastly, we look at how we can extend known results based on Weyl quantization to wider classes of groups using our quantization procedure.