Homogeneous distributions on the Heisenberg group and representations of 𝑆𝑈(2,1)

Fabec, R. C.

doi:10.1090/S0002-9947-1991-1043858-7

Homogeneous distributions on the Heisenberg group and representations of $\textrm {SU}(2,1)$
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by R. C. Fabec

Trans. Amer. Math. Soc. 328 (1991), 351-391

DOI: https://doi.org/10.1090/S0002-9947-1991-1043858-7

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Abstract:

A ’Fourier’ transform of tempered distributions on the Heisenberg group is defined to analyze homogeneous distributions relative the group of dilations $(z,t) \mapsto (r z,{r^2}t)$, $r \in {\mathbf {R}}$. An inversion formula is derived for the abelian central Fourier transform of the distribution. These formulas are applied to the family of homogeneous distributions defining the intertwining operators for the group ${\text {SU}}(2,1)$. Explicit unitary structures are determined on subquotient representations and their spectral decompositions on the minimal parabolic subgroup are obtained.

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