The \(\text{SO}(4,4)\) minimal representation and the Rac representation of \(\text{SO}_0(2,3)^\sim\) (Q2711109)

The author describes geometric realizations of two irreducible unitary representations: the minimal representation of \(SO_0(4,4)\) described by \textit{B. Kostant} in [``Differential geometric methods in theoretical physics'', ed. K. Bleuler and M. Werner, NATO ASI Ser., Ser. C 250, 65-109 (1988; Zbl 0663.22009)], and the Rac representation, which is the spin zero singleton representation of the simply connected cover of the identity component \(SO_0(3,2)\) of \(SO(3,2)\). The Rac is realized as the harmonic component of the space of smooth sections of a certain line bundle over the four-fold cover of a projective quadric in projective four space; the geometric realization of the first representation is quite analogous. NEWLINENEWLINENEWLINEThen the author computes the decomposition into irreducibles of the Rac, when restricted to \(SO_0(1,2)\). Like the restriction of the minimal representation of \(SO_0(4,4)\) to \(SO_0(2,4)\) (this is unpublished work by Vogan), the Rac decomposes into discrete series representations of \(SO_0(1,2)\).NEWLINENEWLINEFor the entire collection see [Zbl 0944.00084].