Analysis of the Brylinski-Kostant model for spherical minimal representations (Q2903303)

From the introduction: In a series of papers [\textit{R. Brylinski}, Can. J. Math. 49, No. 5, 916--943 (1997; Zbl 0907.58021)], [\textit{R. Brylinski} and \textit{B. Kostant}, Proc. Natl. Acad. Sci. USA 91, No. 13, 6026--6029 (1994; Zbl 0803.58023), and Prog. Math. 131, 13--63 (1995; Zbl 0851.22017)] Brylinski and Kostant studied the geometric quantization of minimal nilpotent orbits for simple real Lie Groups that are not of Hermitian type. They constructed the associated irreducible unitary representations on a Hilbert space of half forms on the minimal nilpotent orbit. This can be considered as a Fock model of the minimal representation. In this paper the authors revisit this construction. They start from a pair \((V,Q) \), where \(V \) is a complex vector space and \(Q \) a homogeneous polynomial on \(V \) of degree 4. The structure group \(\text{Str}(V,Q) \) is assumed to have a symmetric open orbit. The main geometric orbit is the orbit \(\Xi \) of \(Q \) under \(K \), a covering of the conformal group \(\text{Conf}(V,Q) \), on a space \(\mathcal W \) of polynomials on \(V \).NEWLINENEWLINEBy a generalized Kantor-Koecher-Tits construction, starting from the Lie algebra \(\mathfrak k \) of \(K \) the authors obtain a simple Lie algebra \(\mathfrak g \), such that the pair \((\mathfrak g,\mathfrak k) \) is non-Hermitian. As a vector space \(\mathfrak g=\mathfrak k\oplus \mathfrak p \), where \(\mathfrak p=\mathcal W \). The main problem here is to define a bracket \(\mathfrak p \oplus \mathfrak p\to\mathfrak k\) such that \(\mathfrak g \) becomes a Lie algebra. Then a representation \(\rho\) of \(\mathfrak g \) is defined on the space \(\mathcal O(\Xi)_{\text{fin}} \) of polynomial functions on \(\mathcal W \). In a first step one defines a representation of an \(\mathfrak s\mathfrak l_2 \)-triple \((E,F,H) \). It turns out that this is possible only under a certain condition \((T) \), which is satisfied by the pairs \(\mathfrak s\mathfrak l(n, \mathbb R), (\mathfrak s\mathfrak o(p,p), \mathfrak s\mathfrak (p))\oplus \mathfrak s\mathfrak o(p), \) and by the exceptional pairs \((\mathfrak e_{6,(6)}, \mathfrak s\mathfrak p(8)), (\mathfrak e_{7(7)}, \mathfrak s\mathfrak u(8)), (\mathfrak e_{8(8)}, \mathfrak s\mathfrak o(16)). \) In these cases the authors obtain a spherical irreducible unitary representation of the connected simply connected group \(\widetilde {G_\mathbb R} \), whose Lie algebra is a real form of \(\mathfrak g \). It is realized on a space of holomorphic functions on \(\Xi\). This Hilbert space is a weighted Bergman space with a weight taking in general both positive and negative values. If \(Q=R^2 \) or \(Q=R^4 \), where \(R \) is semi-invariant, then one can obtain 1 or 3 other (non-spherical) representations of \(\widetilde{G_\mathbb R} \).