Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Abstract

Abstract. We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair (V, Q), where V is a complex vector space and Q a homogeneous polynomial of degree 4 on V . The manifold Ξ is an orbit of a covering of Conf(V, Q), the conformal group of the pair (V, Q), in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra g, and furthermore a real form g R . The connected and simply connected Lie group G R with Lie(G R ) = g R acts unitarily on a Hilbert space of holomorphic functions defined on the manifold Ξ.

Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups

“…We begin with the case |x| = 1. Then x is a primitive idempotent, and therefore 1 2 ) = (r − 1)d, and by [12, Lemma 1.6] we also have dim 1 2 ). For general x ∈ Ξ , we note that x |x| is a primitive idempotent.…”

“…Theorem A(1) and (3) assert that the intrinsic definition (0.1) of the Fock space F(X) coincides with the extrinsic definition built on the embedding X ⊂ V C . This feature is noteworthy even in the classical Fock model of the Weil representation, where V C = Sym(n, C) and X is a submanifold consisting of rank one matrices in Sym(n, C); Theorem A(3) says that any holomorphic, square integrable function on the n-dimensional complex submanifold X extends holomorphically to the 1 2 n(n + 1)-dimensional space Sym(n, C).…”

“…We further remark that on Ξ we have tr(x) = |x|, where |x| = (x|x) 1 2 is the norm on V induced by the trace form (−|−). The trace form is extended C-bilinearly to V C .…”

“…We further define a density ω(z) = K λ−1 |z| , z ∈ X, in terms of the renormalized K-Bessel function. Here, |z| = (z|z) 1 2 and λ is given by (1.12) as explained above. Denoting by O(X) the space of holomorphic functions on the complex manifold X, we then define a 'Fock space' by…”

“…In works of Brylinski and Kostant [6] and Achab and Faraut [1] models of minimal representations on spaces of holomorphic functions were constructed by using the work of Kronheimer and Vergne. Aside from the fact that their cases (non-Hermitian Lie groups) and our cases (Hermitian of tube type) are disjoint, there are one basic common point (0) and two major differences (1), (2) between their construction and our construction of the Fock model (up to a Cayley transform):…”