Minimal representations via Bessel operators

Abstract

We construct an L 2 -model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V ) of simple Jordan algebras V . If V is split and G is not of type A n , then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n, 1) 0 . A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand-Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schrödinger models in L 2 -spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.

Global uniqueness of small representations

“…A representation of a real groups is minimal if the annihilator in U (g) is the Joseph ideal. For the groups considered in this paper, Theorems A and B in [3] imply that the minimal representations satisfy the conditions of Proposition 8.3. In turn, Proposition 8.3 implies that the minimal representations satisfy the conditions of Theorem 10.1.…”

“…If G = Sp 2r (k), then small representations appear naturally in the stable range of theta correspondences, see [4]. For more general G, we have works of [3,7,15], for real groups, and works of [18] and [19] for p-adic groups.…”

Joseph Ideals and Harmonic Analysis for

“…The second equation trivially holds, see equation (18). The left-hand side of the first equation represents D X for a general tensor X ∈ g g. First we exclude the case M = m − 2n = 0.…”

“…This ideal subsequently plays an essential role in the description of the symmetries of the Laplace operator, for which its kernel is exactly this minimal representation, see [11]. The minimal representation for sp(2n) is known as the metaplectic representation, Segal-Shale-Weil representation or the symplectic spinors, see [18,24,31]. This is a representation of sp(2n) on functions on R n .…”

“…In particular it would also be of interest to construct general theories as in [1,2,18,26] for Lie superalgebras. For instance there seems to exist an interesting link with (co)adjoint orbits, see [29].…”