Classification of minimal representations of real simple Lie groups

Abstract

Based on an idea in [Gan-Savin, Represent. Theory (2005)], we give a classification of minimal representations of connected simple real Lie groups not of type A. Actually, we prove that there exist no new minimal representations up to infinitesimal equivalence. Mathematics Subject Classification 2010: 22E46.

Multiplicity in restricting minimal representations

“…The solution space of the Yamabe Laplacian on S p × S q gives the minimal representation of the conformal transformation group O(p + 1, q + 1) when p + q (≥ 6) is even ( [20]). In general, there are at most four minimal representations for each connected simple Lie group G if exist, and they were classified [4,30].…”

“…Let (U(g C )/J ) g ′ C be the algebra of g ′ C -invariant elements in U(g C )/J via the adjoint action. Then one has (U(g C )/J ) g ′ C = C if one of (therefore, all of) the equivalent conditions in Proposition 30 is satisfied, see [30,Lem. 3.4].…”

Conformally invariant differential operators on Heisenberg groups and minimal representations

“…In view of the classification of minimal representations in [52], the construction in Theorem E together with the constructions in [25], [44] and [48] yield L 2 -models for all minimal representations except for the ones of F 4(4) and the complex groups E 8 (C) and F 4 (C). We believe that the case of F 4(4) can be treated with a slight generalization of our methods to the vector-valued case (see Remark 5.2.2).…”

Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules associated to the ideals of minimal nilpotent orbits for simple Lie groups of type $A$

“…The proof of exhaustion uses the facts that highest weights of k-types of a minimal (g, k)-module lie in some specified lattice, and that a-minimal (g, k)-modules are isomorphic to each other if they have a common k-type (see Proposition 3.3). The argument is the same as the classification when g 0 is not of type A [18]. We also need to show the injectivity of some intertwining differential operator for the nonexistence of minimal (sl(3, C), so(3, R))-modules with some k-types (see Proposition 6.1).…”

“…Minimal representations are classified and are known to be infinitesimally equivalent to unitary representations [18,Corollary 5.1]. In the Kirillov-Kostant orbit philosophy, they are considered to be attached to G-orbits in O min ∩ g 0 , and are considered to be a part of building blocks of the unitary dual of G.…”