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 … Show more
“…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.…”
Section: Global Uniqueness Of Small Representationsmentioning
confidence: 98%
“…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.…”
We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
“…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.…”
Section: Global Uniqueness Of Small Representationsmentioning
confidence: 98%
“…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.…”
We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
“…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.…”
Section: The Corresponding Representation Of Osp(m|2n)mentioning
confidence: 99%
“…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 .…”
Section: Introductionmentioning
confidence: 99%
“…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].…”
The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R m−2 . The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra ⊗g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in ⊗g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n) = spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R m−2|2n and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n). As a side result we obtain the proof of a conjecture of M. Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [Bull.
Minimal representations of a real reductive group G are the 'smallest' irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: small representation of a group = large symmetries in a representation space. This viewpoint serves as a driving force to interact algebraic representation theory with geometric analysis of minimal representations, yielding a rapid progress on the program. We give a brief guidance to recent works with emphasis on the Schrödinger model.
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