Integral formula of the unitary inversion operator for the minimal representation of $O(p, q)$

Kobayashi, Toshiyuki; Mano, Gen

doi:10.3792/pjaa.83.27

Cited by 16 publications

(30 citation statements)

References 13 publications

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“…In order to make the proof readable as much as possible, we collect in Appendix the formulas and the properties of various special functions used in this book. A part of the results here was announced in [45] with a sketch of proof.…”

Section: Organization Of This Bookmentioning

confidence: 99%

The Schrödinger model for the minimal representation of the indefinite orthogonal group 𝑂(𝑝,𝑞)

2011

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Section: Organization Of This Bookmentioning

confidence: 99%

The Schrödinger model for the minimal representation of the indefinite orthogonal group 𝑂(𝑝,𝑞)

2011

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“…See e.g. a survey paper [11] for the algebraic theory of 'minimal representations', and [10,[17][18][19][20][21][22] for their analytic aspects.…”

Section: Introductionmentioning

confidence: 99%

Geometric analysis on small unitary representations of GL(N,R)

Kobayashi

Ørsted²,

Pevzner³

2011

Journal of Functional Analysis

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The most degenerate unitary principal series representations π iλ,δ (λ ∈ R, δ ∈ Z/2Z) of G = GL(N, R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction π iλ,δ | H (branching law) with respect to all symmetric pairs (G, H ). For N = 2n with n 2, the restriction π iλ,δ | H remains irreducible for H = Sp(n, R) if λ = 0 and splits into two irreducible representations if λ = 0. The branching law of the restriction π iλ,δ | H is purely discrete for H = GL(n, C), consists only of continuous spectrum for H = GL(p, R) × GL(q, R) (p + q = N), and contains both discrete and continuous spectra for H = O(p, q) (p > q 1). Our emphasis is laid on geometric analysis, which arises from the restriction of 'small representations' to various subgroups.

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“…This is reflected by the fact that in L 2 -models of these representations we cannot expect geometric actions, and consequently the Lie algebra does not act by vector fields. For a general program of L 2 -models and conformal models of minimal representations of real reductive groups, we refer to [24,Chapter 1]. It should be noted that there is no known straightforward way to construct minimal representations.…”

Section: Introductionmentioning

confidence: 99%

“…The Bessel operator was originally studied for euclidean Jordan algebras (see e.g. [10]) in a different context and for V = R p,q in [24]. Using the quadratic representation P of the Jordan algebra, we define in (1.7) the Bessel operator B λ :…”

Section: Introductionmentioning

confidence: 99%

“…which traces back to [8,29,40]. Here we have renormalized the K-Bessel function as K α (z) := ( z 2 ) −α K α (z) following [24,25] and the parameter ν ∈ Z is defined in terms of the structure constants of V (see Section 1.1.3 or Table 3). In the case of V = Sym(k, R) the isotropy subgroup of the structure group on O is disconnected, and we also use the specific generator ψ − 0 (x) := (x|c 1 )…”

Section: Introductionmentioning

confidence: 99%

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Minimal representations via Bessel operators

Hilgert

Kobayashi

Möllers

2014

J. Math. Soc. Japan

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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.

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Integral formula of the unitary inversion operator for the minimal representation of $O(p, q)$

Cited by 16 publications

References 13 publications

The Schrödinger model for the minimal representation of the indefinite orthogonal group 𝑂(𝑝,𝑞)

The Schrödinger model for the minimal representation of the indefinite orthogonal group 𝑂(𝑝,𝑞)

Geometric analysis on small unitary representations of GL(N,R)

Minimal representations via Bessel operators

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