Eigenspaces of invariant differential operators on an affine symmetric space

Oshima, Toshio; Sekiguchi, Jiro

doi:10.1007/bf01389818

Cited by 138 publications

(94 citation statements)

References 27 publications

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“…REMARKS, (i) If r arises from a signature on Δ^ (cf. [23], see also Remark (ii) following Lemma 3.4), then Lemma 3.9 is a consequence of [23,Prop. 3.8].…”

Section: Erik P Van Den Banmentioning

confidence: 88%

A convexity theorem for semisimple symmetric spaces

Ban¹

1986

Pacific J. Math.

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“…REMARKS, (i) If r arises from a signature on Δ^ (cf. [23], see also Remark (ii) following Lemma 3.4), then Lemma 3.9 is a consequence of [23,Prop. 3.8].…”

Section: Erik P Van Den Banmentioning

confidence: 88%

A convexity theorem for semisimple symmetric spaces

Ban¹

1986

Pacific J. Math.

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“…By a similar argument as in [7] we can prove [6] for the precise assumption) there exists a w in W (j) such that P wν is an onto isomorphism. Moreover we have…”

Section: Integral Representations Of Eigenfunctionsmentioning

confidence: 84%

Fourier analysis on semisimple symmetric spaces

Oshima

1981

Lecture Notes in Mathematics

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“…The restricted root system is shown by the notation in [OS1,Appendix] such as BC m1,m2,m3 n and the Lie algebra m Ξ and its complexification for any Ξ ⊂ Ψ(a p ) can be easily read from the Satake diagram as was explained in [OS2,Appendix B]. Namely, if G is semisimple, the subdiagram corresponding to Ψ(Θ) = {α ∈ Ψ(α) ; α| ap ∈ Θ ∪ {0}} is the Satake diagram of m Ξ .…”

Section: We Similarly Define a Character Of Pθ Jθ(λ) And Mθ(λ)mentioning

confidence: 99%

Boundary Value Problems on Riemannian Symmetric Spaces of the Noncompact Type

Oshima

Shimeno

2013

Progress in Mathematics

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Abstract. We characterize the image of the Poisson transform on any distinguished boundary of a Riemannian symmetric space of the noncompact type by a system of differential equations. The system corresponds to a generator system of two-sided ideals of a universal enveloping algebra, which are explicitly given by analogues of minimal polynomials of matrices. IntroductionThe classical Poisson integral of a function on the unit circle in the complex plane gives a harmonic function on the unit disk. More generally, each eigenfunction of the Laplace-Beltrami operator on the Poincaré disk can be represented by a generalized Poisson integral of a hyperfunction on the unit circle.The notion of the Poisson integral has been generalized to a Riemannian symmetric space X = G/K of the noncompact type, where G is a connected real reductive Lie group and K is its maximal compact subgroup. The so-called Helgason conjecture states that every joint eigenfunction of the invariant differential operators on X has a Poisson integral representation by a hyperfunction on the Furstenberg boundary G/P of X, where P is a minimal parabolic subgroup of G. Helgason proved the conjecture for the Poincaré disk. Kashiwara et al. [K-] proved it generally by using the theory of hyperfunctions and the system of differential equations with regular singularities and their boundary value problem due to Kashiwara and Oshima [KO].The Poisson transform is an intertwining operator from the spherical principal series representation to the eigenspace representation. For generic parameter λ of the principal series representation, the Poisson transform P λ gives an isomorphism of the representations. The principal series representation is realized on the space of the sections of a homogeneous line bundle over G/P whose parameter is λ. If λ = ρ, the line bundle is trivial and the representation is realized on the space of functions on G/P . Then the image of P ρ consists of the harmonic functions, that is the functions that are annihilated by the invariant differential operators on the symmetric space that kill the constant functions. We call this the "harmonic case".It is natural to pose the problem of characterizing the image of P λ when the map is not bijective. An interesting case corresponds to the problem of characterizing the image of the Poisson transform from another distinguished boundary of X in one of the Satake compactifications of X (cf. [Sa], [O1, O2]

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Eigenspaces of invariant differential operators on an affine symmetric space

Cited by 138 publications

References 27 publications

A convexity theorem for semisimple symmetric spaces

A convexity theorem for semisimple symmetric spaces

Fourier analysis on semisimple symmetric spaces

Boundary Value Problems on Riemannian Symmetric Spaces of the Noncompact Type

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