On the space of $ K$-finite solutions to intertwining differential operators

Toshihisa Kubo; Bent Orsted

doi:10.1090/ert/527

On the space of $ K$-finite solutions to intertwining differential operators

Toshihisa Kubo^*, Bent Orsted

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

4 Citations (Scopus)

Abstract

In this paper we give Peter-Weyl-type decomposition theorems for the space of K-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of SL(3,ℝ) attached to the minimal nilpotent orbit.

Original language	English
Journal	Representation Theory
Volume	23
Issue	7
Pages (from-to)	213-248
Number of pages	36
ISSN	1088-4165
DOIs	https://doi.org/10.1090/ert/527
Publication status	Published - Sept 2019

Keywords

CONFORMALLY INVARIANT-SYSTEMS
HEISENBERG ULTRAHYPERBOLIC EQUATION
Intertwining differential operators
K-finite solutions
MINIMAL REPRESENTATION
O(P
Peter-Weyl-type formulas
SINGULAR REPRESENTATION
SL(3
Torasso's representation
UNITARY REPRESENTATIONS
duality theorem
generalized Verma modules
hypergeometric polynomials
small representations

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10.1090/ert/527

https://arxiv.org/pdf/1808.10042.pdf

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title = "On the space of $ K$-finite solutions to intertwining differential operators",

abstract = "In this paper we give Peter-Weyl-type decomposition theorems for the space of K-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of SL(3,ℝ) attached to the minimal nilpotent orbit.",

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author = "Toshihisa Kubo and Bent Orsted",

year = "2019",

month = sep,

doi = "10.1090/ert/527",

language = "English",

volume = "23",

pages = "213--248",

journal = "Representation Theory",

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publisher = "American Mathematical Society",

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T1 - On the space of $ K$-finite solutions to intertwining differential operators

AU - Kubo, Toshihisa

AU - Orsted, Bent

PY - 2019/9

Y1 - 2019/9

N2 - In this paper we give Peter-Weyl-type decomposition theorems for the space of K-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of SL(3,ℝ) attached to the minimal nilpotent orbit.

AB - In this paper we give Peter-Weyl-type decomposition theorems for the space of K-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of SL(3,ℝ) attached to the minimal nilpotent orbit.

KW - CONFORMALLY INVARIANT-SYSTEMS

KW - HEISENBERG ULTRAHYPERBOLIC EQUATION

KW - Intertwining differential operators

KW - K-finite solutions

KW - MINIMAL REPRESENTATION

KW - O(P

KW - Peter-Weyl-type formulas

KW - SINGULAR REPRESENTATION

KW - SL(3

KW - Torasso's representation

KW - UNITARY REPRESENTATIONS

KW - duality theorem

KW - generalized Verma modules

KW - hypergeometric polynomials

KW - small representations

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U2 - 10.1090/ert/527

DO - 10.1090/ert/527

M3 - Journal article

SN - 1088-4165

VL - 23

SP - 213

EP - 248

JO - Representation Theory

JF - Representation Theory

IS - 7

ER -

On the space of $ K$-finite solutions to intertwining differential operators

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Keywords

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