Minimal representations via Bessel operators

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Minimal representations via Bessel operators. / Hilgert, Joachim; Kobayashi, Toshiyuki; Möllers, Jan.

I: Journal of the Mathematical Society of Japan, Bind 66, Nr. 2, 2014, s. 349–414.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avisTidsskriftartikelForskningpeer review

Harvard

Hilgert, J, Kobayashi, T & Möllers, J 2014, 'Minimal representations via Bessel operators', Journal of the Mathematical Society of Japan, bind 66, nr. 2, s. 349–414.

APA

Hilgert, J., Kobayashi, T., & Möllers, J. (2014). Minimal representations via Bessel operators. Journal of the Mathematical Society of Japan, 66(2), 349–414.

CBE

Hilgert J, Kobayashi T, Möllers J. 2014. Minimal representations via Bessel operators. Journal of the Mathematical Society of Japan. 66(2):349–414.

MLA

Hilgert, Joachim, Toshiyuki Kobayashi, og Jan Möllers. "Minimal representations via Bessel operators". Journal of the Mathematical Society of Japan. 2014, 66(2). 349–414.

Vancouver

Hilgert J, Kobayashi T, Möllers J. Minimal representations via Bessel operators. Journal of the Mathematical Society of Japan. 2014;66(2):349–414.

Author

Hilgert, Joachim ; Kobayashi, Toshiyuki ; Möllers, Jan. / Minimal representations via Bessel operators. I: Journal of the Mathematical Society of Japan. 2014 ; Bind 66, Nr. 2. s. 349–414.

Bibtex

@article{fe662d88612b4941b5676bbc89b7a588,
title = "Minimal representations via Bessel operators",
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 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 Schr\{"}odinger 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 (\textit{Bessel operators}) which are naturally defined in terms of the Jordan structure. ",
author = "Joachim Hilgert and Toshiyuki Kobayashi and Jan M{\"o}llers",
year = "2014",
language = "English",
volume = "66",
pages = "349–414",
journal = "Journal of the Mathematical Society of Japan",
issn = "0025-5645",
publisher = "The Mathematical Society of Japan",
number = "2",

}

RIS

TY - JOUR

T1 - Minimal representations via Bessel operators

AU - Hilgert, Joachim

AU - Kobayashi, Toshiyuki

AU - Möllers, Jan

PY - 2014

Y1 - 2014

N2 - 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 Schr\"odinger 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 (\textit{Bessel operators}) which are naturally defined in terms of the Jordan structure.

AB - 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 Schr\"odinger 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 (\textit{Bessel operators}) which are naturally defined in terms of the Jordan structure.

M3 - Journal article

VL - 66

SP - 349

EP - 414

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 2

ER -