Plancherel theorem for the affine symmetric space <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mi mathvariant="normal">S</mi><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1,2</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1,2</mn><mo stretchy="false">)</mo></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/DAEEB3F2-AA50-431f-860E-6A357BC5688B-M002.jpg"><?fx-imagestate width="35.56000137" height="4.57200003"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/DAEEB3F2-AA50-431f-860E-6A357BC5688B-M002c.jpg"><?fx-imagestate width="35.56000137" height="4.57200003"?></graphic></alternatives></inline-formula>

JIN Sihai; FAN Xingya

doi:10.13471/j.cnki.acta.snus.2022A091

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Plancherel theorem for the affine symmetric space

S U (1,2) / S O (1,2)

Research articles | 更新时间：2024-01-25

- Plancherel theorem for the affine symmetric space $S U (1,2) / S O (1,2)$
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 63, Issue 1, Pages: 173-180(2024)
- 作者机构：
  
  新疆大学数学与系统科学学院，新疆乌鲁木齐 830017
- 作者简介：
- 基金信息：
- DOI：10.13471/j.cnki.acta.snus.2022A091
  CLC： O152.5
- Published：25 January 2024，
  
  Published Online：15 November 2023，
  
  Received：19 October 2022，
  
  Accepted：20 February 2023
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金四海,范兴亚.仿射对称空间SU(1,2)/SO(1,2)上的Plancherel定理[J].中山大学学报(自然科学版)(中英文),2024,63(01):173-180.

JIN Sihai,FAN Xingya.Plancherel theorem for the affine symmetric space SU(1,2)/SO(1,2)[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2024,63(01):173-180.
金四海,范兴亚.仿射对称空间SU(1,2)/SO(1,2)上的Plancherel定理[J].中山大学学报(自然科学版)(中英文),2024,63(01):173-180. DOI： 10.13471/j.cnki.acta.snus.2022A091.

JIN Sihai,FAN Xingya.Plancherel theorem for the affine symmetric space SU(1,2)/SO(1,2)[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2024,63(01):173-180. DOI： 10.13471/j.cnki.acta.snus.2022A091.

Plancherel theorem for the affine symmetric space

S U (1,2) / S O (1,2)

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摘要

研究了Hilbert空间

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225030&type=

11.00666618

3.30200005

上酉表示的不可约分解，其中

<math id="M4"><mi>Z</mi><mo>=</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53224979&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53224974&type=

28.61733437

2.96333337

是Hermitian型仿射对称空间，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53224989&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53224984&type=

2.03200006

2.87866688

是群

<math id="M6"><mi mathvariant="normal">S</mi><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225000&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53224994&type=

10.92199993

2.96333337

作用在

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225011&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225004&type=

1.94733346

2.28600001

上不变的Haar测度. 利用

<math id="M8"><mi mathvariant="normal">S</mi><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225019&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225015&type=

11.09133339

3.47133350

不变的分布函数，具体的构造了缠结算子，进而得到了

</mo><mi>μ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225033&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225030&type=

11.00666618

3.30200005

上的离散序列表示. 在此基础上，结合离散序列表示的正交补部分，证明了

</mo><mi>μ</mi><mo stretchy="false">)</mo></math>

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225030&type=

11.00666618

3.30200005

上的Plancherel公式.

Abstract

The irreducible decomposition of unitary representations is investigated on the Hilbert space

</mo><mi>μ</mi><mo stretchy="false">)</mo></math>

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225098&type=

12.86933327

3.89466691

， where

<math id="M12"><mi>Z</mi><mo>=</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225051&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225045&type=

33.44333267

3.47133350

， and

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225060&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225057&type=

2.37066650

3.38666677

denotes an

<math id="M14"><mi mathvariant="normal">S</mi><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225067&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225062&type=

12.69999981

3.47133350

-invariant Haar measure on

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225076&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225071&type=

3.21733332

2.62466669

By using the

<math id="M16"><mi mathvariant="normal">S</mi><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1

2</mn><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225086&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225080&type=

12.69999981

3.47133350

-invariant distribution functions， the intertwining operators is constructed concretely， and then the discrete series representations on

</mo><mi>μ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225101&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225098&type=

12.86933327

3.89466691

are obtained. On this basis， combined with the orthogonal complement parts of the discrete series representations， the Plancherel formula on

</mo><mi>μ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225101&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=53225098&type=

12.86933327

3.89466691

is proved.

关键词

仿射对称空间离散序列表示Plancherel定理

Keywords

affine symmetric spacediscrete seriesPlancherel theorem

references

韩威，范兴亚， 2023. 关于仿射对称空间SU（2，2）/SL（2，C）+R非紧分歧离散谱的一点注记［J］. 新疆大学学报（自然科学版）（中英文）， 40（1）： 17-22+29.

钟家庆， 1989. Schur函数的一个展式及其在计数几何中的应用［J］. 中国科学（A辑），（10）： 1018-1029.

DELORME P， KNOP F， KRÖTZ B， et al， 2021. Plancherel theory for real spherical spaces： Construction of the Bernstein morphisms［J］. J Amer Math Soc， 34（3）： 815-908.

DUISTERMAAT J J， KOLK J A C， VARADARAJAN V S， 1979. Spectra of compact locally symmetric manifolds of negative curvature［J］. Invent Math， 52（1）： 27-93.

FLENSTED-JENSEN M， 1980. Discrete series for semisimple symmetric spaces［J］. Ann Math， 111（2）： 253-311.

GEL'FAND I M， GINDIKIN S G， 1977. Complex manifolds whose skeletons are semisimple real Lie groups， and analytic discrete series of representations［J］. Funct Anal Its Appl， 11（4）： 258-265.

HARISH-CHANDRA， 1965. Discrete series for semisimple Lie groups I： Construction of invariant eigendistributions［J］. Acta Math， 113： 241-318.

KRÖTZ B， KUIT J J， OPDAM E M， et al， 2020. The infinitesimal characters of discrete series for real spherical spaces［J］. Geom Funct Anal， 30（3）： 804-857.

MACDONALD I G， 2015. Symmetric functions and Hall polynomials［M］. 2th ed. New York： Oxford University Press.

'OLAFSSON G， ØRSTED B， 1988. The holomorphic discrete series for affine symmetric spaces， I［J］. J Funct Anal， 81（1）： 126-159.

ZHANG G， 1992. A weighted Plancherel formula II： The case of the ball［J］. Studia Math， 102（2）： 103-120.

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⁰

Plancherel theorem for the affine symmetric space SU(1,2)/SO(1,2)

DOI：10.13471/j.cnki.acta.snus.2022A091

摘要

Abstract

关键词

Keywords

references

Plancherel theorem for the affine symmetric space $S U (1,2) / S O (1,2)$