"Dual reductive pair"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 6개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
==introduciton== | ==introduciton== | ||
| − | * In the mid-1970s, Howe | + | * In the mid-1970s, Howe introduced the notion of dual pairs in <math>Mp(W)</math>: these are subgroups of <math>Mp(W)</math> of the form <math>G \times H</math> where <math>G</math> and <math>H</math> are mutual centralisers of each other. |
* He gave a classification and construction of all such possible dual pairs. They basically take the following form: | * He gave a classification and construction of all such possible dual pairs. They basically take the following form: | ||
| − | * (i) if | + | * (i) if <math>U</math> is a quadratic space with corresponding orthogonal group <math>O(U)</math> and <math>V</math> a symplectic space with corresponding metaplectic group <math>Mp(V)</math>, then <math>W = U \otimes V</math> is naturally a symplectic space, and <math>O(U)\times Mp(V)</math> is a dual pair in <math>Mp(W) = Mp(U \otimes V)</math>. |
| − | * (ii) | + | * (ii) <math>U(V)\times U(V')</math>, where <math>V</math> and <math>V'</math> are Hermitian and skew-Hermitian spaces respectively for a quadratic extension <math>E/F</math>. |
| − | * (iii) | + | * (iii) <math>GL(U) \times GL(V)</math>, where <math>U</math> and <math>V</math> are vector spaces over <math>F</math>. |
* The dual pairs in (i) and (ii) are called Type I dual pairs, while those in (iii) are called Type II. | * The dual pairs in (i) and (ii) are called Type I dual pairs, while those in (iii) are called Type II. | ||
==Type II dual pairs== | ==Type II dual pairs== | ||
| − | * It is particularly easy to describe the Weil representation | + | * It is particularly easy to describe the Weil representation <math>\Omega</math> for Type II dual pairs. |
| − | * The group | + | * The group <math>GL(U) \times GL(V)</math> acts naturally on <math>U \otimes V</math> and hence on the space <math>S(U \otimes V)</math> of Schwarz functions: this is the Weil representation <math>\Omega</math>. |
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| + | ==related items== | ||
| + | * [[Howe duality]] | ||
| + | * [[Schur-Weyl duality for general linear groups]] | ||
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| + | [[분류:duality]] | ||
| + | [[분류:theta]] | ||
| + | [[분류:migrate]] | ||
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| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q7306388 Q7306388] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'reductive'}, {'LOWER': 'dual'}, {'LEMMA': 'pair'}] | ||
2021년 2월 17일 (수) 02:02 기준 최신판
introduciton
- In the mid-1970s, Howe introduced the notion of dual pairs in \(Mp(W)\): these are subgroups of \(Mp(W)\) of the form \(G \times H\) where \(G\) and \(H\) are mutual centralisers of each other.
- He gave a classification and construction of all such possible dual pairs. They basically take the following form:
- (i) if \(U\) is a quadratic space with corresponding orthogonal group \(O(U)\) and \(V\) a symplectic space with corresponding metaplectic group \(Mp(V)\), then \(W = U \otimes V\) is naturally a symplectic space, and \(O(U)\times Mp(V)\) is a dual pair in \(Mp(W) = Mp(U \otimes V)\).
- (ii) \(U(V)\times U(V')\), where \(V\) and \(V'\) are Hermitian and skew-Hermitian spaces respectively for a quadratic extension \(E/F\).
- (iii) \(GL(U) \times GL(V)\), where \(U\) and \(V\) are vector spaces over \(F\).
- The dual pairs in (i) and (ii) are called Type I dual pairs, while those in (iii) are called Type II.
Type II dual pairs
- It is particularly easy to describe the Weil representation \(\Omega\) for Type II dual pairs.
- The group \(GL(U) \times GL(V)\) acts naturally on \(U \otimes V\) and hence on the space \(S(U \otimes V)\) of Schwarz functions: this is the Weil representation \(\Omega\).
메타데이터
위키데이터
- ID : Q7306388
Spacy 패턴 목록
- [{'LOWER': 'reductive'}, {'LOWER': 'dual'}, {'LEMMA': 'pair'}]