"Dual reductive pair"의 두 판 사이의 차이
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imported>Pythagoras0 (새 문서: ==introduciton== * In the mid-1970s, Howe [8] 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 cent...) |
imported>Pythagoras0 |
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| 2번째 줄: | 2번째 줄: | ||
* In the mid-1970s, Howe [8] 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. | * In the mid-1970s, Howe [8] 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: | * 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 \ | + | * (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 $E/F$. | * (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 $F$. | * (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. | + | * The dual pairs in (i) and (ii) are called Type I dual pairs, while those in (iii) are called Type II. |
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==Type II dual pairs== | ==Type II dual pairs== | ||
* It is particularly easy to describe the Weil representation $\Omega$ for 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 $\Omega$. | * 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 $\Omega$. | ||
2017년 1월 16일 (월) 23:14 판
introduciton
- In the mid-1970s, Howe [8] 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 $\Omega$.