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Zbl 0729.22023
Schmidt, Martin U.
Lowest weight representations of some infinite dimensional groups on Fock spaces.
(English)
[J] Acta Appl. Math. 18, No.1, 59-84 (1990). ISSN 0167-8019; ISSN 1572-9036/e

If V and W are two separable Hilbert spaces, define $U(V,W)$ to be the subgroup of $GL(V\oplus W)$, the group of all invertible bounded operators of the Hilbert space $V\oplus W$ which leave invariant the Hermitian form defined by the operator $J=\pmatrix 1&0\\0&-1 \endpmatrix$ on $V\oplus W.$ \par Let $U\sb{res}(V,W)$ be the subgroup of $U(V,W)$ consisting of those elements $\pmatrix a&b\\b&d \endpmatrix\in U(V,W)$ for which $b: W\to V$ and $c: V\to W$ are Hilbert-Schmidt operators. If $A\sb i=\pmatrix a\sb i&b\sb i \\ c\sb i&d\sb i \endpmatrix$, $(i=1,2,3)$, $A\sb 3=A\sb 1A\sb 2$, are in $U\sb{res}(V,W)$ it turns out that $c(A\sb 1,A\sb 2)=\det\sp{- 1}(d\sb 1\sp{-1}d\sb 3d\sb 2\sp{-1})is$ a co-cycle, which induces a central extension of $U\sb{res}(V,W)$, $\tilde U\sb{res}(V,W)=\{\pmatrix a&b\\ c&d \endpmatrix,z\}$, with $z\bar z=\det (1-d\sp{*\sp{- 1}}b\sp*bd\sp{-1}).$ \par The author determines the unitary lowest weight representations of $\tilde U\sb{res}(V,W)$ namely the irreducible components of the k-fold tensor product of the Segal-Shale-Weil representation [see e.g. {\it G. B. Segal}, Commun. Math. Phys. 80, 301-342 (1981; Zbl 0495.22017)]. \par Let H be a Hilbert space, let $S\sp n(H)$ denote the Hilbert space completion of the vector space of nth powers of the symmetric algebra of H with the Hermitian form $$<h\sb 1h\sb 2...h\sb n,h'\sb 1h'\sb 2...h'\sb n>=\sum\sb{n}\prod\sp{n}\sb{i=1}<h\sb{\sigma (i)}\cdot h'\sb i>$$ where $\sigma$ runs through all permutations of 1,2,...,n. Define $S(H)=\oplus S\sp n(H)$ endowed with the final topology defined by the inclusions $i\sb n: S\sp n(H)\to S(H)$, and $\check{S}(H)$ the Hilbert space completion of S(H). Let $\hat S(H)=\prod\sb{n}S\sp n(H)$ endowed with the initial topology defined by the projections $p\sb n: \hat S(H)\to S\sp n(H)$. $\check{S}(H)$ and $\hat S(H)$ are the antilinear dual spaces of each other and $\check{S}(H)\subseteq S(H)\subseteq \hat S(H)$ with continuous and dense inclusions. $d\Gamma$ is the canonical representation of the Lie algebra L(H) on S(H). By means of $d\Gamma$, a representation $d{\tilde\Gamma}$ of $L\sb{res}(V,W)$ is constructed. Herein L(H) is the Lie algebra of bounded operators of H and C(H) the subalgebra of all compact operators of H; and $L\sb{res}(V,W)$ denotes the subalgebra of $L(V\oplus W)$ consisting of elements of the form $\pmatrix a&b \\ b&d \endpmatrix$, $b: W\to V$, $c: V\to W$, b and c being Hilbert-Schmidt operators. For $A\sb 1$, $A\sb 2$ in $L\sb{res}(V,W)$, $A\sb i=\pmatrix a\sb i&b\sb i \\ c\sb i&d\sb i \endpmatrix$, $i=1,2$, the co-cycle $(A\sb 1,A\sb 2)=tr.(c\sb 2b\sb 1- c\sb 1b\sb 2)$ induces a central extension $\tilde L\sb{res}(V,W)$ of $L\sb{res}(V,W).$ \par The author proves (1.6 Theorem, p. 67): The representation of $L\sb{res}(V,W)$ can be lifted to a unitary representation of $U\sb{res}(V,W)$ on S(V$\oplus \bar W)$. In the case when V and W are finite-dimensional, Kashiwara and Vergne showed that this representation is completely reducible and determined all the minimal weight vectors. By methods which are purely algebraic, the author generalizes their results to the infinite-dimensional case. \par In Section 2 these questions are answered for the case of the symplectic group Sp(V), V a complex Hilbert space. The decomposition of the tensor products of the Segal-Shale-Weil representation of the metaplectic group is obtained.
MSC 2000:
*22E65 Infinite-dimensional Lie groups
22E70 Appl. of Lie groups to physics

Keywords: separable Hilbert spaces; invertible bounded operators; Hilbert-Schmidt operators; unitary lowest weight representations; Segal-Shale-Weil representation; Lie algebra; completely reducible; minimal weight vectors; symplectic group; tensor products; metaplectic group

Citations: Zbl 0495.22017

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