Lowest weight representations of some infinite dimensional groups on Fock spaces. (English) Zbl 0729.22023

Schmidt, Martin U.

Acta Appl. Math. 18, No. 1, 59-84 (1990).

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 \(J=\begin{pmatrix} 1&0\\0&-1 \end{pmatrix}\) on \(V\oplus W.\)
Let \(U_{res}(V,W)\) be the subgroup of \(U(V,W)\) consisting of those elements \(\begin{pmatrix} a&b\\b&d \end{pmatrix}\in U(V,W)\) for which \(b: W\to V\) and \(c: V\to W\) are Hilbert-Schmidt operators. If \(A_ i=\begin{pmatrix} a_ i&b_ i \\ c_ i&d_ i \end{pmatrix}\), \((i=1,2,3)\), \(A_ 3=A_ 1A_ 2\), are in \(U_{res}(V,W)\) it turns out that \(c(A_ 1,A_ 2)=\det^{- 1}(d_ 1^{-1}d_ 3d_ 2^{-1})is\) a co-cycle, which induces a central extension of \(U_{res}(V,W)\), \(\tilde U_{res}(V,W)=\{\begin{pmatrix} a&b\\ c&d \end{pmatrix},z\}\), with \(z\bar z=\det (1-d^{*^{- 1}}b^*bd^{-1}).\)
The author determines the unitary lowest weight representations of \(\tilde U_{res}(V,W)\) namely the irreducible components of the k-fold tensor product of the Segal-Shale-Weil representation [see e.g. G. B. Segal, Commun. Math. Phys. 80, 301-342 (1981; Zbl 0495.22017)].
Let H be a Hilbert space, let \(S^ n(H)\) denote the Hilbert space completion of the vector space of nth powers of the symmetric algebra of H with the Hermitian form \[ <h_ 1h_ 2...h_ n,h'_ 1h'_ 2...h'_ n>=\sum_{n}\prod^{n}_{i=1}<h_{\sigma (i)}\cdot h'_ i> \] where \(\sigma\) runs through all permutations of 1,2,...,n. Define \(S(H)=\oplus S^ n(H)\) endowed with the final topology defined by the inclusions \(i_ n: S^ n(H)\to S(H)\), and \(\check{S}(H)\) the Hilbert space completion of S(H). Let \(\hat S(H)=\prod_{n}S^ n(H)\) endowed with the initial topology defined by the projections \(p_ n: \hat S(H)\to S^ n(H)\). \(\check{S}(H)\) 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 \(L_{res}(V,W)\) 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 \(L_{res}(V,W)\) denotes the subalgebra of \(L(V\oplus W)\) consisting of elements of the form \(\begin{pmatrix} a&b \\ b&d \end{pmatrix}\), \(b: W\to V\), \(c: V\to W\), b and c being Hilbert-Schmidt operators. For \(A_ 1\), \(A_ 2\) in \(L_{res}(V,W)\), \(A_ i=\begin{pmatrix} a_ i&b_ i \\ c_ i&d_ i \end{pmatrix}\), \(i=1,2\), the co-cycle \((A_ 1,A_ 2)=tr.(c_ 2b_ 1- c_ 1b_ 2)\) induces a central extension \(\tilde L_{res}(V,W)\) of \(L_{res}(V,W).\)
The author proves (1.6 Theorem, p. 67): The representation of \(L_{res}(V,W)\) can be lifted to a unitary representation of \(U_{res}(V,W)\) 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.
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.

Reviewer: T.S.Bhanu Murthy (Madras)

MSC:

22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties
22E70	Applications of Lie groups to the sciences; explicit representations

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

References:

[1]	SegalG. B.: Unitary representations of some infinite dimensional groups,Commun. Math. Phys. 80 (1981), 301-342. · Zbl 0495.22017
[2]	CareyA. L. and RuijsenaarsS. N. M.: On fermion gauge groups, current algebras and Kac-Moody algebras,Acta Appl. Math. 10 (1987), 1-86. · Zbl 0644.22012
[3]	ShaleD.: Linear symmetries of free boson fields,Trans. Amer. Math. Soc. 103 (1962), 149-167. · Zbl 0171.46901
[4]	SchroerB., SeilerR., and SwiecaJ.: Problems of stability for quantum fields in external time-dependent potentials,Phys. Rev. D2 (1970), 2927-2937.
[5]	RuijsenaarsS. N. M.: On Bogoliubov transformations II,Ann. of Phys. 116 (1978), 105-134.
[6]	MacdonaldI. G.:Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
[7]	KashiwaraM. and VergneM.: On the Segal-Shale-Weil representations and harmonic polynomials,Invent. Math. 44 (1978), 1-47. · Zbl 0375.22009
[8]	JakobsenH.: On singular holomorphic representations,Invent. Math. 62 (1980), 67-78. · Zbl 0466.22016
[9]	EnrightT. and ParthasarathyR.: A proof of a conjecture of Kashiwara and Vergne, in J. Carmona and M. Vergne,Non Commutative Harmonic Analysis and Lie Groups, Lecture Notes in Mathematics 880, Springer-Verlag, Berlin, 1981, pp. 74-90.
[10]	EnrightT., HoweR., and WallachN.: A classification of unitary highest weight modules, in P. C.Trombi (ed.),Representation Theory of Reductive Groups, Birkh?user-Verlag, Boston, 1983, pp. 97-143.
[11]	JakobsenH.: The last possible place of unitarity for certain highest weight modules,Math. Ann. 256 (1981), 439-447. · Zbl 0478.22007
[12]	JakobsenH.: Hermitian symmetric spaces and their unitary highest weight modules,J. Funct. Anal. 52 (1983), 385-412. · Zbl 0517.22014