# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Lowest weight representations of some infinite dimensional groups on Fock spaces. (English) Zbl 0729.22023

If V and W are two separable Hilbert spaces, define $U\left(V,W\right)$ to be the subgroup of $GL\left(V\oplus W\right)$, the group of all invertible bounded operators of the Hilbert space $V\oplus W$ which leave invariant the Hermitian form defined by the operator $J=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ on $V\oplus W·$

Let ${U}_{res}\left(V,W\right)$ be the subgroup of $U\left(V,W\right)$ consisting of those elements $\left(\begin{array}{cc}a& b\\ b& d\end{array}\right)\in U\left(V,W\right)$ for which $b:W\to V$ and $c:V\to W$ are Hilbert-Schmidt operators. If ${A}_{i}=\left(\begin{array}{cc}{a}_{i}& {b}_{i}\\ {c}_{i}& {d}_{i}\end{array}\right)$, $\left(i=1,2,3\right)$, ${A}_{3}={A}_{1}{A}_{2}$, are in ${U}_{res}\left(V,W\right)$ it turns out that $c\left({A}_{1},{A}_{2}\right)={det}^{-1}\left({d}_{1}^{-1}{d}_{3}{d}_{2}^{-1}\right)is$ a co-cycle, which induces a central extension of ${U}_{res}\left(V,W\right)$, ${\stackrel{˜}{U}}_{res}\left(V,W\right)=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),z\right\}$, with $z\overline{z}=det\left(1-{d}^{{*}^{-1}}{b}^{*}b{d}^{-1}\right)·$

The author determines the unitary lowest weight representations of ${\stackrel{˜}{U}}_{res}\left(V,W\right)$ 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 ${S}^{n}\left(H\right)$ denote the Hilbert space completion of the vector space of nth powers of the symmetric algebra of H with the Hermitian form

$<{h}_{1}{h}_{2}···{h}_{n},{h}_{1}^{\text{'}}{h}_{2}^{\text{'}}···{h}_{n}^{\text{'}}>=\sum _{n}\prod _{i=1}^{n}<{h}_{\sigma \left(i\right)}·{h}_{i}^{\text{'}}>$

where $\sigma$ runs through all permutations of 1,2,...,n. Define $S\left(H\right)=\oplus {S}^{n}\left(H\right)$ endowed with the final topology defined by the inclusions ${i}_{n}:{S}^{n}\left(H\right)\to S\left(H\right)$, and $\stackrel{ˇ}{S}\left(H\right)$ the Hilbert space completion of S(H). Let $\stackrel{^}{S}\left(H\right)={\prod }_{n}{S}^{n}\left(H\right)$ endowed with the initial topology defined by the projections ${p}_{n}:\stackrel{^}{S}\left(H\right)\to {S}^{n}\left(H\right)$. $\stackrel{ˇ}{S}\left(H\right)$ and $\stackrel{^}{S}\left(H\right)$ are the antilinear dual spaces of each other and $\stackrel{ˇ}{S}\left(H\right)\subseteq S\left(H\right)\subseteq \stackrel{^}{S}\left(H\right)$ 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 $d\stackrel{˜}{{\Gamma }}$ of ${L}_{res}\left(V,W\right)$ 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 ${L}_{res}\left(V,W\right)$ denotes the subalgebra of $L\left(V\oplus W\right)$ consisting of elements of the form $\left(\begin{array}{cc}a& b\\ b& d\end{array}\right)$, $b:W\to V$, $c:V\to W$, b and c being Hilbert-Schmidt operators. For ${A}_{1}$, ${A}_{2}$ in ${L}_{res}\left(V,W\right)$, ${A}_{i}=\left(\begin{array}{cc}{a}_{i}& {b}_{i}\\ {c}_{i}& {d}_{i}\end{array}\right)$, $i=1,2$, the co-cycle $\left({A}_{1},{A}_{2}\right)=tr·\left({c}_{2}{b}_{1}-{c}_{1}{b}_{2}\right)$ induces a central extension ${\stackrel{˜}{L}}_{res}\left(V,W\right)$ of ${L}_{res}\left(V,W\right)·$

The author proves (1.6 Theorem, p. 67): The representation of ${L}_{res}\left(V,W\right)$ can be lifted to a unitary representation of ${U}_{res}\left(V,W\right)$ on S(V$\oplus \overline{W}\right)$. 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.

##### MSC:
 2.2e+66 Infinite-dimensional Lie groups and their Lie algebras: general properties 2.2e+71 Applications of Lie groups to physics; explicit representations
##### References:
 [1] SegalG. B.: Unitary representations of some infinite dimensional groups,Commun. Math. Phys. 80 (1981), 301-342. · Zbl 0495.22017 · doi:10.1007/BF01208274 [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 · doi:10.1007/BF00046582 [3] ShaleD.: Linear symmetries of free boson fields,Trans. Amer. Math. Soc. 103 (1962), 149-167. · doi:10.1090/S0002-9947-1962-0137504-6 [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. · doi:10.1016/0003-4916(78)90006-4 [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 · doi:10.1007/BF01389900 [8] JakobsenH.: On singular holomorphic representations,Invent. Math. 62 (1980), 67-78. · Zbl 0466.22016 · doi:10.1007/BF01391663 [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 · doi:10.1007/BF01450539 [12] JakobsenH.: Hermitian symmetric spaces and their unitary highest weight modules,J. Funct. Anal. 52 (1983), 385-412. · Zbl 0517.22014 · doi:10.1016/0022-1236(83)90076-9