Tuong Ton-That:
Reciprocity theorems for holomorphic representations of some infinite-dimensional groups


Submission: 1999, Mar. 25

Let $\mu$ denote the \emph{Gaussian measure} on $\mathbb{C}^{n\times k}$ defined by $d\mu\left( Z\right) =\pi^{-nk}\exp\left[ -\operatorname *{Tr}\left( ZZ^{\dag}\right) \right] \,dZ$, where $\operatorname *{Tr}$ denotes the trace function, $Z^{\dag}=\bar{Z}^{T}$, and $dZ$ denotes the Lebesgue measure on $\mathbb{C}^{n\times k}$. Let $\mathcal{F}_{n\times k}$ denote the Barg\-mann--Segal--Fock space of holomorphic entire functions on $\mathbb{C}^{n\times k}$ which are also square-integrable with respect to $\mu$. Fix $n$ and let $\mathcal{F}_{n\times\infty}$ denote the Hilbert-space completion of the \emph{inductive limit} $\lim_{k\rightarrow\infty}\mathcal {F}_{n\times k}$. Let $G_{k}$ and $H_{k}$ be compact groups such that $H_{k}\subset G_{k}\subset\mathrm{GL}_{k}\left( \mathbb{C}\right) $. Let $G_{\infty}$ (resp.\ $H_{\infty}$) denote the inductive limit $\bigcup _{k=1}^{\infty}G_{k}$ (resp.\ $\bigcup_{k=1}^{\infty}H_{k}$). Then the representation $R_{G_{\infty}}$ (resp.\ $R_{H_{\infty}}$) of $G_{\infty}$ (resp.\ $H_{\infty}$), obtained by right translation on $\mathcal{F}_{n\times\infty}$, is a \emph{holomorphic representation} of $G_{\infty}$ (resp.\ $H_{\infty}$) in the sense defined by Ol'shanskii. Then $R_{G_{\infty }}$ and $R_{H_{\infty}}$ give rise to the dual representations $R_{G_{n}^{\prime}}^{\prime}$ and $R_{H_{n}^{\prime}}^{\prime}$ of the \emph{dual pairs} $\left( G_{n}^{\prime},G_{\infty}^{}\right) $ and $\left( H_{n}^{\prime},H_{\infty}^{}\right) $, respectively. The generalized \emph{Bargmann--Segal--Fock space} $\mathcal{F}_{n\times\infty}$ can be considered as both a $\left( G_{n}^{\prime},G_{\infty}^{}\right ) $\emph{-dual module} and an $\left( H_{n}^{\prime},H_{\infty}^{}\right ) $-dual module. It is shown that the following multiplicity-free decompositions of $\mathcal {F}_{n\times\infty}$ into \emph{isotypic components} $\mathcal{F}% _{n\times\infty }=\sum\limits_{\left( \lambda\right) }{\!{\oplus}\,}\mathcal {I}_{n\times\infty}^{\left( \lambda\right) }= \sum\limits_{\left( \mu\right) }{\!{\oplus}\,}\mathcal{I}_{n\times\infty }^{\left( \mu\right) }$ hold, where $\left( \lambda\right) $ is a common \emph{irreducible signature} of the pair $\left( G_{n}^{\prime},G_{\infty}^{}\right) $ and $\left( \mu\right) $ a common irreducible signature of the pair $\left( H_{n}^{\prime},H_{\infty}^{}\right) $, and $\mathcal{I}_{n\times\infty }^{\left( \lambda\right) }$ (resp.\ $\mathcal{I}_{n\times\infty}^{\left( \mu\right) }$) is both the isotypic component of the equivalence classes $\left( \lambda\right) _{G_{\infty}}$ (resp.\ $\left( \mu\right) _{H_{\infty}}$) and $\left( \lambda^{\prime}\right) _{G_{n}^{\prime}}$ (resp.\ $\left( \mu^{\prime}\right) _{H_{n}^{\prime}}$). A \emph{reciprocity theorem,} giving the multiplicity of $\left( \mu\right) _{H_{\infty}}$ in the restriction to $H_{\infty}$ of $\left( \lambda\right) _{G_{\infty}}$ in terms of the multiplicity of $\left( \lambda^{\prime}\right) _{G_{n}^{\prime}}$ in the restriction to $G_{n}^{\prime}$ of $\left( \mu^{\prime}\right) _{H_{n}^{\prime}}$, constitutes the main result of this paper. Several applications of this theorem to Physics are also discussed.

Subject Classification: (PACS codes): 02.20.Tw, 02.20.Qs, 03.65.Fd

Keywords and Phrases:

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