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Transcending classical invariant theory. (English) Zbl 0716.22006

Let $\stackrel{˜}{S}p$ denote the twofold cover of the real symplectic group $S{p}_{2n}\left(ℝ\right)=Sp$. The theory of $\theta$-series as developed by Siegel is ruled by a certain representation $\omega$ of $\stackrel{˜}{S}p$, the so called oscillator representation first constructed by Shale and Weil. Let $\left(G,{G}^{\text{'}}\right)$ be a reductive dual pair, that is, G and ${G}^{\text{'}}$ are reductive subgroups of Sp one being the centralizer of the other. Let $\stackrel{˜}{G},$ ${\stackrel{˜}{G}}^{\text{'}}$ be their inverse images in $\stackrel{˜}{S}p$. Then ${\omega |}_{\stackrel{˜}{G}{\stackrel{˜}{G}}^{\text{'}}}$ may be viewed as a representation of $\stackrel{˜}{G}×{\stackrel{˜}{G}}^{\text{'}}$. Let for any subgroup E of $\stackrel{˜}{S}p$ the notion R(E) stand for the set of irreducible representations of E modulo equivalence and $R\left(E,\omega$) for the set of all $\sigma \in R\left(E\right)$ that occur as quotients of $\omega |E$. In [Proc. Symp. Pure Math. 33, No.1, 275- 285 (1979; Zbl 0423.22016)] R. Howe stated the conjecture that $R\left(\stackrel{˜}{G}{\stackrel{˜}{G}}^{\text{'}},\omega \right)$ as a subset of $R\left(\stackrel{˜}{G}×{\stackrel{˜}{G}}^{\text{'}}\right)=R\left(\stackrel{˜}{G}\right)×R\left({\stackrel{˜}{G}}^{\text{'}}\right)$ is the graph of a bijection between $R\left(\stackrel{˜}{G},\omega \right)$ and $R\left({\stackrel{˜}{G}}^{\text{'}},\omega \right)$. This correspondence may be viewed as underlying mechanism of the theta-correspondences in the theory of automorphic forms.

The paper under consideration contains a proof of this conjecture. At first the problem is algebraized by reformulating it in terms of $\left(𝔊,K\right)$-modules. Then by noting that the maximal compact subgroup K of G also is a member of a reductive dual pair one constructs new pairs out of old ones and uses properties of compact pairs and classical invariant theory. There are some facts needed which only can be read off from the classification of reductive dual pairs. For this the classification over $ℝ$ is also included.

Reviewer: A.Deitmar

##### MSC:
 2.2e+48 Representations of Lie and real algebraic groups: algebraic methods