On the dual pairs \((O(p,q), SL(2,\mathbb{R}))\), \((U(p,q), U(1,1))\) and \((Sp(p,q), O^*(4))\)
Let \(Sp(2k,\mathbb R)\) denote the symplectic group acting on \(\mathbb R^{2k}\) and consider a dual pair \((G,G')\), i.e., two reductive subgroups of \(Sp(2k,\mathbb R)\) which are their mutual centralizers. In addition, we assume that \(G\) and \(G'\) are invariant under the standard Cartan involution of \(G\), so that \(K := G \cap U(k)\) and \(K' := G' \cap U(k)\) are their maximal compact subgroups. If \(H\) is a reductive subgroup of \(Sp(2k,\mathbb R)\) with maximal compact subgroup \(K_H \subseteq U(k)\) and \(\widetilde K_H\) the corresponding subgroup of the twofold covering group \(\widetilde Sp(2k,\mathbb R)\) of \(Sp(2k,\mathbb R)\), then we write \({\mathcal R}(\mathbf h,\widetilde K_H, \omega)\) for the set of equivalence classes of simple \((\mathbf h,\widetilde K_H)\) modules which can be obtained as simple quotients of submodules of the oscillator representation \(\omega\) on the space \({\mathcal P}\) of holomorphic polynomials on \(\mathbb C^k\) (here we use the realization in the Fock model). The theta correspondence between \({\mathcal R}(\mathbf g,\widetilde K, \omega)\) and \({\mathcal R}(\mathbf g',\widetilde K', \omega)\) is now established as follows. For \(\rho \in {\mathcal R}(\mathbf g',\widetilde K', \omega)\) we consider the space \(\Omega(\rho) := {\mathcal P}/{\mathcal N}_\rho\), where \({\mathcal N}_\rho\) is the intersection of all \((\mathbf g', \widetilde K')\)-submodules \({\mathcal N}\) of \({\mathcal P}\) such that \({\mathcal P}/{\mathcal N} \cong \rho\). According to a result of R.\ Howe, we then have \(\Omega(\rho) \cong \rho \otimes \rho'\), where \(\rho'\) is a \((\mathbf g,\widetilde K)\)-module of finite length with a unique simple quotient \(\theta(\rho)\) (the so-called theta lift of \(\rho\)). A central problem in the theory of dual pairs is to make the theta correspondence, i.e., the map \(\theta\), more explicit. In the paper under review, the authors use quite elementary methods and explicit computations with polynomials to make \(\theta\) explicit for the dual pairs \((O(p,q), SL(2,\mathbb R))\), \((U(p,q), U(1,1))\), and \((Sp(p,q), O^*(4))\). In particular their computations provide quite explicit information on the \(K\)-highest weight vectors of the representations.
scientific article; zbMATH DE number 1344301