Gradient-type differential operators and unitary highest weight representations of SU(p,q)
Let \(G=SU(p,q)\) and K a maximal compact subgroup of G. Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) denote the Cartan decomposition of \({\mathfrak g}=Lie(G)\), \(\{X_ j\}\) an orthonormal basis of \({\mathfrak p}_ C\) and \({\mathfrak p}^{\pm}_ C\) the upper and lower triangular matrices of \({\mathfrak p}_ C\). For \((\tau_{\lambda},H_{\lambda})\in \hat K\) with highest weight \(\lambda\) let \(C^{\infty}(G,\tau_{\lambda})=\{f\in C^{\infty}(G,H_{\lambda})\); \(f(gk)=\tau_{\lambda}(k)^{-1}f(g)\) \((k\in K\), \(g\in G)\}\), \(\nabla\) the gradient operator defined by \(\nabla f(g)=\sum r(X_ j)f(g)\otimes \bar X_ j\) and \({\mathcal P}_{\lambda}: H_{\lambda}\otimes {\mathfrak p}_ C\to W(\lambda)^{\perp}\) a projection onto the orthogonal complement of a K-invariant subspace \(W(\lambda)\) of \(H_{\lambda}\otimes {\mathfrak p}_ C\). Then we call \({\mathcal D}_{\lambda}={\mathcal P}_{\lambda}\nabla\) a differential operator of gradient type. When \(W(\lambda)=H_{\lambda}\otimes {\mathfrak p}^-_ C\), \({\mathcal D}_{\lambda}\) is the Schmid operator and the discrete series arises as the \(L^ 2\)-kernel of \({\mathcal D}_{\lambda}.\) In this paper, noting the Fock realization of unitary highest weight representations \(F_{\lambda}\) (\(\lambda\in \Lambda)\), the authors construct \(W(\lambda)\subset H_{\lambda}\otimes {\mathfrak p}^-_ C\) as the direct sum of K-irreducible subspaces \(H^{\lambda-\beta}\) with highest weights \(\lambda-\beta\) \((\beta \in \Phi^+_ n)\) which satisfy \((\lambda +\delta,2\beta)/(\beta,\beta)<1\). Let \(\Lambda_ 1\) be the subset of \(\Lambda\) for which there exists \(H^{\lambda -\beta}\) with \((\lambda +\delta,2\beta)/(\beta,\beta)=1\). Then \({\mathcal D}_{\lambda}\) \((\lambda \in \Lambda_ 1)\) is not the Schmid operator and the unitary highest weight representation with highest weight \(\lambda\) contained in Ker(\({\mathcal D}_{\lambda})\) corresponds to a point at the far right of the discrete set, opposite the discrete series, in the classification scheme. They also give an equivalent operator \(\tilde {\mathcal D}_{\lambda}\) on \(C^{\infty}(G/K,H_{\lambda})\) and show that Ker(\(\tilde {\mathcal D}_{\lambda})\) \((\lambda \in \Lambda_ 1)\) is properly contained in Ker(\({\bar \partial})\), especially, functions in Ker(\(\tilde {\mathcal D}_{\lambda})\) are not only holomorphic but also satisfy additional first order differential equations, which completely determine the representation spaces for ladder representations of G.
scientific article; zbMATH DE number 4110966
scientific article