Intertwining operators into cohomology representations for semisimple Lie groups
Let \(G\) be a semisimple connected real Lie group with finite center, let \(K\) be a maximal compact subgroup of \(G\), and let \({\mathfrak g}_0\), \({\mathfrak k}_0\) be, respectively, the Lie algebras of \(G,K\), with complexifications \({\mathfrak g},{\mathfrak k}\). Consider the Cartan decomposition \({\mathfrak g}_0={\mathfrak k}_0\oplus {\mathfrak p}_0\) of \({\mathfrak g}_0\) such that \({\mathfrak p}_0\) is the orthogonal complement of \({\mathfrak k}_0\) with respect to the Killing form of \({\mathfrak g}_0\) and let \(\theta\) be the Cartan involution of \({\mathfrak g}_0\) defined by \({\mathfrak k}_0\). Choose a nonzero \(X\in {\mathfrak t}_0\), where \({\mathfrak t}_0\) is the compact part of a \(\theta\)-stable Cartan subalgebra \({\mathfrak h}_0={\mathfrak t}_0\oplus{\mathfrak a}_0\) of \({\mathfrak g}_0\), with \({\mathfrak a}_0\) assumed to be maximally Abelian in \({\mathfrak p}_0\) and \(\dim{\mathfrak t}_0>0\). The centralizer \(L\) of \(X\) in \(G\) is connected and \(\Theta\)-stable, where \(\Theta\) is the Cartan involution of \(G\) corresponding to \(\theta\). Let \({\mathfrak l}_0\) be the Lie algebra of \(L\) with complexification \({\mathfrak l}\) and fix a \(\theta\)-stable parabolic subalgebra \({\mathfrak q}={\mathfrak l}\oplus{\mathfrak u}\) of \({\mathfrak g}\). Then \(G/L\) is an open complex submanifold of the compact complex manifold \(G^C/Q\), where \(G^C\) is the complexification of the adjoint group of \(G\) and \(Q\) is the analytic subgroup of \(G^C\) with Lie algebra \({\mathfrak q}\). The author constructs an explicit intertwining operator from a nonunitary principal series representation of \(G\) into the group \(Z_{\text{Dol}}^{0,s} (G/L,W)\) of Dolbeault \((0,s)\)-cocycles of \(G/L\) with values in the carrier space \(W\) of an induced representation of \(L\), where \(s= \dim({\mathfrak u}\cap {\mathfrak k})\). The image of this operator lies in the space of strongly harmonic \((0,s)\) forms. Results of \textit{L. Barchini, A. W. Knapp}, and \textit{R. Zierau} [ibid. 107, 302-341 (1992; Zbl 0761.22015)] are generalized by replacing the one-dimensional fiber over \(G/L\) by certain Langlands quotients [\textit{A. W. Knapp}, Representation theory of semisimple groups: An overview based on examples, Princeton (1986; Zbl 0604.22001)].
scientific article; zbMATH DE number 1106592
scientific article