Strongly harmonic forms for representations in the discrete series
Let \(G\) be a semisimple connected Lie group with finite centre, \(K\) a maximal compact subgroup with the same rank as \(G\) and let \(T \subset K\) be a Cartan subgroup of \(G\). In this case \(G\) has a nonempty discrete series, having representations which are associated to regular elliptic coadjoint orbits \(\mathcal O = G/T\). The quotient \(G/T\) is endowed with two \(G\)-invariant hermitian forms, one indefinite, the other positive definite. \textit{W. Schmid} used the latter metric in [Representation theory and harmonic analysis on semisimple Lie groups, Math. Surv. Monogr. 31, 223--286 (1989; Zbl 0744.22016) and Ann. Math. (2) 103, 375--394 (1976; Zbl 0333.22009)] to construct discrete series representations in the space of \(L^{2}\) harmonic forms. In this paper the author considers the formal adjoint operator \(\overline \partial^{*}_{\text{inv}}\) to the standard Dolbeault operator \(\overline {\partial}\) with respect to the invariant indefinite hermitian form on \(G/T\) and she proves that every \(K\)-finite Dolbeault cohomology class admits a strongly harmonic representative, i.e. a representative in \(\ker \overline {\partial} \cap \ker \overline {\partial}^{*}_{\text{inv}}\). This provides explicit integral formulas for the harmonic representatives.
scientific article; zbMATH DE number 1271584