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Strongly harmonic differential forms on elliptic orbits. (English) Zbl 0865.22010
Eastwood, Michael (ed.) et al., The Penrose transform and analytic cohomology in representation theory. AMS-IMS-SIAM summer research conference, June 27 - July 3, 1992, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 154, 77-88 (1993).
From the abstract: “This paper can be thought as an introduction based on examples to the works ‘Intertwining operators into Dolbeault cohomology representations’ by L. Barchini, A. W. Knapp, and R. Zierau [J. Funct. Anal. 107, 302-341 (1992; Zbl 0761.22015)] and ‘Szegö Mappings, Harmonic Forms and Dolbeault Cohomology’ [J. Funct. Anal., to appear].” Let \(G\) be a semisimple linear Lie group and \(G/L\) an elliptic coadjoint orbit of \(G\), i.e., \(L\) is the centralizer of a subtorus of \(G\). Elliptic orbits carry several natural \(G\)-invariant complex structures which are determined by parabolic subalgebras \(q\) of the complexification \(g\) of the Lie algebra of \(G\) whose Levi algebra \(l\) is the complexification of the Lie algebra of \(L\). Now there exist several ways to associate to \(G/L\) representations of \(G\) resp. its Lie algebra \(g_0\). There is an algebraic construction using derived functors leading to the Zuckerman modules \(A_q(\lambda)\), where \(\lambda\) is a linear functional on an abelian subalgebra of \(l\) containing the Lie algebra of the central torus in \(L\). On the other hand the complex geometry of \(G/L\) given by the invariant complex structure suggests another construction leading to a representation of \(G\) in a Dolbeault cohomology space \(H^s(G/L,\xi^\#)\), where \(s\) is the complex dimension of a maximal compact subspace of \(G/L\) given as the orbit \(K/(K\cap L)\) of a maximal compact subgroup \(K\) of \(L\), and \(\xi^\#\) is a certain one-dimensional representation of \(L\) defining a holomorphic line bundle \(G \times_{L\xi^\#}\). According to results of H.-W. Wong [Dolbeault cohomologies and Zuckerman modules associated with finite rank representations, Ph. D. thesis (Harvard University, 1991)] both constructions lead to the same Harish-Chandra module whenever \(\lambda\) and \(\xi^\#\) are related in the appropriate way, and, in addition, \(\lambda\) satisfies a certain dominance condition. The main purpose of the paper under review is to describe intertwining operators between these two kinds of representations. The key idea is to start with the Principal Series construction which leads to \(A_q(\lambda)\) in terms of its Langlands parameters, and then describe an intertwining operator from the Principal Series to strongly harmonic \(s\)-forms on \(G\times_{L\xi^\#}\), i.e., forms which are annihilated by the \(\overline\partial\)-operator and its formal adjoint. Once it is shown that such a non-zero operator exists, the irreducibility of the Harish-Chandra module of \(H^s(G/L,\xi^\#)\) implies that each \(K\)-finite cohomology class can be represented by a strongly harmonic form which in turn helps to understand the unitary structure on this module.
For the entire collection see [Zbl 0780.00026].
22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
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