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The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations. (English) Zbl 0896.17002
Let \(G\) be a simply connected complex simple Lie group and write \({\mathfrak g}\) for its Lie algebra. Fix a Borel subalgebra \({\mathfrak b}\) of \({\mathfrak g}\). This paper is a summary of recent results in the study of the set of abelian ideals in \({\mathfrak b}\). Detailed proofs will appear elsewhere.
Let \(\Xi\) be an index set parameterizing the set of abelian ideals in \({\mathfrak b}\). The first section of the paper is chiefly a summary of two earlier theorems due to Kostant. The first result deals with the connection between \(\Xi\) and the set \({\mathcal C}\) of all commutative Lie subalgebras in \({\mathfrak g}\). There is a natural way to attach to \({\mathcal C}\) a graded \({\mathfrak g}\)-submodule \(C\) of the exterior algebra \(\bigwedge {\mathfrak g}\). The first result says that \(\Xi\) parameterizes the irreducible components of \(C\). The second result tells how \(C\) relates to the general structure of \(\bigwedge {\mathfrak g}\).
The second section sketches a result of Peterson that says the cardinality of \(\Xi\) is \(2^l\) where \(l\) is the rank of \({\mathfrak g}\).
The third section generalizes some ideas of Peterson. Fix a Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak b}\), a choice of positive roots \(\Delta_+\) corresponding to \({\mathfrak b}\), and let \(Z=\{ z \in {\mathfrak h}_{\mathbb R} \mid\langle z, \phi \rangle \in \{2,1,0,-1\}\), \(\forall \phi \in \Delta_+ \}\). Then a natural bijection is established, \(\zeta:Z \rightarrow \Xi \times \text{Cent}(G)\).
The fourth section uses aspects of this bijection to construct any abelian ideal and to give its structure.
The natural bijection above makes each ideal correspond to a subset \(Z_\xi\) of \(Z\) with cardinality \(| \text{Cent}(G)| \). One result in section five says that these subsets are orbits of a specific subgroup in the extended affine Weyl group.
Section six relates the theory to the discrete series. In particular, the dimension admitting nontrivial \(L^2\)-cohomology in Schmid’s construction is expressible in terms of an associated abelian ideal. Moreover, the highest weight of the corresponding minimal \(K\)-type is also expressible in terms of weights associated to abelian ideals.

MSC:
17B05 Structure theory for Lie algebras and superalgebras
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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