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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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