The Kazhdan-Lusztig Conjecture for Real Reductive Groups
Let G be a reductive linear real Lie group with abelian Cartan subgroups. The Kazhdan-Lusztig conjecture of  provides an algorithm for computing explicitly the distribution characters of the irreducible (admissible) representations of G. The history and status of this conjecture will not be related here (see , , , , and ); suffice it to say that there is (at least) a very detailed program for proving it, although there is no complete account in print. Our purpose here is simply to state the conjecture in as elementary a way as possible. There are at least two motivations for this, both a little tenuous. First, the industry of computing irreducible characters was created largely to assist in the study of unitary representations. It may therefore be useful to have the Kazhdan-Lusztig conjecture written down without the ponderous baggage of its proof. There are no examples of applications in this direction as yet, however. Second, the existing proof of the conjecture is (to a narrow-minded group representer) unsatisfactory. In section 8, we explain what kind of serious theorems must be combined with the formal results to give a proof of the conjecture. They are tantalizingly simple; but they have resisted all attempts at representation-theoretic proof for several years.
KeywordsSimple Root Recursion Formula Irreducible Character Cartan Subalgebra Distribution Character
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