On the structure of Kazhdan-Lusztig cells for branched Dynkin diagrams
Because of the complexity of the Kazhdan-Lusztig conjectures, they are of limited value for calculation, and it is often valuable to have less information in a more explicit form. For instance, the \(\tau\)-invariant of \textit{W. Borho} and \textit{J. Jantzen} [Invent. Math. 39, 1-53 (1977; Zbl 0327.17002)] and \textit{M. Duflo} [Ann. Math., II. Ser. 105, 107-120 (1977; Zbl 0346.17011)] provides useful (but incomplete) information about the action of a simple reflection in a coherent continuation representation. Roughly, the information it provides comes from little \(SL(2)\)'s inside \(G\), the reductive Lie group under consideration. At the next level of sophistication is the generalized \(\tau\)-invariant, where one exploits some of the relations in a Weyl group arising from rank two Levi subgroups, and gets somewhat better information about coherent continuation representations [\textit{D. Vogan}, Duke Math. J. 46, 61-108 (1979; Zbl 0398.22021), Math. Ann. 242, 209-224 (1979; Zbl 0387.17007)]. It is natural to stop with rank two subgroups, since all the defining relations of a Weyl group appear at that level. However, coherent continuation representations have more structure than arbitrary Weyl group representations: one may expect the structure of coherent continuation representations to be restricted by the presence of a Levi factor of type \(D_ 4\) in ways that are difficult to predict with more elementary considerations. It is a purpose of the present paper to describe explicitly some restrictions on the nature of coherent continuation arising from Levi factor of type \(D_ 4\).
scientific article; zbMATH DE number 130474