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On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory. (English) Zbl 1035.20004
Let \(V\) be a finite-dimensional complex vector space. A nilpotent linear map \(N\colon V\to V\) is said to fix a flag \(F=\{F_0\subset F_1\subset\cdots\subset F_{n-1}\subset F_n=V\}\) if \(NF_i\subseteq F_{i-1}\) for each \(i\), where \(F_i\) is a subspace of \(V\) of dimension \(i\). The variety \({\mathcal B}_N\) of all flags in \(V\) fixed by a nilpotent map \(N\) is called a Springer fiber. The topology of the components of \({\mathcal B}_N\) and their pairwise intersections was studied by a number of people such as N. Spaltenstein, J. A. Vargas, …. The present paper extends some of their results to describe the homological structure of components of \({\mathcal B}_N\) and their pairwise intersections in the cases where the nilpotent maps \(N\) correspond to the hook and two-row shape partitions. In these cases, the components are nonsingular, and are iterated bundles of flag manifolds and Grassmannians.
The author also relates his computations to the structure of the Kazhdan-Lusztig bases of certain representations of Iwahori-Hecke algebras of type \(A\). The suitable normalized inner products of these basis vectors are polynomials in \(t\) and \(t^{-1}\) that are invariant under the map \(t\mapsto t^{-1}\). The author shows that for irreducible representations labeled by a hook or two-row shape, the suitable normalized inner products equal the intersection homology Poincaré polynomials of pairwise intersections of irreducible components of Springer fibers of the general linear group.

MSC:
20C08 Hecke algebras and their representations
20G05 Representation theory for linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
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