Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections
Abstract
We study the space of all bimodule homomorphisms RxR R(t)R Ry RzR R(t')R Rw as a one-sided module, where Rx,Ry,Rz,Rw are standard twisted bimodules and R(t) and R(t') are the Bott-Samelson bimodules corresponding to sequences of reflections t and t' respectively. We prove that this module is always reflexive under some reasonable restrictions on the representation of the underlying Coxeter group. However, unlike the case where t and t' contain only simple reflections, this module does not need any longer to be free. We provide a series of counterexamples already for the symmetric groups Sn, where n4. The projective dimension of the modules dual to them is n-3 and thus serves to measure the deviation from the free modules. When placed within a geometric framework, these examples show how to find fibers of points fixed by the compact torus in the Bott-Samelson resolutions (as in the original definition by Raoul Bott and Hans Samelson) with non-vanishing odd cohomology.
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