Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections

Abstract

We study the space of all bimodule homomorphisms $R_x{\otimes }_{R}R(\underline{t}){\otimes }_R{R}_y\to R_z{\otimes }_{R}R({\underline{t}}^{\prime }){\otimes }_R{R}_w$ as a one-sided module, where $R_x,R_y,R_z,R_w$ are standard twisted bimodules and $R(\underline{t})$ and $R({\underline{t}}^{\prime })$ are the Bott-Samelson bimodules corresponding to sequences of reflections $\underline{t}$ and ${\underline{t}}^{\prime }$ 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 $\underline{t}$ and ${\underline{t}}^{\prime }$ 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 $S_n$, where $n\ge 4$. 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.

An improved version