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Intersection cohomology of $$B\times B$$-orbit closures in group compactifications. (English) Zbl 1110.14047
From the introduction: An adjoint semi-simple group $$G$$ has a “wonderful” compactification $$X$$, which is a smooth projective variety, containing $$G$$ as an open subvariety. $$X$$ is acted upon by $$G\times G$$ and, $$B$$ denoting a Borel subgroup of $$G$$, the group $$B\times B$$ has finitely many orbits in $$X$$. The main results of this paper concern the intersection cohomology of the closures of the $$B\times B$$-orbits. Examples of such closures are the “large Schubert varieties,” the closures in $$X$$ of the double cosets $$BwB$$ in $$G$$.
After recalling some basic results about the wonderful compactification, we discuss in Section 1 the description of the $$B\times B$$-orbits, and establish some basic results.
In Section 2 the “Bruhat order” of the set $$V$$ of orbits is introduced and described explicitly. As an application we obtain cellular decompositions of the large Schubert varieties.
Let $${\mathcal H}$$ be the Hecke algebra associated to $$G$$, it is a free module over an algebra of Laurent polynomials $$\mathbb{Z}[u, u-1]$$. As a particular case of results of [(*) J. G. M. Mars and T. A. Springer, Represent. Theory 2, 33–69 (1998; Zbl 0887.14026)], the spherical $$G\times G$$-variety $$X$$ defines a representation of the Hecke algebra associated to $$G\times G$$, i.e. $${\mathcal H}\otimes_{\mathbb{Z}[u,u^{-1}]}{\mathcal H}$$, in a free module $${\mathcal M}$$ over an extension of $$\mathbb{Z}[u, u^{-1}]$$, with a basis $$(m_v)$$ indexed by $$V$$. The definition of $${\mathcal M}$$ is sheaf-theoretical, working over the algebraic closure of a finite field. This is discussed in Section 3. On the model of G. Lusztig and D. A. Vogan, jun. [Invent. Math. 71, 365–379 (1983; Zbl 0544.14035)] a duality map $$\Delta$$ is introduced on $${\mathcal M}$$, coming from Verdier duality in sheaf theory. The matrix coefficients of $$\Delta$$ relative to the basis $$(m_v)$$ are discussed at the end of Section 3.
In Section 4 it is shown that the intersection cohomology of an orbit closure $$\bar v$$ leads to “Kazhdan-Lusztig” elements in $${\mathcal M}$$. The results about matrix coefficients of Section 3 together with results of [(*)] imply the evenness of local intersection cohomology, and the existence of Kazhdan-Lusztig polynomials.
We also prove evenness of global intersection cohomology of closures $$\overline v$$. The results on intersection cohomology, proved in the first instance in positive characteristics, then also follow in characteristic 0, and over $$\mathbb{C}$$.
Section 5 contains a brief discussion of the extension of results of the previous sections to intersection cohomology of an orbit closure $$\overline v$$, for certain non-constant sheaves on $$v$$.
We have formulated the constructions of the paper (e.g. of the $${\mathcal H}\otimes{\mathcal H}$$-module $${\mathcal M}$$) in such a manner that they also make sense for general Coxeter groups.
Section 6 contains some remarks about the constructions for such groups.
Computation by hand of our Kazhdan-Lusztig polynomials turns out to be quite cumbersome, the only manageable case (for the author) being $$G=\text{PGL}_2$$. The Appendix A by W. van der Kallen gives a number of numerical examples, ohtained by computer calculations.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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##### References:
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