Intersection cohomology of \(B\times B\)-orbit closures in group compactifications.

*(English)*Zbl 1110.14047From 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.

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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