Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

Abstract

Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of the torus. MSC:14M15

Residue formulas for push-forwards in equivariant cohomology: a symplectic approach

“…In [BS12], Bérczi and Szenes used iterated residues at infinity to compute the push-forward of a certain characteristic class over the complete flag variety. In [Zie14] we prove similar residue-type formulas for push-forwards over Grassmannians (classical, Lagrangian and orthogonal). For example, the push-forward of a characteristic class α of the tautological bundle R over the Grassmannian Grass m (C n ), which at the fixed points of the action is given by the symmetric polynomial V , can be expressed as…”

“…In [Zie14] we have obtained the above formulas in the case of classical Grassmannians, Lagrangian Grassmannians, orthogonal Grassmannians. The case of complete flag varieties is covered in [BS12].…”

“…In [Zie14] we have proven that if α(R) restricted to the fixed points of the action is given by a symmetric polynomial V , this expression is equal to the following residue…”

“…Chapter 2 describes the notation conventions used in this paper, trying to find a compromise between the notations of the two influential papers [JK95] and [GK96]. Chapter 3 briefly describes localization theorems in equivariant cohomology, the results obtained in [Zie14] and the various approaches to nonabelian localization. In Chapter 4 we adapt the Jeffrey-Kirwan theorem to the S-equivariant setting, for a torus S, by modifying the proof of the Jeffrey-Kirwan theorem given by Guillemin and Kalkman in [GK96].…”

“…In Chapter 4 we adapt the Jeffrey-Kirwan theorem to the S-equivariant setting, for a torus S, by modifying the proof of the Jeffrey-Kirwan theorem given by Guillemin and Kalkman in [GK96]. Two examples at the end of Chapter 4 relate the equivariant version of Jeffrey-Kirwan theorem to the formulas in [Zie14]. The Appendix shows the details of the adaptation of the Guillemin-Kalkman proof of the nonabelian localization theorem to the case of S 1 -action.…”