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

Magdalena Zielenkiewicz

Open Mathematics (2014)

  • Volume: 12, Issue: 4, page 574-583
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Magdalena Zielenkiewicz. "Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity." Open Mathematics 12.4 (2014): 574-583. <http://eudml.org/doc/269799>.

@article{MagdalenaZielenkiewicz2014,
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.},
author = {Magdalena Zielenkiewicz},
journal = {Open Mathematics},
keywords = {Grassmannian; Equivariant cohomology theory; Localization; Berline-Vergne formula; equivariant cohomology; integration; iterated residue},
language = {eng},
number = {4},
pages = {574-583},
title = {Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity},
url = {http://eudml.org/doc/269799},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Magdalena Zielenkiewicz
TI - Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity
JO - Open Mathematics
PY - 2014
VL - 12
IS - 4
SP - 574
EP - 583
AB - 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.
LA - eng
KW - Grassmannian; Equivariant cohomology theory; Localization; Berline-Vergne formula; equivariant cohomology; integration; iterated residue
UR - http://eudml.org/doc/269799
ER -

References

top
  1. [1] Atiyah M.F., Bott R., The moment map and equivariant cohomology, Topology, 1984, 23(1), 1–28 http://dx.doi.org/10.1016/0040-9383(84)90021-1 Zbl0521.58025
  2. [2] Bérczi G., Szenes A., Thom polynomials of Morin singularities, Ann. of Math., 2012, 175(2), 567–629 http://dx.doi.org/10.4007/annals.2012.175.2.4 Zbl1247.58021
  3. [3] Berline N., Vergne M., Zéros d’un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J., 1983, 50(2), 539–549 http://dx.doi.org/10.1215/S0012-7094-83-05024-X Zbl0515.58007
  4. [4] Borel A., Seminar on Transformation Groups, Ann. of Math. Stud., 46, Princeton University Press, Princeton, 1960 
  5. [5] Fehér L.M., Rimányi R., Thom series of contact singularities, Ann. of Math., 2012, 176(3), 1381–1426 http://dx.doi.org/10.4007/annals.2012.176.3.1 Zbl1264.32023
  6. [6] Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991 http://dx.doi.org/10.1007/978-1-4612-0979-9 
  7. [7] Ginzburg V.A., Equivariant cohomology and Kähler geometry, Functional Anal. Appl., 1987, 21(4), 271–283 http://dx.doi.org/10.1007/BF01077801 Zbl0656.53062
  8. [8] Hsiang W., Cohomology Theory of Topological Transformation Groups, Ergeb. Math. Grenzgeb., 85, Springer, New York-Heidelberg, 1975 http://dx.doi.org/10.1007/978-3-642-66052-8 
  9. [9] Jeffrey L.C., Kirwan F.C., Localization for nonabelian group actions, Topology, 1995, 34(2), 291–327 http://dx.doi.org/10.1016/0040-9383(94)00028-J 
  10. [10] Jeffrey L.C., Kirwan F.C., Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math., 1998, 148(1), 109–196 http://dx.doi.org/10.2307/120993 Zbl0949.14021
  11. [11] Kazarian M., On Lagrange and symmetric degeneracy loci, preprint available at http://www.newton.ac.uk/preprints/NI00028.pdf 
  12. [12] Quillen D., The spectrum of an equivariant cohomology ring: I, Ann. of Math., 1971, 94(3), 549–572 http://dx.doi.org/10.2307/1970770 Zbl0247.57013
  13. [13] Rimányi R., Quiver polynomials in iterated residue form, preprint available at http://arxiv.org/abs/1302.2580 Zbl06376534
  14. [14] Weber A., Equivariant Chern classes and localization theorem, J. Singul., 2012, 5, 153–176 Zbl1292.14009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.