Group Theory
On quiver varieties and affine Grassmannians of type A
[Sur les variétés carquois et grassmannienes affines de type A]
Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 207-212.

Nous construisons les variétés carquois de Nakajima de type A en termes de Grassmanniennes affines de type A. Ceci fournit une compactification de ces variétés carquois et une décomposition de ces Grassmanniennes affines en une union disjointe de variétés carquois. En conséquence, les singularités des variétés carquois, des orbites nilpotentes et des Grassmanniennes affines sont les mêmes en type A. La construction fournit aussi un cadre géométrique pour la dualité (GL(m),GL(n)) extérieure et permet d'identifier la base naturelle des espaces de poids dans la construction de Nakajima avec la base naturelle des espaces de multiplicité des produits tensoriels dans la construction géométrique en termes de Grassmanienne affine.

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians.

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DOI : 10.1016/S1631-073X(03)00022-0
Mirković, Ivan 1 ; Vybornov, Maxim 2

1 Department of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst, MA 01003-4515, USA
2 Department of Mathematics, MIT, 77 Massachusetts Ave, Cambridge, MA 02139-4307, USA
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Mirković, Ivan; Vybornov, Maxim. On quiver varieties and affine Grassmannians of type A. Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 207-212. doi : 10.1016/S1631-073X(03)00022-0. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00022-0/

[1] V. Baranovsky, V. Ginzburg, A. Kuznetsov, Wilson's Grassmannian and a noncommutative quadric, Preprint, 2002, | arXiv

[2] A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, Preprint

[3] Chriss, N.; Ginzburg, V. Representation Theory and Complex Geometry, Birkhäuser, Boston, 1997

[4] Ginzburg, V. Lagrangian construction of the enveloping algebra U(sln), C. R. Acad. Sci. Paris Sér. I Math., Volume 312 (1991) no. 12, pp. 907-912

[5] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, Preprint, 1995, | arXiv

[6] Howe, R. Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur Lectures, Israel Math. Conf. Proc., 8, Tel Aviv, Bar-Ilan University, Ramat Gan, 1992, 1995, pp. 1-182

[7] Kostant, B. Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2), Volume 74 (1961), pp. 329-387

[8] Lusztig, G. Green polynomials and singularities of unipotent classes, Adv. Math., Volume 42 (1981) no. 2, pp. 169-178

[9] Lusztig, G. On quiver varieties, Adv. Math., Volume 136 (1998), pp. 141-182

[10] A. Maffei, Quiver varieties of type A, Preprint, 2000, | arXiv

[11] Mirković, I.; Vilonen, K. Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett., Volume 7 (2000) no. 1, pp. 13-24

[12] Nakajima, H. Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J., Volume 76 (1994) no. 2, pp. 365-416

[13] Nakajima, H. Quiver varieties and Kac–Moody algebras, Duke Math. J., Volume 91 (1998) no. 3, pp. 515-560

[14] Slodowy, P. Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math., 815, Springer, Berlin, 1980

[15] Wilson, G. Collisions of Calogero–Moser particles and an adelic Grassmannian. With an appendix by I.G. Macdonald, Invent. Math., Volume 133 (1998) no. 1, pp. 1-41

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