Group Theory
On quiver varieties and affine Grassmannians of type A
[Sur les variétés carquois et grassmannienes affines de type A]

Présenté par : Pierre Deligne

Mirković, Ivan1 ; Vybornov, Maxim2
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
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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Accepté le :
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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://www.numdam.org/articles/10.1016/S1631-073X(03)00022-0/

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