The equivariant Riemann–Roch theorem and the graded Todd class

Vergne, Michèle

doi:10.1016/j.crma.2017.01.009

Geometry/Algebra

The equivariant Riemann–Roch theorem and the graded Todd class
[Le théorème de Riemann–Roch équivariant et la classe de Todd graduée]

Présenté par : Michèle Vergne

Vergne, Michèle ¹

¹ Université Denis-Diderot–Paris-7, Institut de Mathématiques de Jussieu, C.P. 7012, 4 place Jussieu, Boite Courrier 247, 75252 Paris Cedex 05, France

Comptes Rendus. Mathématique, Tome 355 (2017) no. 5, pp. 563-570.

Résumé
Abstract

Soit G un tore d'algèbre de Lie $g$ agissant de manière hamiltonienne sur une variété M. Soit $L$ un fibré de Kostant tel que l'application moment associée soit propre. Soit $Λ \subset g^{⁎}$ le réseau des poids de G. On considère un paramètre $k \geq 1$ et la multiplicité $m (λ, k)$ de la représentation quantifiée $R R_{G} (M, L^{k})$ . On définit la distribution $〈 Θ (k), f 〉 = \sum_{λ \in Λ} m (λ, k) f (λ / k)$ pour f une fonction test sur $g^{⁎}$ . La distribution $Θ (k)$ admet un développement asymptotique $〈 Θ (k), f 〉 \sim k^{\dim M / 2} \sum_{n = 0}^{\infty} k^{- n} 〈 D H_{n}, f 〉$ où les distributions $D H_{n}$ sont des distributions associées aux composantes homogènes de la classe de Todd équivariante de M. Lorsque M est compacte et f polynomiale, cette série est finie et exacte.

Let G be a torus with Lie algebra $g$ and let M be a G-Hamiltonian manifold with Kostant line bundle $L$ and proper moment map. Let $Λ \subset g^{⁎}$ be the weight lattice of G. We consider a parameter $k \geq 1$ and the multiplicity $m (λ, k)$ of the quantized representation $R R_{G} (M, L^{k})$ . Define $〈 Θ (k), f 〉 = \sum_{λ \in Λ} m (λ, k) f (λ / k)$ for f a test function on $g^{⁎}$ . We prove that the distribution $Θ (k)$ has an asymptotic development $〈 Θ (k), f 〉 \sim k^{\dim M / 2} \sum_{n = 0}^{\infty} k^{- n} 〈 D H_{n}, f 〉$ where the distributions $D H_{n}$ are the twisted Duistermaat–Heckman distributions associated with the graded equivariant Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.

Reçu le : 2017-01-17
Accepté le : 2017-01-17
Publié le : 2017-04-18

DOI : 10.1016/j.crma.2017.01.009

Affiliations des auteurs :

Vergne, Michèle ¹

¹ Université Denis-Diderot–Paris-7, Institut de Mathématiques de Jussieu, C.P. 7012, 4 place Jussieu, Boite Courrier 247, 75252 Paris Cedex 05, France

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     title = {The equivariant {Riemann{\textendash}Roch} theorem and the graded {Todd} class},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {563--570},
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Vergne, Michèle. The equivariant Riemann–Roch theorem and the graded Todd class. Comptes Rendus. Mathématique, Tome 355 (2017) no. 5, pp. 563-570. doi : 10.1016/j.crma.2017.01.009. http://www.numdam.org/articles/10.1016/j.crma.2017.01.009/

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