L’objet de cet article est de calculer la cohomologie et la K-théorie équivariantes des variétés de Bott-Samelson (théorèmes 3.3 et 4.3) et d’en déduire des résultats sur les variétés de drapeaux des groupes de Kac-Moody. Dans la section 3, on retrouve la formule de restriction aux points fixes de la base ${\left\{{\widehat{\xi }}^w\right\}}_{w\in W}$ de $H_T^{*}(G/B)$ (théorème 3.9) prouvée par Sara Billey dans [4]. Dans la section 4, on donne l’expression explicite de la restriction aux points fixes de la base ${\left\{{\widehat{\psi }}^w\right\}}_{w\in W}$ de $K_T(G/B)$ définie par Kostant et Kumar dans...

We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring $K_T(G/B)$ of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_T(G/B)$ to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of $K{\left(X\right)}_{\mathbb{Q}}$ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of...

Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M,\Omega )$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $\Phi$ is proper so that the reduced space $M_{\mu }:={\Phi }^{-1}(K·\mu )/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as$${𝒬}_K^{-\infty }\left(M\right):=\sum _{\mu \in \widehat{K}}𝒬\left(M_{\mu }\right)V_{\mu }^K.$$The aim of this article is to study the functorial properties of the assignment $(M,K)\to {𝒬}_K^{-\infty }\left(M\right)$.

We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant $K$-theory of generalized flag varieties $G/P$. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant...

In this paper we show that the multiplicities of holomorphic discrete series representations relative to reductive subgroups satisfy the credo “quantization commutes with reduction”.

The classical Serre-Swan's theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially...

Cet article est consacré à l’étude de la structure d’anneau du groupe de Grothendieck équivariant d’une courbe projective munie d’une action d’un groupe fini. On explicite cette structure en introduisant un groupe de classes de cycles à coefficients dans les caractères et une notion d’auto-intersection pour ces cycles. De ce résultat, on déduit une expression de la caractéristique d’Euler équivariante d’un $G$-faisceau.