Geometric Invariant Theory and Generalized Eigenvalue Problem II

Ressayre, Nicolas

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1467-1491.

Soit $G$ un sous-groupe fermé réductif et connexe d’un groupe réductif complexe et connexe $\widehat{G}$. On fixe des tores maximaux et des sous-groupes de Borel de $G$ et $\widehat{G}$. De cette manière les représentations irréductibles de $G$ et $\widehat{G}$ sont paramétrées par des poids dominants. On s’intéresse au cône ${ℒR}^{\circ }(G,\widehat{G})$ engendré par les paires $(\nu ,\widehat{\nu })$ de poids dominants réguliers tels que $V_{\nu }^{*}$ est un sous-$G$-module de $V_{\widehat{\nu }}$. Nous obtenons ici une paramétrisation bijective des faces de ${ℒR}^{\circ }(G,\widehat{G})$, en étudiant plus généralement les GIT-cônes des $G$-variétés projectives. Nous montrons aussi comment les relations d’inclusions entre les faces de ${ℒR}^{\circ }(G,\widehat{G})$ se lisent sur notre paramétrisation.

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\widehat{G}$. Fix maximal tori and Borel subgroups of $G$ and $\widehat{G}$. Consider the cone $ℒ{ℛ}^{\circ }(G,\widehat{G})$ generated by the pairs $(\nu ,\widehat{\nu })$ of strictly dominant characters such that $V_{\nu }^{*}$ is a submodule of $V_{\widehat{\nu }}$. We obtain a bijective parametrization of the faces of $ℒ{ℛ}^{\circ }(G,\widehat{G})$ as a consequence of general results on GIT-cones. We show how to read the inclusion of faces off this parametrization.

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