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Transformation groups 2017. Conference dedicated to Prof. Ernest B. Vinberg on the occasion of his 80th birthday
15 декабря 2017 г. 11:00, г. Москва, Сколковский институт науки и технологий (Сколково, ул. Нобеля, д. 3), к. 303


On some modules of covariants for a reflection group

C. De Concini

La Sapienza University, Romе, Italy

Аннотация: This is joint work with Paolo Papi. Let $W$ be an irreducible finite reflection group, $\mathfrak{h}$ its (complexified) reflection module. $\mathcal{H} = C[\mathfrak{h}]/I$, where $I$ is the ideal generated by polynomial invariants of positive degree. $A = (\Lambda (\mathfrak{h})\otimes \mathcal{H})^W$ is an exterior algebra and we completely determine the $A$-module structure of $N := hom_W (\mathfrak{h},\Lambda (\mathfrak{h})\otimes \mathcal{H})$.
When $\mathfrak{h}$ is the Cartan subalgebra of a simple Lie algebra $\mathfrak{g}$, it is well known and easy that $A$ is canonically isomorphic to $(\Lambda(\mathfrak{g}))^{\mathfrak{g}}$ and we verify that $N = hom_{\mathfrak{g}}(\mathfrak{g}, \Lambda(\mathfrak{g})$ as an $A$-module.
Finally if $V$ is an irreducible $g$-module whose zero weight space we denote by $V_0$, we construct a degree preserving map
$$hom_{\mathfrak{g}}(V,\Lambda(\mathfrak{g}))\to hom_W (V_0,\Lambda (\mathfrak{h})\otimes \mathcal{H})$$
which we conjecture to be injective. This conjecture implies a well known conjecture by Reeder.

Язык доклада: английский


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