Geodesic
ISOMORPHIC INDUCED MODULES AND DYNKIN DIAGRAM AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS
Glasgow mathematical journal, Tome 58 (2016) no. 1, pp. 187-203

Voir la notice de l'article provenant de la source Cambridge University Press

Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities. 
GUILHOT, JÉRÉMIE; LECOUVEY, CÉDRIC. ISOMORPHIC INDUCED MODULES AND DYNKIN DIAGRAM AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS. Glasgow mathematical journal, Tome 58 (2016) no. 1, pp. 187-203. doi: 10.1017/S0017089515000142
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