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# Isomorphic induced modules and Dynkin diagram automorphisms of semisimple Lie algebras

Abstract : Consider a simple Lie algebra g and (g) over bar subset of g a Levi subalgebra. Two irreducible (g) over bar -modules yield isomorphic inductions to g when their highest weights coincide up to conjugation by an element of the Weyl group W of g which is also a Dynkin diagram automorphism of (g) over bar. In this paper, we study the converse problem: given two irreducible (g) over bar -modules of highest weight mu and nu whose inductions to g are isomorphic, can we conclude that mu and nu are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of (g) over bar ? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing mu and nu 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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01109350
Contributor : Jeremie Guilhot Connect in order to contact the contributor
Submitted on : Monday, January 26, 2015 - 11:00:24 AM
Last modification on : Wednesday, January 19, 2022 - 4:50:43 PM

### Citation

Jeremie Guilhot, Cédric Lecouvey. Isomorphic induced modules and Dynkin diagram automorphisms of semisimple Lie algebras. Glasgow Mathematical Journal, Cambridge University Press (CUP), 2016, 58 (1), pp.187-203. ⟨10.1017/S0017089515000142⟩. ⟨hal-01109350⟩

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