Reductions for branching coefficients

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ar X iv :1 10 2. 01 96 v1 [ ma th. AG ] 1 F eb 20 11 Reductions for branching coefficients N. Ressayre February 2, 2011 Abstract Let G be a connected reductive subgroup of a complex connected reductive group G. We are interested in the branching problem. Fix maximal tori and Borel subgroups of G and G. Consider the cone LR(G, G) generated by the pairs (?, ?) of dominant characters such that V ?? is a submodule of V? . It is known that LR(G, G) is a closed convex polyhedral cone. In this work, we show that every regular face of LR(G, G) gives rise to a reduction rule for multiplicities. More precisely, we prove that for (?, ?) on such a face, the multiplicity of V ?? in V? equal to a similar multiplicity for representations of Levi subgroups of G and G. This generalizes, by different methods, results obtained by Brion, Derksen-Weyman, Roth. . . 1 Introduction Let G be a connected reductive subgroup of a complex connected reductive group G. We are interested in the branching problem: Decompose irreducible representations of G as sum of irreducible G-modules. We fix maximal tori T ? T and Borel subgroups B ? T and B ? T of G and G.

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Publié par

profil-urra-2012

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ReductionsforbranchingcoeﬃcientsN.RessayreFebruary2,2011AbstractLetGbeaconnectedreductivesubgroupofacomplexconnectedreductivegroupGˆ.Weareinterestedinthebranchingproblem.FixmaximaltoriandBorelsubgroupsofGandGˆ.ConsidertheconeLR(G,Gˆ)generatedbythepairs(ν,νˆ)ofdominantcharacterssuchthatVν∗isasubmoduleofVνˆ.ItisknownthatLR(G,Gˆ)isaclosedconvexpolyhedralcone.Inthiswork,weshowthateveryregularfaceofLR(G,Gˆ)givesrisetoareductionruleformultiplicities.Moreprecisely,weprovethatfor(ν,νˆ)onsuchaface,themultiplicityofVν∗inVνˆequaltoasimilarmultiplicityforrepresentationsofLevisubgroupsofGandGˆ.Thisgeneralizes,bydiﬀerentmethods,resultsobtainedbyBrion,Derksen-Weyman,Roth...1IntroductionLetGbeaconnectedreductivesubgroupofacomplexconnectedreductivegroupGˆ.Weareinterestedinthebranchingproblem:DecomposeirreduciblerepresentationsofGˆassumofirreducibleG-modules.WeﬁxmaximaltoriT⊂TˆandBorelsubgroupsB⊃TandBˆ⊃TˆofGandGˆ.LetX(T)denotethegroupofcharactersofTandletX(T)+denotethesetofdominantcharacters.Forν∈X(T)+,wedenotebyVνtheirreduciblerepresentationofhighestweightν.Similarly,weusenotationX(Tˆ),X(Tˆ)+,VνˆrelativelytoGˆ.ForanyG-moduleV,wedenotebyVGthesubspaceofG-ﬁxedvectors.Considerthefollowingintegerscννˆ(G,Gˆ)=dim(Vν⊗Vνˆ)G.(1)1

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