Reductions for branching coefficients

Description

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.

• introduction let


• uy ?


• let ?


• groups uy


• maximal torus


• weyl group


• schubert varieties


• c? ?

Sujets

• Maximal torus
• Weyl group
• Schubert variety

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 13
• Langue: English

Extrait

Reductionsforbranchingcoefficients N.Ressayre February2,2011 Abstract LetGbeaconnectedreductivesubgroupofacomplexconnected reductivegroupG ˆ .Weareinterestedinthebranchingproblem.Fix maximaltoriandBorelsubgroupsofGandG ˆ .Considerthecone LR(G,G ˆ )generatedbythepairs(ν,νˆ)ofdominantcharacterssuch thatV ν ∗ isasubmoduleofV ν ˆ .ItisknownthatLR(G,G ˆ )isaclosed convexpolyhedralcone.Inthiswork,weshowthateveryregular faceofLR(G,G ˆ )givesrisetoareductionruleformultiplicities.More precisely,weprovethatfor(ν,νˆ)onsuchaface,themultiplicityof V ν ∗ inV ν ˆ equaltoasimilarmultiplicityforrepresentationsofLevi subgroupsofGandG ˆ .Thisgeneralizes,bydifferentmethods,results obtainedbyBrion,Derksen-Weyman,Roth... 1Introduction LetGbeaconnectedreductivesubgroupofacomplexconnectedreductive groupG ˆ .Weareinterestedinthebranchingproblem: DecomposeirreduciblerepresentationsofG ˆ assumofirreducible G-modules. WefixmaximaltoriT⊂T ˆ andBorelsubgroupsB⊃TandB ˆ ⊃T ˆ ofGandG ˆ .LetX(T)denotethegroupofcharactersofTandletX(T) + denotethesetofdominantcharacters.Forν∈X(T) + ,wedenotebyV ν the irreduciblerepresentationofhighestweightν.Similarly,weusenotation X(T ˆ ),X(T ˆ ) + ,V νˆ relativelytoG ˆ .ForanyG-moduleV,wedenotebyV G thesubspaceofG-fixedvectors.Considerthefollowingintegers c ννˆ (G,G ˆ )=dim(V ν ⊗V νˆ ) G .(1) 1