Fusion process and representations of affine Hecke algebras Paolo Papi SapienzaUniversit`adiRoma joint work with V. Guizzi and M. Nazarov PaoloPapi FusionprocessandaffineHeckealgebras Upshot We want to provide a detailed account of a combinatorial construction (after Cherednik) which exhibits the irreducible (finite dimensional) modules of the affine Hecke algebra of GL as cyclic n modules over the Hecke algebra of GL . n The main theorem we are going to discuss has been (roughly) stated by Cherednik without proof. PaoloPapi FusionprocessandaffineHeckealgebras Affine Hecke algebras: Bernstein presentation The affine Hecke algebra H(cid:101)n (of GLn) is the C(q)-algebra with the following presentation PaoloPapi FusionprocessandaffineHeckealgebras Affine Hecke algebras: Bernstein presentation Generators for H(cid:101)n T ,...,T 1 n−1 X±1,...,X±1 1 n Relations H(cid:101)n T T T = T T T , 1 ≤ k ≤ n−2, k k+1 k k+1 k k+1 T T = T T , 1 < |k −l|, k l l k (T −q)(T +1) = 0, 1 ≤ k ≤ n−1, k k X X = X X , 1 ≤ k,l ≤ n, k l l k X T = T X , l (cid:54)= k,k +1, l k k l T X T = qX , 1 ≤ k ≤ n−1. k k k k+1 PaoloPapi FusionprocessandaffineHeckealgebras Affine Hecke algebras: Bernstein presentation The “finite” Hecke algebra Hn is the subalgebra of H(cid:101)n generated by the T ; it has the following presentation, which shows i it as a q deformation of C[S ]. n PaoloPapi FusionprocessandaffineHeckealgebras Affine Hecke algebras: Bernstein presentation Generators for H n T ,...,T 1 n−1 Relations H n T T T = T T T , 1 ≤ k ≤ n−2, k k+1 k k+1 k k+1 T T = T T , 1 < |k −l|, k l l k (T −q)(T +1) = 0. k k PaoloPapi FusionprocessandaffineHeckealgebras The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n. H(cid:101)n = Hn ⊗An as vector spaces Theorem The center of H(cid:101)n consists precisely of symmetric polynomials in X±1,...,X±1. 1 n Corollary Any irreducible H(cid:101)n-module is finite dimensional Remarks Recall the defining relations or H(cid:101)n: T T T =T T T , T T =TT , k k+1 k k+1 k k+1 k l l k (T −q)(T +1)=0, X X =XX , k k k l l k XT =T X, T X T =qX l k k l k k k k+1 PaoloPapi FusionprocessandaffineHeckealgebras Theorem The center of H(cid:101)n consists precisely of symmetric polynomials in X±1,...,X±1. 1 n Corollary Any irreducible H(cid:101)n-module is finite dimensional Remarks T T T =T T T , T T =TT , k k+1 k k+1 k k+1 k l l k (T −q)(T +1)=0, X X =XX , k k k l l k XT =T X, T X T =qX l k k l k k k k+1 The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n. H(cid:101)n = Hn ⊗An as vector spaces PaoloPapi FusionprocessandaffineHeckealgebras Corollary Any irreducible H(cid:101)n-module is finite dimensional Remarks T T T =T T T , T T =TT , k k+1 k k+1 k k+1 k l l k (T −q)(T +1)=0, X X =XX , k k k l l k XT =T X, T X T =qX l k k l k k k k+1 The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n. H(cid:101)n = Hn ⊗An as vector spaces Theorem The center of H(cid:101)n consists precisely of symmetric polynomials in X±1,...,X±1. 1 n PaoloPapi FusionprocessandaffineHeckealgebras Remarks T T T =T T T , T T =TT , k k+1 k k+1 k k+1 k l l k (T −q)(T +1)=0, X X =XX , k k k l l k XT =T X, T X T =qX l k k l k k k k+1 The Xi±1 generate a (maximal) commutative subalgebra An of H(cid:101)n. H(cid:101)n = Hn ⊗An as vector spaces Theorem The center of H(cid:101)n consists precisely of symmetric polynomials in X±1,...,X±1. 1 n Corollary Any irreducible H(cid:101)n-module is finite dimensional PaoloPapi FusionprocessandaffineHeckealgebras