Induced Representations of Affine Hecke Algebras and Singularities of R-matrices

We give an explicit criterion for the irreducibility of some induction products of evaluation modules of affine Hecke algebras of type A. This allows to describe the form of the zeros and poles of the trigonometric R-matrix associated to any evaluation module of Uv(ŝlN ). Abridged english version — Let Ĥn(u) be the affine Hecke algebra associated to GL(n) and Hn(u) its finite-dimensional subalgebra of type An−1. We assume that the parameter u ∈ C ∗ is not a root of unity. The simple Hn(u)-modules Sλ are parametrized by partitions λ of n. Let Sλ(z) be the Ĥn(u)-module obtained from Sλ by evaluation at z ∈ C . Write λ = (λ1, . . . , λr) and λ = (l1, . . . , lk) (the conjugate of λ). Let Eλ = {e = λi + lj − i− j + 1, 1 ≤ i ≤ r, 1 ≤ j ≤ k} denote the set of hook-lengths of λ and set Zλ = {u , e ∈ Eλ}. Given finite-dimensional Ĥni(u)modules Mi, i = 1, 2, we write M1 ⊙M2 for the induction product of M1 and M2 (this is then an Ĥn1+n2(u)-module). Theorem 1 Let z1, . . . , zm ∈ C . The module Sλ(z1) ⊙ · · · ⊙ Sλ(zm) is simple if and only if zj/zi 6∈ Zλ for all i, j. The proof of Theorem 1 is reduced to a problem of canonical bases in the following way. Let Rn be the complexified Grothendieck group of the category Cn of Ĥn(u)-modules for which the generators yi of the maximal commutative subalgebra have eigenvalues of the form u , k ∈ Z. By Zelevinsky’s classification [13], the simple modules Lm of Cn are parametrized by the multisegments m = ∑ i≤j mij [i, j] over Z such that ∑ i<j mij(j − i+1) = n. Let R = ⊕ n≥0 Rn. Following Zelevinsky we consider R as a bialgebra with multiplication and comultiplication corresponding respectively to induction and restriction with respect to the maximal parabolic subalgebras Ĥk(u)⊗ Ĥn−k(u). Zelevinsky has shown that R is a polynomial ring in the generators [L[i,j]], i ≤ j associated with segments [i, j]. We denote by Mm the standard induced module corresponding to m, so that [Mm] = ∏ i≤j [L[i,j]] mij . On the other hand, let A = A[N ∞] denote the ring of polynomial functions on the group N − ∞ of lower unitriangular Z × Z-matrices with a finite number of non-zero off-diagonal entries. It is a polynomial ring in the coordinate functions tji, i < j, and has a natural bialgebra structure. We put tm := ∏ i≤j t mij j+1,i. It is well known that A − and the enveloping algebra U = U(n∞)