2.3. IWAHORI-SPHERICAL REPRESENTATIONS 11 The typical situation encountered in this work is that F is a number field, ι is an isomorphism from Q to C and ρ : Gal(F/F ) −→ GLn(Q ) a continuous homomorphism and v a finite place of F . Hence, via the above de- scribed equivalence of categories, one obtains a Weil-Deligne representation of WF v over Q and via ι a Weil-Deligne representation of WF v over C. If W = (r, N) is a Weil-Deligne representation over Q we will denote by Wι the correspond- ing Weil-Deligne representation over C via ι. We let W F-ss denote the Frobenius semi-simplification of W and let W ss denote the Frobenius semi-simplification of r. Let ρ1 = (r1,N1) and ρ2 = (r2,N2) be two Weil-Deligne representations of WK, with underlying vector spaces V1 and V2. The direct sum ρ1 ⊕ ρ2 = (r , N ) of ρ1 and ρ2 is defined to have underlying vector-space V1 ⊕ V2 and r = r1 ⊕ r2 and N is defined via N ((v1,v2)) = (N1(v1),N2(v2) for v1 ∈ V1 and v2 ∈ V2. The tensor product (r, N) := ρ1 ⊗ ρ2 of ρ1 and ρ2 is defined to be the Weil-Deligne representation with underlying vector space V1 ⊗ V2 and for every σ ∈ WK one has r(σ)(v1 ⊗ v2) = r1(σ)v1 ⊗ r2(σ)v2 and N(v1 ⊗ v2) = N1v1 ⊗ v2 + v1 ⊗ N2v2 The i’th exterior product ∧iρ1 of ρ1 is defined by viewing ∧iV1 as a sub-space of the i-fold tensor product V ⊗i 1 . 2.3. Iwahori-spherical representations The local components of the automorphic representations to which we will later apply arguments involving families of automorphic representations are of a specific type: They are Iwahori-spherical representations. Hence we now describe now for later use some known results about Iwahori-spherical and closely related types of representations. Let us first fix some notation: Let K be a p-adic field, let O be its valuation ring and let k be its residue field. Let G = GLn(K) for some n ≥ 2. Let B be the upper triangular Borel subgroup of GLn(K). Let I denote the Iwahori subgroup of GLn(O) associated to B and let I1 denote the subgroup of I corresponding to unipotent matrices in the reduction modulo the maximal ideal of O. Let T be the diagonal torus of GLn. Let δB : B → C× be the modulus character, which takes b ∈ B to | detb |K where detb denotes the determinant of the conjugation action of b on the set n of strictly upper triangular matrices in Mn(K). Definition 2.1. Let π be an irreducible admissible representation of GLn(K). • Let πI := {v ∈ π i · v = v for all i ∈ I} • For any character ρ : I → C× that is trivial on I1 define πρ := {v ∈ π i · v = ρ(i)v for all i ∈ I} The representation π is called Iwahori-spherical if πI = (0). Note that automorphic representations whose local components at some finite place satisfy πρ = (0) for some non-trivial character ρ as above will be used in the potential level-lowering results that we prove in Section 7.1.

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