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The application arises because of a very strong homological property enjoyed by certain cell filtrations forq-permutation modules.
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The Direct Summand Property in Modular Invariant Theory
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Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k.
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The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property.
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On Orthogonal Wavelets with the Oversampling Property
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