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We define an imbedding of the Hecke algebra module carried by the involutionsin a Weyl group W (defined by the author and Vogan) into a completion of theHecke algebra. An analogous result is proved for any Coxeter group. A variantof the construction gives an imbedding of the biregular representation of aHecke algebra of W in a completion of the Hecke algebra of W^2.

An involution based left ideal in the Hecke algebra