An involution based left ideal in the Hecke algebra

TL;DR

This paper introduces an embedding of the involution module of a Hecke algebra into its completion, extending the result to all Coxeter groups and relating the biregular representation to a completion of W^2's Hecke algebra.

Contribution

It provides a new embedding construction for the involution module in Hecke algebras, generalizing previous work and connecting to the biregular representation.

Findings

Embedding of the involution module into the Hecke algebra completion.

Extension of the embedding result to all Coxeter groups.

Embedding of the biregular representation into a completion of W^2's Hecke algebra.

Abstract

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