On certain infinite dimensional contragredient modules (Q801158)

Let G be a group and FG be the group algebra of G over a field F, M be an FG-module and \(M^*=Hom_ F(M,F)\) be the contragredient module of M. Let \(H_ i\) be a subgroup of G and \(L_ i\) a one-dimensional \(FH_ i\)- module \((i=1,2)\). In this note, the author deals with the structure of \(Hom_{FG}(FG\otimes_{FH_ 2}L_ 2,(FG\otimes_{FH_ 1}L_ 1)^*)\) and gives an explicit base of the space. In particular, let G be a group with a BN-pair with Weyl group W. Let L be the trivial one dimensional FB-module. It was shown that if G is finite the dimension of the Hecke algebra \(Hom_{FG}(FG\otimes_{FB}L,FG\otimes_{FB}L)\) is equal to \(| W|\), and on the contrary, if G is infinite, in most cases the dimension of the algebra is equal to one. The author shows as an example that if \(| W|\) is finite \(\dim Hom_{FG}(FG\otimes_{FB}L,(FG\otimes_{FB}L)^*)=| W|\) even if G is infinite. Thus the algebra \(Hom_{FG}(FG\otimes_{FB}L,(FG\otimes_{FB}L)^*)\) seems to be a natural generalization of the Hecke algebra of the group G with BN-pairs.