A Mackey formula in the group ring of a Coxeter group

for certain positive integers m(s, s’). If J is a subset of S, let W, be the subgroup generated by J. Let yJ be the character of W induced by the principal character of W, . If J and K are subsets of S, the Mackey formula for the product of induced characters [3, Theorem 44.31 tells us that ~~~ is a sum of certain induced characters, where the summation index ranges over the double coset space W,\W/W, . In a Coxeter group we have precise information about these double cosets so that the Mackey formula may be written in a more explicit way. For w E W let Z(w) denote the length of w as a word in the elements of S. Each (W, , W,) double coset contains a unique element of minimal length [l, Chap. 4, Exercise 1.31. Thus W,\W/ W, has a distinguished cross section which we denote X,, . In case J is empty we write X, = X,, for the distinguished cross section of W/W, . We shall prove in Lemma 2 of Section 2 that if x E X,, , then x-lW,x n W, = W, for some subset L of S. Thus the Mackey formula says