Kazhdan-Lusztig cells and the Frobenius-Schur indicator

Let W be a finite Coxeter group. It is well known that the number of involutions in W is equal to the sum of the degrees of the irreducible characters of W. Following a suggestion of Lusztig, we show that this equality is compatible with the decomposition of W into Kazhdan-Lusztig cells. The proof uses a generalisation of the Frobenius-Schur indicator to symmetric algebras, which may be of independent interest.