  Lusztig's fake degree is the generating polynomial for the major index of
standard Young tableaux of a given shape. Results of Springer and James &
Kerber imply that, mysteriously, its evaluation at a $k$-th primitive root of
unity yields the number of border strip tableaux with all strips of size $k$,
up to sign. This is essentially the special case of the Murnaghan-Nakayama rule
for evaluating an irreducible character of the symmetric group at a rectangular
partition.
  We refine this result to standard Young tableaux and border strip tableaux
with a given number of descents. To do so, we introduce a new statistic for
border strip tableaux, extending the classical definition of descents in
standard Young tableaux. Curiously, it turns out that our new statistic is very
closely related to a descent set for tuples of standard Young tableaux
appearing in the quasisymmetric expansion of LLT polynomials given by Haglund,
Haiman and Loehr.
