We do not have a function for the generic degrees of an Iwahori-Hecke algebra since they are not always defined (for example, if the parameters of the algebra are roots of unity). If we have a generic multi-parameter Iwahori-Hecke algebra H then the generic degrees are certainly defined, and they can be computed with the command:
List( SchurElements( H ), x - PoincarePolynomial( H ) / x );
(See PoincarePolynomial and SchurElement.)
GAP 3.4.4