Last update: 20 September 2012
In the following, for simplicity of exposition we shall assume that we are not in the special case of [15, (2.16)] where the root system is type and is the (co)root lattice. All our results and proofs are valid in this special case but the definition of the double affine Hecke algebra and the formulas in (2.32) and (2.34) may need some slight modification. See Remark 2.3 for details.
Let be the set of coroots and dix parameters indexed by such that for all and
The double affine Hecke algebra is the group algebra of the double braid group with the additional relations
The double affine Hecke algebra has bases
(see [6, Prop. 5.4 and Cor. 5.8]).
In the presence of (2.31) the relations (2.29) are equivalent to
Using that the satisfy the braid relations and that
Let and let be a reduced word for For let
so that the sequence is the sequence of labels of the hyperplanes crossed by the walk For example, in Type with the picture is
Let An alcove walk of type beginning at is a sequence of steps, where a step of type is
Let be the set of alcove walks of type beginning at For a walk let
Let let be a reduced word for and let be as defined in (2.35). Then, in
where the sum is over all alcove walks of type beginning at
The proof is by induction on the length of the base case being the formulas in (2.34). To do the induction step let
(by a crossing and a fold, respectively). Let By induction, a term in is
The last step of is
In some special cases when the affine root system is nonreduced (see [15, (2.1.6)]) the formulas in (2.32) and (2.34) need modification and the definition of the double affine Hecke algebra may need an additional relation. The most involved of these cases is type (see [15, (1.4.3)]) where the double affine Hecke algebra needs additional parameters and (in the notation of , and ) and additional relations
and the formulas for and need to be changed to
The statement and proof of the analogue of Theorem 2.2 for this case is the same, except with the factors associated to the 0-folds and -folds replaced by the rational functions in which appear in the expressions of and in Equations (2.40), (2.41), and (2.39).
This page is taken from a paper entitled A combinatorial formula for Macdonald Polynomials by Arun Ram and Martha Yip.