p] and p is not a

root of unity then we can write [I.

By studying special features of our construction when the quantization parameter is a

root of unity, realize the Verlinde algebra as a module over the DAHA, shedding new light on fundamental results of Cherednik and Witten.

An n-th

root of unity, where n is a positive integer (i.

0) For every

root of unity [zeta] there is a [theta]-function [theta] [sub.

If d [greater than or equal to] 1 divides n and w is a primitive d-th

root of unity, then

This notion has been generalized to the notion of graded q-differential algebra, where q is a primitive Nth

root of unity (see papers [7,8,10]).

If e is invertible on Z, Z/l (1) denotes the sheaf of l-th

root of unity and for any integer i, we denote Z/l (i) = [(Z/l (1)).

A complex number [omega] is said to be an n-th

root of unity if [omega.

Since every root of [mu](x) mod P coincides with [theta] mod P for some choice of primitive nth

root of unity [[xi].

n](x), which is defined as the minimal polynomial over Q for any primitive nth

root of unity [zeta] in C.

Since all the roots of the numerator and the denominator of X(q) are roots of unity, this follows from the fact that for each d-th primitive

root of unity [[omega].

In [12], this formula was studied in the case where q is an arbitrary

root of unity, and higher order analogs of the peak algebra were obtained.