The Classification of Elements in T × *

In this paper, we give an explicit description of the good, semi-good, and bad elements in T × *. These three concepts were introduced by Xi for proving the Deligne–Langlands conjecture for complex affine Hecke algebras when the order of the nonzero complex parameter q is not too small. First, we define the root graph of a root system and the degenerate paths associated with an element (s, q) T × *. Then we establish a criterion for (s, q) being good in terms of the non-existence of the degenerate paths associated with (s, q) in the corresponding root graph. Finally, we distinguish the bad elements from the semi-good ones by the vanishing of the Poincaré polynomial at q.