This paper is devoted to the Harish-Chandra-type decomposition of the global nonsymmetric spherical functions (defined via nonsymmetric Macdonald polynomials) in terms of their asymptotic expansions; the
q
,
t
q,t
–generalization of the celebrated Harish-Chandra
c
c
–function is an important ingredient. This is for any reduced root systems in the
q
,
t
q,t
–setting; we pay special attention to the case of
A
1
A_1
, where this decomposition is very explicit.