In this paper, we give an explicit realization of all irreducible modules in Chari’s category
O
~
\widetilde {\mathcal O}
for the affine Kac-Moody algebra
A
1
(
1
)
A_{1}^{(1)}
by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of
s
l
2
(
C
)
\mathfrak {sl}_2(\mathbb {C})
with finite weight multiplicities, including the polynomial current algebra
s
l
2
(
C
)
⊗
C
[
t
]
\mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t]
, the loop algebra
s
l
2
(
C
)
⊗
C
[
t
,
t
−
1
]
\mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}]
and the three-point Lie algebra
s
l
2
(
C
)
⊗
C
[
t
,
t
−
1
,
(
t
−
1
)
−
1
]
\mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}]
arisen in the work by Kazhdan-Lusztig.