Let
U
n
(
q
)
U_{n}(q)
denote the unitriangular group of degree
n
n
over the finite field with
q
q
elements. In a previous paper we obtained a decomposition of the regular character of
U
n
(
q
)
U_{n}(q)
as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character
ξ
D
(
φ
)
\xi _{{\mathcal {D}}}(\varphi )
of
U
n
(
q
)
U_{n}(q)
. We prove that
ξ
D
(
φ
)
\xi _{ {\mathcal {D}}}(\varphi )
is induced from a linear character of an algebra subgroup of
U
n
(
q
)
U_{n}(q)
, and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of
ξ
D
(
φ
)
\xi _{{\mathcal {D}}}(\varphi )
as characters induced from an algebra subgroup of
U
n
(
q
)
U_{n}(q)
. Finally, we identify a special irreducible constituent of
ξ
D
(
φ
)
\xi _{{\mathcal {D}}}(\varphi )
, which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption
p
≥
n
p \geq n
where
p
p
is the characteristic of the field) that gives a necessary and sufficient condition for
ξ
D
(
φ
)
\xi _{{\mathcal {D}}}(\varphi )
to have a unique irreducible constituent.