Let
W
W
be a Weyl group and let
W
′
W’
be a parabolic subgroup of
W
W
. Define
A
A
as follows:
\[
A
=
R
⊗
Q
[
u
]
A
(
W
)
A = R{ \otimes _{{\mathbf {Q}}[u]}}\mathcal {A}(W)
\]
where
A
(
W
)
\mathcal {A}(W)
is the generic algebra of type
A
n
{A_n}
over
Q
[
u
]
{\mathbf {Q}}[u]
an indeterminate, associated with the group
W
W
, and
R
R
is a
Q
[
u
]
{\mathbf {Q}}[u]
-algebra, possibly of infinite rank, in which
u
u
is invertible. Similarly, we define
A
′
A’
associated with
W
′
W’
. Let
M
M
be an
A
−
A
A - A
bimodule, and let
b
∈
M
b \in M
. Define the relative norm [14]
\[
N
W
,
W
′
(
b
)
=
∑
t
∈
T
u
−
l
(
t
)
a
t
−
1
b
a
t
{N_{W,W’}}(b) = \sum \limits _{t \in T} {{u^{ - l(t)}}{a_{{t^{ - 1}}}}b{a_t}}
\]
where
T
T
is the set of distinguished right coset representives for
W
′
W’
in
W
W
. We show that if
b
∈
Z
M
(
A
′
)
=
{
m
∈
M
|
m
a
′
=
a
′
m
∀
a
′
∈
A
′
}
b \in {Z_M}(A’) = \{ m \in M|ma’ = a’m\quad \forall a’ \in A’\}
, then
N
W
,
W
′
(
b
)
∈
Z
M
(
A
)
{N_{W,W’}}(b) \in {Z_M}(A)
. In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Hecke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott. Let
α
=
(
k
1
,
k
2
,
…
,
k
z
)
\alpha = ({k_1},{k_2}, \ldots ,{k_z})
be a partition of
n
n
and let
S
α
=
Π
i
=
1
Z
S
k
i
{S_\alpha } = \Pi _{i = 1}^Z{S_{{k_i}}}
be a "left-justified" parabolic subgroup of
S
n
{S_n}
of shape
α
\alpha
. Define
\[
b
α
=
N
S
n
,
S
α
(
N
α
)
{b_\alpha } = {N_{{S_n},{S_\alpha }}}({\mathcal {N}_\alpha })
\]
, where
\[
N
α
=
∏
i
=
1
z
N
S
k
i
−
1
,
S
1
(
a
w
i
)
{\mathcal {N}_\alpha } = \prod \limits _{i = 1}^z {{N_{{S_{{k_i} - 1}},{S_1}}}({a_{{w_i}}})}
\]
with
w
i
{w_i}
a
k
i
{k_i}
-cycle of length
k
i
−
1
{k_i} - 1
in
S
k
i
{S_{{k_i}}}
. Then the main result of this paper is Theorem. The set
{
b
α
|
α
⊢
n
}
\{ {b_\alpha }|\alpha \vdash n\}
is a basis for
Z
A
(
S
n
)
(
A
(
S
n
)
)
{Z_{A({S_n})}}(A({S_n}))
over
Q
[
u
,
u
−
1
]
{\mathbf {Q}}[u,{u^{ - 1}}]
. Remark. The norms
b
α
{b_\alpha }
in
Z
A
(
S
n
)
(
A
(
S
n
)
)
{Z_{A({S_n})}}(A({S_n}))
are analogs of conjugacy class sums in the center of
Q
S
n
{\mathbf {Q}}{S_n}
and, in fact, specialization of these norms at
u
=
1
u = 1
gives the standard conjugacy class sum basis of the center of
Q
S
n
{\mathbf {Q}}{S_n}
up to coefficients from
Q
{\mathbf {Q}}
.