Centers of generic Hecke algebras
Authors: Jones Lenny K. 
Published: June 2012

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]
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript upper W comma upper W Sub Superscript prime Subscript Baseline left-parenthesis b right-parenthesis equals sigma-summation Underscript t element-of upper T Endscripts u Superscript minus l left-parenthesis t right-parenthesis Baseline a Subscript t Sub Superscript negative 1 Baseline b a Subscript t">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b element-of upper Z Subscript upper M Baseline left-parenthesis upper A prime right-parenthesis equals StartSet m element-of upper M vertical-bar m a Superscript prime Baseline equals a prime m for-all a Superscript prime Baseline element-of upper A prime EndSet">

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 This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott.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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript alpha Baseline equals product Underscript i equals 1 Overscript z Endscripts upper N Subscript upper S Sub Subscript k Sub Sub Subscript i Sub Subscript minus 1 Subscript comma upper S 1 Baseline left-parenthesis a Subscript w Sub Subscript i Subscript Baseline right-parenthesis">

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}}

.