Let
π
\pi
be a self-dual supercuspidal representation of
G
L
(
N
,
F
)
GL(N,F)
and
ρ
\rho
a supercuspidal representation of
S
p
(
2
k
,
F
)
Sp(2k,F)
, with
F
F
a local nonarchimedean field of odd residual characteristic. Given a type, indeed a
S
p
(
2
N
+
2
k
,
F
)
Sp(2N+2k,F)
-cover, for the inertial class
[
G
L
(
N
,
F
)
×
S
p
(
2
k
,
F
)
,
π
⊗
ρ
]
S
p
(
2
N
+
2
k
,
F
)
[GL(N,F) \times Sp(2k,F), \pi \otimes \rho ]_{Sp(2N+2k,F)}
satisfying suitable hypotheses, we produce a type, indeed a
S
p
(
2
t
N
+
2
k
,
F
)
Sp(2tN+2k,F)
-cover, for the inertial class
[
G
L
(
N
,
F
)
×
t
×
S
p
(
2
k
,
F
)
,
π
⊗
t
⊗
ρ
]
S
p
(
2
t
N
+
2
k
,
F
)
[GL(N,F)^{\times t} \times Sp(2k,F), \pi ^{\otimes t } \otimes \rho ]_{Sp(2tN+2k,F)}
, for any positive integer
t
t
. We describe the corresponding Hecke algebra as a convolution algebra over an affine Weyl group of type
C
~
t
\tilde C_t
with quadratic relations inherited from the case
t
=
1
t=1
and the structural data for
π
\pi
.