Propagation de paires couvrantes dans les groupes symplectiques

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