Description:
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 .