  Let $\pi $ be an irreducible smooth complex representation of a general
linear $p$-adic group and let $\sigma $ be an irreducible complex supercuspidal
representation of a classical $p$-adic group of a given type, so that
$\pi\otimes\sigma $ is a representation of a standard Levi subgroup of a
$p$-adic classical group of higher rank. We show that the reducibility of the
representation of the appropriate $p$-adic classical group obtained by
(normalized) parabolic induction from $\pi\otimes\sigma $ does not depend on
$\sigma $, if $\sigma $ is "separated" from the supercuspidal support of $\pi
$. (Here, "separated" means that, for each factor $\rho $ of a representation
in the supercuspidal support of $\pi $, the representation parabolically
induced from $\rho\otimes\sigma $ is irreducible.) This was conjectured by E.
Lapid and M. Tadi\'c. (In addition, they proved, using results of C. Jantzen,
that this induced representation is always reducible if the supercuspidal
support is not separated.)
  More generally, we study, for a given set $I$ of inertial orbits of
supercuspidal representations of $p$-adic general linear groups, the category
$\CC _{I,\sigma}$ of smooth complex finitely generated representations of
classical $p$-adic groups of fixed type, but arbitrary rank, and supercuspidal
support given by $\sigma $ and $I$, show that this category is equivalent to a
category of finitely generated right modules over a direct sum of tensor
products of extended affine Hecke algebras of type $A$, $B$ and $D$ and
establish functoriality properties, relating categories with disjoint $I$'s. In
this way, we extend results of C. Jantzen who proved a bijection between
irreducible representations corresponding to these categories. The proof of the
above reducibility result is then based on Hecke algebra arguments, using
Kato's exotic geometry.
