  In the case of p-adic general linear groups, each irreducible representation
is parabolically induced by a tensor product of irreducible representations
supported by cuspidal lines. One gets in this way a parameterization of the
irreducible representations of p-adic general linear groups by irreducible
representations supported by cuspidal lines. It is obvious that in this
correspondence an irreducible representation of a p-adic general linear group
is unitarizable if and only if all the corresponding irreducible
representations supported by cuspidal lines are unitarizable. C. Jantzen has
defined an analogue of such correspondence for irreducible representations of
classical p-adic groups. It would have interesting consequences if one would
know that the unitarizability is also preserved in this case. A purpose of this
paper and its sequel, is to give some very limited support for possibility of
such preservation of the unitarizability. More precisely, we show that if we
have an irreducible unitarizable representation $\pi$ of a classical p-adic
group whose one attached representation $\pi_L$ supported by a cuspidal line
$L$ has the same infinitesimal character as the generalized Steinberg
representation supported by that cuspidal line, then $\pi_L$ is unitarizable.
