On unitarity of some representatations of classical p-adic groups I

Abstract

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.