A family of square integrable representations of classical p-adic groups in the case of general half-integral reducibilities

Abstract

The main aim of this paper is a presentation of a construction of a large family of non-cuspidal irreducible square integrable representations ( 1 , . . ., k , ) of symplectic and odd-orthogonal p-adic groups, starting from the cuspidal representations of the Levi subgroups.The only information that we need about these irreducible cuspidal representations are the generalized rank one reducibilities.We also get a number of interesting facts about these square integrable representations.In C. Moeglin and M. Tadi's paper "Construction of discrete series for classical p-adic groups" (J.Amer.Math.Soc. 15 (2002), 715-786) is given a general construction of all the irreducible square integrable representations of the classical p-adic groups modulo cuspidal data (under a natural assumption).The construction of the family that we present in this paper preceded the general construction.Although the general construction gives a construction of all the square integrable representations of classical p-adic groups, it is interesting to have also available this former construction.Namely, the construction that we present here is much more direct than the general construction, and it gives a number of explicit information about representations.These facts may be useful in further study of the representations of the family that we construct in this paper.It is for expecting that we shall deal a lot in the future with the representations of this family, since this family includes all the generic irreducible square integrable representations (for example).From the Shahidi's conjecture on existence of a generic representation in each L 2 L-packet, would follow that each L 2 L-packet contains some of the representation from the family whose construction we present in this paper.