#### Glasnik Matematicki, Vol. 37, No.1 (2002), 21-57.

### A FAMILY OF SQUARE INTEGRABLE REPRESENTATIONS
OF CLASSICAL *p*-ADIC GROUPS IN THE CASE OF GENERAL
HALF-INTEGRAL REDUCIBILITIES

### Marko Tadić

Department of Mathematics, University of Zagreb, Bijenička 30,
10000 Zagreb, Croatia

*e-mail:* `tadic@math.hr`

**Abstract.** The main aim of this paper is a
presentation of a large family of non-cuspidal irreducuble square
integrable representations δ(Δ_{1}, ... ,
Δ_{k}, σ)_{τ} 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 fact about these square
integrable representations.
In C. Moeglin and M. Tadic'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 construczion 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 preset 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 future
with the representations of this family, since this family includes all
the generic irreducible square integrable representations
(for example). From Shahidi's conjecture on exsistence 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.

**2000 Mathematics Subject Classification.**
22E50, 22E35.

**Key words and phrases.** Classical groups, symplectic
groups, orthogonal groups, *p*-adic fields, irreducible square
integrable representations.

**Full text (PDF)** (free access)

*Glasnik Matematicki* Home Page