In this paper the author studies two problems. In previous works of Mœglin and C. Mœglin and the author [J. Am. Math. Soc. 15, No. 3, 715–786 (2002; Zbl 0992.22015)], the discrete series of classical groups over \(p\)-adic fields were classified in terms of admissible triples consisting of partial cuspidal support (of that discrete series representation), a set called the Jordan block of that representation, and of a partially defined function (defined, roughly, on the Jordan block). The author gives a parametrization of tempered representations of classical groups extending the notion of admissible triples to the notion of tempered triples (again, one has a partial cuspidal support, now a multiset called the Jordan block) and again the partially defined function on the Jordan block. The author has two reductions in this problem. First, he studies the problem of parameterizing two tempered representations induced from the discrete series representation of a maximal Levi subgroup. He describes the two representations, denoted by “plus” and “minus” describing certain subquotients of the appropriate Jacquet module existing only in one of these representations; the ”plus” tempered representation. Some Bernstein center considerations then turn this description into the existence of some embeddings of this tempered representation. The second reduction concerns the tempered representations induced from the discrete series of a non-maximal Levi subgroup, but in the case which, for symplectic and odd-orthogonal groups, coincides with the elliptic tempered representations. Now, the general tempered representations of classical groups are irreducibly induced from these (in the case of symplectic and odd-orthogonal groups elliptic) tempered representations.
The second problem which the author studies in this paper to give a Jacquet module interpretation of the partially defined function (one of the invariants of the discrete series representations described above) on the Jordan block of a discrete series. He then explicitly elaborates this situation in the case when a discrete series is obtained from another discrete series (of a smaller classical group of the same type) by the so-called deformation of the Jordan block. In the appendix, the author settles one reducibility result for the parabolic induction of the generalized principal series (induction from the essentially square-integrable representation of a maximal Levi subgroup). This result extends the author’s previous result in the similar situation when the induction was from a maximal Levi subgroup where on the part belonging to the smaller classical group one has a cuspidal representation.