DISCRETE SERIES FOR p-ADIC SO(2n) AND RESTRICTIONS OF REPRESENTATIONS OF O(2n)

In [M-T] (also, cf. [Mœ2]), Mœglin and Tadić construct the discrete series for a number of families of classical groups. However, they do not address discrete series for the split classical group SO(2n, F ), only the nonsplit ones. The basic reason for this is that the Weyl groups are different: the groups considered by Mœglin and Tadić all have Weyl groups of the form W ∼= { permutations and sign changes on n letters }, whereas the Weyl group for SO(2n, F ) requires the number of sign changes to be even. This introduces a number of complications, which we take a moment to discuss. The complications go beyond simple changes in the combinatorics. For example, one datum which appears in the admissible triples used by Mœglin and Tadić in the classification of discrete series is the partial cuspidal support of an irreducible representation. For SO(2n, F ), there is not a corresponding notion of partial cuspidal support (or more precisely, the corresponding partial cuspidal support can consist of more than one representation–cf. Example 8.1). At a subtler level, for the groups they consider, the Jordρ (where Jordρ = {(ρ′, a) ∈ Jord | ρ′ ∼= ρ}) for different ρ are essentially independent of each other (cf. section 14.5 [M-T] for a more detailed discussion). From the standpoint of [J1],[J4], this has its roots in the observation (cf. [G1],[G2]) that if ρ1, . . . , ρk are irreducible unitary supercuspidal representations of general linear groups and σ is an irreducible supercuspidal representation of an appropriate classical group, then Ind ((ρ1 ⊗ · · · ⊗ ρ1)⊗ · · · ⊗ (ρk ⊗ · · · ⊗ ρk)⊗ σ) has 2 components, where m = |{i | Ind(ρi ⊗ σ) is reducible }|. For SO(2n, F ), the situation is different (cf. [G1])–e.g., one can have Ind(ρ1 ⊗ ρ2 ⊗ σ0) reducible even if both Ind(ρ1 ⊗ σ0) and Ind(ρ2 ⊗ σ0) are irreducible. At a more practical level, the μ∗ structure of [T2], which figures prominently in the paper, did not have an SO(2n, F ) counterpart (though note the subsequent development of such in [J5]). This leaves two obvious strategies for the classification of discrete series for the groups SO(2n, F ). One approach is to emulate the work of Mœglin and Tadić, making the requisite changes along the way. Another approach is to start with the Mœglin-Tadić classification of discrete series for the groups O(2n, F ) and study restrictions to SO(2n, F ). By a lemma of [G-K] (essentially Mackey theory–see Lemma 2.3 below), this is equivalent to studying when ĉπ ∼= π, where ĉ denotes the character of O(2n, F ) which is 1 on SO(2n, F ) and −1 on O(2n, F ) \ SO(2n, F ). We adopt the latter approach. Note that this requires retaining the Basic Assumption (BA) of [M-T], which we do (though the former approach would certainly require something like this as well). Owing to its somewhat technical nature, we