On the Howe correspondence for symplectic–orthogonal dual pairs

Abstract

We reformulate some of Moeglin's results on the correspondence for the dual pairs (Sp(2n, R), O(p, q)) with p and q even, and fill in the cases where p and q are both odd. We arrive at a complete and detailed description, in terms of Langlands parameters, of the dual pair correspondence for the cases p + q = 2n and p + q = 2n + 2. In addition, we point out and suggest a way to correct an error in Moeglin's paper.

The local Langlands conjecture for GSp(4)

“…Moreover, for a finite place, (iv) follows directly from our definition of L-parameters for GSp 4 and Theorem 5.6(iii)(b). For an infinite place, (iv) follows by [AB95] and [Pau05]. Thus, the weak lifting is strong.…”

Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs

“…The non-archimedean case immediately follows from the two theorems in this section. For F ¼ R, this is [26], Theorem 6.2. For F ¼ C, this can be read o¤ from [1], Theorem 2.8, although the reader has to be careful in that the complex valuation in [1] is the usual absolute value i.e.…”

On the theta correspondence for (𝐺𝑆𝑝(4),𝐺𝑆𝑂(4,2)) and Shalika periods

“…In Section 6, we compute the local theta correspondence for GSp + 4 , GSO 4,2 following a similar computation in [13]. In Section 7, for the sake of completeness and for the sake of convenience for the reader, we recall the computation of local theta lifts from Sp 4 (R) to SO 4,2 (R) for discrete series representations by Paul [28].…”

“…In this section, for the sake of completeness of a study of local theta correspondence for the dual pair (GSp 4 , GSO 4,2 ) and also for the sake of convenience of the reader, we shall recall a result by Paul [28] on the local theta correspondence for discrete series representations over the real field R.…”