The theta correspondence for similitudes

Quadratic base change for p-adic SL(2) as a theta correspondence II: Jacquet modules

“…Cognet's statements are at the level of the similitude groups G 1 and H 1 . However, it is a straightforward argument using results relating similitude theta correspondences to regular theta correspondences (see, e.g., [B] or [Ro1]) to obtain the result above from [C].…”

The local Langlands conjecture for GSp(4)

“…Note that this differs from the normalization used in [Rob96]. Observe in particular that the central elements (t, t −1 ) ∈ R 0 act by the quadratic character…”

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

“…It was shown by Roberts [30] that this follows from the Howe conjecture for isometry groups, and thus holds if the residual characteristic is not 2.…”

“…By [1,Theorem 1.4] and [30,Theorem 4.4 (3)], this multiplicity one theorem implies that θ(σ) is multiplicity free. Thus, θ(σ) should be irreducible whenever Θ(σ) is nonzero.…”

“…Let us recall briefly the local theta correspondence for similitude groups (cf. [30] and [13,Section 2]). Let W be a symplectic space over F and V an even dimensional nondegenerate symmetric bilinear space over F .…”