On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups

Abstract

Abstract. The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint γ-factor of its L-parameter. In this paper, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality.

The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case

“…Moreover, if we set φ 0 = L ψ ( π 0 ) and we denote the representation of WD F corresponding to τ i by φ i , we have γ Sh (s i , τ i × π 0 , ψ) = γ(s i , φ i × φ 0 , ψ) up to an invertible function. This follows from [24,Proposition 4.5] and [2]. Therefore, the absolute value of C Sp(W2n) ψ ((s 1 , .…”

Comparison of Local Relative Characters and the Ichino–ikeda Conjecture for Unitary Groups

“…Proof: The proof is the same as in [ILM,lemma A.2] the key fact being that the real exponents of tempered representations of unitary groups are all half integers. In loc.…”

The characterization of theta-distinguished representations of GL(n)

“…By [JS04], [ACS14] and [PR12] (the appendix), there is a generic supercuspidal representation ρ ′ of G n satisfying equalities similar to (4.9). Now according to Ichino, Lapid and Mao [ILM14] (Corollary A.6, which actually holds for the types of G n stated above, when G n is split, by replacing Proposition A.5 with [CKPSS04] Corollary 10.1), there exist a number field with a ring of adèles A and a cuspidal globally generic representation Π ′ of G n (A), such that Π ′ v 1 = π ′ and Π ′ v 2 = ρ ′ for some pair of places v 1 and v 2 .…”

“…First, the functorial lift in the quasi-split case was only described globally ([CPSS11]), leaving out the local lift, which we use. Second, in appealing to [ILM14] (Corollary A.6), we need the following result obtained in [CKPSS04] (Corollary 10.1) for split groups: for a globally generic cuspidal representation Π ′ of G n (A), at all places v, the local Langlands parameters of Π ′ v (the Satake parameters when data are unramified) are bounded in absolute value by 1 2 − 1 N 2 +1 , where Π ′ functorially lifts to a representation of GL N (A). This result follows from the local lift and the strong bounds of Luo, Rudnick and Sarnak [LRS99], and will be obtained once the details of the local lift are provided.…”