A generalized Theta lifting, CAP representations, and Arthur parameters

Abstract

We study a new lifting of automorphic representations using the theta representation Θ on the 4-fold cover of the symplectic group, Sp 2r (A). This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to Θ.We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to Sp 2r (A). Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of Sp 2r (A).Proof. See [29, Section7.1]. 7.2. Local Root Exchange. Suppose now that F is a non-archimedean local field, and let G be the F -rational points of a split algebraic group over F or finite cover thereof. As in the global setting, we consider unipotent subgroups C, X, Y ⊂ G, and a nontrivial characterAs before, set B = Y C, D = XC, and A = BX = DY . We also assume that these subgroups satisfy the local analogue of the six properties preceding Lemma 7.1 (For a precise statement and proof, see [13, Section 6.1]).Lemma 7.2. Suppose that π is a smooth representation of A, and extend the character ψ C trivially to ψ B on B and ψ D on D. Then we have an isomorphism of C-modules J B,ψ B (π) ∼ = J D,ψ D (π).Moreover, J C,ψ C (π) = 0 ⇐⇒ J B,ψ B (π) = 0 ⇐⇒ J D,ψ D (π) = 0.

On the wavefront sets associated with theta representations

“…(2) 2r+1 , we also show that Conjecture 2.5 agrees with the respective work [Les19], [BFG03] and [Kap17], as mentioned above; see the end of §4. (iii) In §5, we consider the archimedean analogue of Conjecutre 2.5, and recapitulate the relevant work in the literature, see Theorem 5.6 for the main result.…”

“…(2) 2r+1 was also carried out in [Kap17]. For degree-four cover of Sp 2r , a theory of theta lifting was investigated by Leslie [Les19]. For Sp (n) 2r with odd n, the theta representations were studied in depth by Friedberg and Ginzburg towards a theory of generalized theta liftings and descent on such high degree covers of Sp 2r , see the series of papers [FG18,FG17a,FG17b,FG].…”

Theta liftings of non-generic representations on double covers of orthogonal groups

“…Hence, J Z(Hs 1 ) (Θ 4k+2l+3 ) ∼ = Θ 4k+2l+3 . By Proposition 5.12(d) of [1] or Lemma A.1 of [16], there exists a short exact sequence of Q s 1 (F ) × SO 2k+1 (F )-modules (6.15) 0 → ind…”

“…Jiang [13] proposed a conjecture that relates Arthur parameters to the maximal unipotent orbit that supports an automorphic representation. In Section 13 of [16], Leslie conjectured an extension of Arthur parameters to the metapletic groups. In view of these works, the results of this paper are conjecturally related to the question of how Arthur parameters behave under the non-minimal theta liftings introduced in [5].…”

Kazhdan–Lusztig representations and Whittaker space of some genuine representations

“…The representation π C + is the unique irreducible subrepresentation of I(χ), and is the covering analogue of the Steinberg representation. Meanwhile, π C − is the unique irreducible Langlands quotient of I(χ), the so-called theta representation Θ(G, χ) (in the notation of [Gao17]) associated to the exceptional character χ, see also [KP84,Cai,Les].…”