The Shimura-Waldspurger Correspondence for Mp(2n)

The Shimura-Waldspurger Correspondence for Wee Teck Gan (Joint with Atsushi Ichino) April 11, 2016

The Problem Let 1 µ 2 Sp(2n) 1 be the metaplectic group (both locally and globally). Consider the genuine automorphic discrete spectrum Ã disc := Ã disc ()

The Problem Let 1 µ 2 Sp(2n) 1 be the metaplectic group (both locally and globally). Consider the genuine automorphic discrete spectrum Ã disc := Ã disc () Problem: Describe decomposition of Ã in the style of Arthur s conjecture When n = 1, this was done by Waldspurger around 1980.

Local Shimura Correspondence Let k C be a local field and fix a nontrivial character Set ψ : k C. Irr(Mp(2n, k)) = {irred. genuine representations of Mp(2n, k)}.

Local Shimura Correspondence Let k C be a local field and fix a nontrivial character Set ψ : k C. Irr(Mp(2n, k)) = {irred. genuine representations of Mp(2n, k)}. Theorem (Adams-Barbasch, G-Savin) There is a bijection, depending on ψ: θ ψ : Irr(Mp(2n, k)) V n Irr(SO(V n )) as V n runs over all 2n + 1-dimensional quadratic space of discriminant 1. The bijection is defined by local theta correspondence with respect to ψ.

LLC for By combining with LLC for SO(2n + 1) (Arthur, Moeglin), one gets: Corollary One has: Irr(Mp(2n, k) {(φ, η)} where φ : WD k Sp 2n (C) is an L-parameter for SO(2n + 1);

LLC for By combining with LLC for SO(2n + 1) (Arthur, Moeglin), one gets: Corollary One has: Irr(Mp(2n, k) {(φ, η)} where φ : WD k Sp 2n (C) is an L-parameter for SO(2n + 1); η Irr(S φ ), where S φ is the component group of φ. Given φ, have local L-packet Π φ,ψ = {σ φ,η : η Irr(S φ )}.

Global Results F a number field and ψ : F \A C. Theorem (A) There is a decomposition Ã = Ψ Ã Ψ,ψ where Ψ runs over elliptic global A-parameters of and each Ã Ψ is a near equivalence class determined by Ψ and ψ. Here, elliptic global A-parameters of = elliptic global A-parameters of SO(2n + 1).

Elliptic A-parameters Ψ = i Π i S di satisfying: Π i cuspidal rep. of GL(2n i ) which is of symplectic or orthogonal type if d i is odd or even.

Elliptic A-parameters Ψ = i Π i S di satisfying: Π i cuspidal rep. of GL(2n i ) which is of symplectic or orthogonal type if d i is odd or even. S di is irred. rep. of SL 2 (C) of dim. d i

Elliptic A-parameters Ψ = i Π i S di satisfying: Π i cuspidal rep. of GL(2n i ) which is of symplectic or orthogonal type if d i is odd or even. S di is irred. rep. of SL 2 (C) of dim. d i n 1 d 1 +... + n r d r = n;

Near Equivalence Class Each Ψ gives rise to a near equivalence class of representations of Mp(2n, A) via the local Shimura correspondence For almost all v, Π i,v is unramified for all i, with L-parameter φ i,v. Set φ Ψv = (d i 1)/2 φ i,v (d i 3)/2... i (d i 1)/2 This defines an unramified L-parameter for SO(2n + 1, F v ) and determines an unramified representation of Mp(2n, F v ) by local Shimura corr.

The Tempered Part Theorem (B) We have an explicit description of Ã Ψ,ψ when Ψ = i Π I is tempered (i.e. d i = 1 for all i)

The Tempered Part Theorem (B) We have an explicit description of Ã Ψ,ψ when Ψ = i Π I is tempered (i.e. d i = 1 for all i) Namely, one has: for each v, Ψ v gives rise to the local L-packet Π Ψv,ψ v the global L-packet Π Ψ,ψ = v Π Ψv,ψ v. global and local component groups: : S Ψ = i Z/2Za i S Ψ,A := v S Ψv. a quadratic character given Wee Teck bygan (Joint with Atsushi Ichino) ɛ Ψ : S Ψ {±1}

Multiplicity Formula Then each η Irr(S Ψ,A ) gives rise to σ η = v σ ηv IrrMp(2n, A). One has with Ã Ψ,ψ = m η = η IrrS Ψ,A m η σ η { 1 if (η) = ɛ Ψ ; 0 otherwise.

Theta Lifting Proof of Theorems relies on global theta correspondence between and split SO(2n + 2r + 1) with r > n. This has been studied by J.S. Li. Given abstract rep. set σ = v σ v Irr(Mp(2n, A)), m disc (σ) = dim Hom (σ, Ã disc ) m(σ) = dim Hom (σ, Ã).

Theta Lifting Proof of Theorems relies on global theta correspondence between and split SO(2n + 2r + 1) with r > n. This has been studied by J.S. Li. Given abstract rep. set Have local theta lift Set σ = v σ v Irr(Mp(2n, A)), m disc (σ) = dim Hom (σ, Ã disc ) m(σ) = dim Hom (σ, Ã). 0 θ ψv (σ v ) IrrSO(2n + 2r + 1, F v ). θψ abs (σ) = v θ ψv (σ v ) IrrSO(2n + 2r + 1, A).

Results of J.S. Li Theorem (J.S. Li) One has: m disc (σ) m disc (θ abs (σ)) m(θ abs (σ)) m(σ).

Results of J.S. Li Theorem (J.S. Li) One has: m disc (σ) m disc (θ abs (σ)) m(θ abs (σ)) m(σ). Corollary Let Σ Ã disc be a near equivalence class, say Σ = i m i σ i. Set θψ abs (Σ) = i m i θψ abs (σ i). Then θ abs ψ (Σ) A disc(so(2n + 2r + 1))

Assignment of A-parameters Thus, to θψ abs (Σ), one can assign by Arthur an A-parameter of SO(2n + 2r + 1). Proposition Ψ r = i Π i S di Ψ r = Ψ S 2r for an elliptic A-parameter Ψ of SO(2n + 1). Hence, one defines Ψ to be the A-parameter of Σ, so that This gives Thm. A. Σ = Ã Ψ.

Proof of Prop Consider in two different ways: One one hand, it is equal to L S (s, θψ abs (σ)) L S ψ (s, σ) ζs (s + r 1 2 ) ζs (s + r 3 2 )... ζs (s r + 1 2 ). This has largest pole at s = r + 1 2 > n + 1.

Proof of Prop Consider in two different ways: One one hand, it is equal to L S (s, θψ abs (σ)) L S ψ (s, σ) ζs (s + r 1 2 ) ζs (s + r 3 2 )... ζs (s r + 1 2 ). This has largest pole at On the other hand, it is equal to i d j j=1 s = r + 1 2 > n + 1. L S (s + d i + 1 2 j, Π i ). For this to have largest pole at s = r + 1/2, need S 2r Ψ r.

Equality? What is the difference between θ abs ψ (Σ) = θabs ψ ( A Ψ,ψ ) A Ψ r? If equality holds, then one can simply transport the description of A Ψ r to get a description of Ã Ψ,ψ.

Equality? What is the difference between θ abs ψ (Σ) = θabs ψ ( A Ψ,ψ ) A Ψ r? If equality holds, then one can simply transport the description of A Ψ r to get a description of Ã Ψ,ψ. Lemma Given π A Ψ r, one has for some σ on Mp(2n, A). π = θψ abs (σ) Since m disc (π) > 0, J.S. Li s in equality implies m(σ) > 0, i.e. σ is automorphic.

Key Prop for Tempered Ψ Proposition If Ψ is tempered, then with σ as above m cusp (σ) = m disc (σ) = m(σ), so that m disc (σ) = m disc (θ abs (σ)). In particular θ abs (Ã Ψ,ψ ) = A Ψ+S2r This implies that the structure of Ã Ψ,ψ is the same as that of A Ψ+S2r. For example, one can transport the Arthur multiplicity formula for Ψ + S 2r to Ψ. Note: ɛ Ψ+S2r = ɛ Ψ.

Local packets Only remaining issue for Thm. B is: show that the local packets of Mp 2n inherited from Ψ + S 2r on SO(2n + 2r + 1) is equal to the local L-packet defined by the local Shimura correspondence, i.e. Ã Ψv,ψ v := θ ψv (A Ψ r v ) = Π Ψv? Is the labelling by IrrS Ψv the same on both sides? These are purely local questions, but is the most difficult part of our argument. We use local results of Moeglin on explicating the local A-packets of SO and global arguments involving the global multiplicity formula of Arthur for all inner forms of SO(2n + 1).

Proof of Key Prop We know m cusp (σ) m disc (σ) m(σ). Need to show: any σ Ã has image in Ã cusp.

Proof of Key Prop We know m cusp (σ) m disc (σ) m(σ). Need to show: any σ Ã has image in Ã cusp. Suppose not. Near equiv. class of σ given by Ψ = σ has weak lift to GL(2n) given by i Π i a multiplicity-free sum of cuspidal reps of symplectic type.

Proof continued If σ not cuspidal, then σ Ind P ρ, with ρ cuspidal. If M = GL(k 1 ) µ2... µ2 GL(k r ) µ2 Mp(2n 0 ), then ρ = i τ i σ 0. Then σ has weak lift to GL(2n) of the form i (τ i τ i ) Ψ 0

Proof continued If σ not cuspidal, then σ Ind P ρ, with ρ cuspidal. If M = GL(k 1 ) µ2... µ2 GL(k r ) µ2 Mp(2n 0 ), then ρ = i τ i σ 0. Then σ has weak lift to GL(2n) of the form i (τ i τ i ) Ψ 0 This is not a multiplicity-free sum of symplectic-type cuspidal reps. This gives the desired contradiction.

Endoscopy Wen-Wei Li has developed a theory of endoscopy for with elliptic endoscopic groups H a,b = SO(2a + 1) SO(2b + 1) with a + b = n. The local L-packets should satisfy local character identities: this is ongoing thesis work of my student Caihua Luo. Wen-Wei has stabilised the elliptic part of the invariant trace formula of. Using this, one should be able to define local L-packets for. So the point is to relate the two notions of L-packets. I suspect we will hear more about this in Wen-Wei s talk.

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