LOCAL THETA CORRESPONDENCES FOR U(n, n) AND R-GROUPS
ATSUSHI ICHINO
1. Introduction
In this article, we discuss the local theory of the theta lifting. This theory is called the local theta correspondence or the Howe duality [3].
Besides its own interest, it is also necessary to study the correspondence for automorphic representations.
Let G = G′ =U(n, n) be the quasi-split unitary groups in 2n vari- ables over ap-adic field. Then (G, G′) is a reductive dual pair inSp4n2. Let π be an irreducible admissible representation of G. By the local theta correspondence, which is proved by Waldspurger [11] whenp̸= 2, there exists at most one irreducible admissible representation θ(π) of G′ such that
HomG×G′(ω⊗π, θ(π))̸= 0.
Hereω is the Weil representation ofG×G′. In general, it remains still difficult to describe the correspondence explicitly, but we can determine it completely for certain tempered representations.
In terms of the Arthur conjecture [1], we can give a brief account of our result. The local theta correspondence preserves the Langlands parameter in this case, hence induces a twist inside of the (tempered) L-packet. Remarkably, this twist is non-trivial and determined by root numbers!
2. The main theorem
We now give a more precise description of the main theorem. Let F be a p-adic field with p̸= 2 and E a quadratic extension of F. Fix δ∈E× such that trE/F(δ) = 0. We realizeG(resp.G′) as the isometry group of the hermitian (resp. skew-hermitian) form given by
( −δ1n δ1n
) ( resp.
( 1n
−1n
)) .
In fact, G = G′. Then we have the Weil representation ω of G×G′ for a fixed non-trivial additive character ψF of F. Note that ω also depends on the choice of δ.
1
Let π be an irreducible tempered representation of G. Then π is realized as an irreducible component of an induced representation I(σ) = IndGP σ, whereP is a parabolic subgroup ofGandσ is a discrete series representation of the Levi component L of P. In the following, we assume that L≃GLn1(E)× · · · ×GLnr(E) with n1+· · ·+nr =n.
By the induction principle of Kudla [6], we see that θ(π) is an irre- ducible component of an induced representation I′(σ) = IndGP′′σ when σ is supercuspidal. Here we regard P′ = P as a parabolic subgroups of G′. However, this principle does not determine which component corresponds to π.
On the other hand, we can parameterize the irreducible components of I(σ) by the theory of R-groups, which is due to Harish-Chandra [10], Knapp-Stein [5], and Silberger [9]. We now review this theory and recall the computation of R-groups by Goldberg [2].
We writeσ =σ1⊗· · ·⊗σr whereσi is a discrete series representation of GLni(E) on the space Vi for 1 ≤ i ≤ r. Let µi be the central character of σi. For λ ∈ Cr, put I(σ, λ) = IndGP σ| |λ. In particular, I(σ) = I(σ,0). Let W be the Weyl group of G with respect to the split component of the center of L. Note that W is isomorphic to a subgroup of (Z/2Z)r⋊Sr. For a representative w ∈ G of an element of W and a holomorphic section f(λ) of I(σ, λ), put
M(w, σ, λ)f(λ)(g) =
∫
U∩wU w−1\U
f(λ)(w−1ug)du.
Here U is the unipotent radical of P. This integral is absolutely con- vergent if Re(λ1)≫ · · · ≫Re(λr)≫0, and extends to a meromorphic function of λ on Cr. It is well-known that there exists a normaliza- tion factor r(w, σ, λ) of the intertwining operator M(w, σ, λ). Namely, this factor is a meromorphic function of λ such that the normalized intertwining operator
N(w, σ, λ) =r(w, σ, λ)−1M(w, σ, λ) is holomorphic on √
−1Rr. Moreover, we can choose normalization factors so that the cocycle condition
N(ww′, σ,0) =N(w, w′σ,0)N(w′, σ,0)
holds for representatives w, w′ ∈ G of elements of W. For 1 ≤ i ≤ r, let ri ∈ W be the sign change at the i-th component. We put ki =
n1 +· · ·+ni−1,mi =ni+1+· · ·+nr, and we take
wi =
1ki
1ni 1mi
1ki
−1ni
1mi
∈G
for a representative of ri. Let I be the set of i such that
• σi ≃tσ¯i−1,
• σi ̸≃σj for all j > i,
• the Asai L-functionLAsai(s, σi) is holomorphic at s= 0.
Let R be the subgroup of W generated by ri for all i ∈ I. Note that R ≃ (Z/2Z)#I. For each i ∈ I, we fix an isomorphism Ai : Vi → V∼ i
such that A2i = idVi and Aiσi(a) =tσ¯−i 1(a)Ai for a∈GLni(E). Then N(ri, σ) =µi(δ)AiN(wi, σ,0)
is a self-intertwining operator of I(σ). For ri = ri1· · ·rik ∈ R with i={i1, . . . , ik} ⊂I, put
N(ri, σ) = N(ri1, σ)· · · N(rik, σ).
Then r7→ N(r, σ) is a homomorphism. For each κ∈R, letˆ πκ ={f ∈I(σ)| N(r, σ)f =κ(r)f for all r ∈R}. By the theory of R-groups, we see that πκ is irreducible and
I(σ) =⊕
κ∈Rˆ
πκ.
Moreover, replacingAiwith−Ai if necessary, we may assume thatπ1is χ-generic. Hereχ is a fixed non-degenerate character of the unipotent radical of the standard Borel subgroup of G.
Similarly, we writeI′(σ, λ) = IndGP′′σ| |λandI′(σ) =I′(σ,0). Here we regardP′ =P as a parabolic subgroup ofG′. We also regard W′ =W as the Weyl group of G′, wi′ = wi as an element of G′, ri′ = ri as an element ofW′,R′ =Ras a subgroup ofW′, and χas a non-degenerate character of the unipotent radical of the standard Borel subgroup of G′. For each κ′ ∈Rˆ′, let
π′κ′ ={f′ ∈I′(σ)| N(r′, σ)f′ =κ′(r′)f′ for all r′ ∈R′}.
Here we regard N(r′, σ) as a self-intertwining operator ofI′(σ). Then πκ′′ is irreducible, π1′ is χ-generic, and
I′(σ) = ⊕
κ′∈Rˆ′
πκ′′.
Then our main result is as follows.
Theorem 2.1. Let κ∈R. Thenˆ θ(πκ) =πκ′′ where κ′(r′i) =κ(ri)·ϵ(1/2, σi, ψF ◦trE/F)µi(δ)−1 for i∈I.
Remark 2.2. This is consistent with the conjecture of Prasad [8].
Remark 2.3. Even if p= 2, we can prove that HomG×G′(ω,˜πκ ⊗πθ(κ)′ )̸= 0.
Remark 2.4. For the case G=U(n, n+ 1) and G′ =U(n, n), it seems that root numbers do not appear in the local theta correspondence. In fact, the correspondence from Gto G′ preserves the genericity.
3. The proof
The idea of the proof is very simple and straightforward. In fact, we construct a certain element T of
HomG×G′(ω⊗I(σ, λ), I′(σ, λ)) which satisfies the following conditions:
(i) For any f ∈I(σ), there exists Φ∈ω such that T(Φ, f)̸= 0.
(ii) Let i∈I, Φ∈ω, and f(λ) ∈I(σ, λ). Then M(w′i, σ, λ)T(Φ, f(λ))
=|δ|niλiµi(δ)−1ϵ(−λi+ 1/2, σi, ψ)T(Φ, M(wi, σ, λ)f(λ)).
From these properties, we can easily deduce our main result. See [4]
for more details.
Remark 3.1. There will be one difficulty in extending our result to gen- eral parabolic subgroups. In fact, it is necessary to compare the nor- malization factors (or the Plancherel measures) forGandG′. However, if the inducing data is generic, then we might overcome this difficulty by the result of Mui´c and Savin [7].
References
[1] J. Arthur, Unipotent automorphic representations: conjectures, Ast´erisque 171-172(1989), 13–71.
[2] D. Goldberg, R-groups and elliptic representations for unitary groups, Proc.
Amer. Math. Soc.123(1995), 1267–1276.
[3] R. Howe, θ-series and invariant theory, Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math. 33-1, Amer. Math. Soc., 1979, pp. 275–285.
[4] A. Ichino,On the local theta correspondence and R-groups, preprint, 2001.
[5] A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, II, Invent. Math.60(1980), 9–84.
[6] S. S. Kudla,On the local theta-correspondence, Invent. Math.83(1986), 229–
255.
[7] G. Mui´c and G. Savin, Symplectic-orthogonal theta lifts of generic discrete series, Duke Math. J.101(2000), 317–333.
[8] D. Prasad, Theta correspondence for unitary groups, Pacific J. Math. 194 (2000), 427–438.
[9] A. J. Silberger, The Knapp-Stein dimension theorem for p-adic groups, Proc.
Amer. Math. Soc.68(1978), 243–246;Correction, Proc. Amer. Math. Soc.76 (1979), 169–170.
[10] ,Introduction to harmonic analysis on reductivep-adic groups, Prince- ton University Press, 1979.
[11] J.-L. Waldspurger, D´emonstration d’une conjecture de dualit´e de Howe dans le cas p-adique, p ̸= 2, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Weizmann, 1990, pp. 267–324.
Graduate school of mathematics, Kyoto University, Kitashirakawa, Kyoto, 606-8502, Japan
E-mail address: [email protected]