Lifting of Characters on Orthogonal and Metaplectic Groups Jeffrey Adams

Notes of my talk at the AMS conference at Northeastern University, Boston, October 7-8, 1995. These are the transparencies of my talk, condensed and slightly edited. They are available on the web at http://www.wam.umd.edu/ jda/preprints. The main results appear in a preprint Lifting of Characters on Orthogonal and Metaplectic Groups.

ABSTRACT

We define lifting of characters between the split orthogonal group SO(n + 1, n) and the non-linear metaplectic group S̃p(2n, ℝ). This is analogous to endoscopic lifting from the the quasisplit form of a linear group G. For tempered representations it agrees with theta-lifting. It is analogous to Kazhdan and Patterson’s theory of metaplectic forms.

Sp(2n, ℝ):=symplectic group in 2n variables over ℝ, S̃p(2n, ℝ):=metaplectic (non-linear, two-fold) cover of Sp(2n, ℝ), O(p, q):=orthogonal group in 2n + 1 variables, signature p, q, SO(p, q):= special orthogonal group The following theorem confirms the real case of a conjecture of Kudla and generalizes a result of Waldspurger [Forum. Math. 1991].

Theorem. (joint with D. Barbasch): The dual pair correspondence induces a bijection

S̃p(2n, ℝ)_{genuine} ⟷ ⋃_{p,q} SO(p, q)

between the irreducible, admissible, genuine representations of S̃p(2n, ℝ) and the irreducible admissible representations of the groups SO(p, q) (here and elsewhere the union is over the real forms of SO(2n + 1), i.e. p + q = 2n + 1 with (−1)^{q} fixed.)

Question: Suppose π ↔ π′. What is the relation between the global characters of π and π′?

Lemma.

(1) There is a natural bijection (the orbit correspondence) between the strongly regular semisimple conjugacy classes of Sp(2n, ℝ) and {SO(p, q)}.

(2) This bijection induces a bijection between stable strongly regular semisimple conjugacy classes. Two such classes correspond if and only if they have the same non-trivial eigenvalues.

This is the basic setup of lifting. For (2), the real forms of SO(2n + 1) are taken together. Notation: g ⟷ g′. Definition: G_{0} denotes the strongly regular semisimple elements of G.

Typeset by AMS-TEX

We identify a representation π with its global character Θ_{π}, and Θ_{π} with the function on G_{0} representing it. Example: Discrete Series Let π_{SO}(λ) be the discrete series representation of SO(n + 1, n) with Harish-Chandra parameter

λ = (a_{1}, …, a_{k}; b_{1}, …, b_{ℓ})

(a_{i}, b_{j} ∈ ℤ + \frac{1}{2}, a_{1} > ⋯ > a_{k} > 0, b_{1} > ⋯ > b_{ℓ} > 0). Then the theta-lift of π_{SO}(λ) is the discrete series representation π_{Sp}(λ′) of S̃p(2n, ℝ) with Harish-Chandra parameter

λ′ = (a_{1}, …, a_{k}, −b_{ℓ}, …, −b_{1}).

The global characters of π_{Sp}(λ′) and π_{SO}(λ) have the following formulas on a compact Cartan subgroup:

π_{SO}(λ) = \frac{∑_{w∈W_{K}} sgn(w)e^{wλ}}{D_{SO}}

π_{Sp}(λ′) = \frac{∑_{w∈W_{K′}} sgn(w)e^{wλ′}}{D_{Sp}}

where W_{K} is the Weyl group of K = S(O(n + 1) × O(n)), W_{K′} is the Weyl group of K′ = U(n), and

D = ∏(e^{α/2} − e^{−α/2})

is the Weyl denominator. Remark: This is very imprecise: the Weyl denominators depend on a choice of positive roots, and D_{SO} is only well-defined on a covering group. Stabilize: define

\overline{π}_{SO}(λ) = ∑_{w∈W_{K}\W} π_{SO}(wλ)
\overline{π}_{Sp}(λ′) = ∑_{w∈W_{K′} \W} π_{Sp}(wλ′)

where W is the Weyl group of type B_{n}/C_{n}. These have character formulas (on the compact Cartans):

\overline{π}_{SO}(λ) = \frac{∑_{w∈W} sgn(w)e^{wλ}}{D_{SO}}
\overline{π}_{Sp}(λ′) = \frac{∑_{w∈W} sgn(w)e^{wλ′}}{D_{Sp}}

The compact Cartan subgroups of SO(n + 1, n) and S̃p(2n, ℝ) are isomorphic, and this isomorphism can be chosen to take λ to λ′.

Thus \overline{π}_{Sp}(λ′) may be obtained from \overline{π}_{SO}(λ) by multiplying by the transfer factor

\frac{D_{SO}}{D_{Sp}}

Problem: Make this precise (cf. the remark), extend to other Cartan subgroups and other representations. Let ω = ω_{even} ⊕ ω_{odd} be the oscillator representation of S̃p(2n, ℝ), with global character Ω = Ω_{even} + Ω_{odd}.

Definition.

Φ = Ω_{even} − Ω_{odd}

(a function on the regular elements).

Proposition. For all regular semisimple g ∈ S̃p(2n, ℝ):

|\frac{D_{SO}(g′)}{D_{Sp}(g)}| = |Φ(g)| (g′ ⟷ g).

Remarks:

(1) This is a basic requirement of transfer factors [Shelstad, Math. Annalen 1982].

(2) Φ(g)^{2} = \frac{±1}{det(1 + g)} (Howe). Without the absolute value both sides of the

proposition are in ℝ^{∗} ∪ iℝ^{∗}, so there is equality up to a fourth root of 1.

(3) By (2) Φ(g) is non-singular near the identity.

(4) Φ is stable (stable: an ad hoc definition for S̃p(2n, ℝ)). (Properties (3) and (4) do not hold for Ω). By the Proposition it makes sense to define the transfer factor to be Φ on any Cartan subgroup.

Main Definition:. Let Θ be a stable, invariant function on SO(n + 1, n)_{0}. For g′ ∈ Sp(2n, ℝ)_{0} let

τ(Θ)(g′) = Θ(g) (g ⟷ g′)

(independent of the choice of g). For g′ ∈ S̃p(2n, ℝ)_{0} let

Γ(Θ)(g′) = τ(Θ)(p(g′))Φ(g′)

This is a genuine, stable invariant function on S̃p(2n, ℝ)_{0}.

In other words, letting Θ′ = Γ(Θ),

Θ′(g′) = Φ(g′)Θ(g) (g′ ⟷ g).

For example taking Θ = 1 gives Θ′ = Ω_{e} − Ω_{o}. This is a useful principal: given any lifting theory of this form, taking Θ = 1 shows that the transfer factor is the character of a (virtual) representation.

Main Theorem.

(1) The map Γ : Θ → Θ′ = Γ(Θ) is a bijection between stable invariant eigendistributions on SO(n+1, n) and genuine stable invariant eigendistributions on S̃p(2n, ℝ).

(2) Θ′ is a virtual character if and only if Θ is a virtual character.

(3) Θ′ is tempered if and only if Θ is tempered.

(4) Γ takes the (stable) discrete series of SO(2n+1) to the (stable) genuine discrete series of S̃p(2n, ℝ).

(5) Γ commutes with parabolic induction and coherent continuation.

(6) If Θ is a tempered virtual character then Θ′ is the (normalized) theta-lift of Θ.

Conjecture. An analogous result holds over a p-adic field.

This result holds, and is very easy, in the case of complex groups. Remarks:

(1) The proof uses the acceptable group Spin(p, q): the inverse image of SO(p, q) in Spin(2n + 1, ℂ). The Cartan subgroups of Spin(p, q) and S̃p(2n, ℝ) are isomorphic.

(2) Key Fact:

Θ and Θ′ have the same (normalized) Weyl numerators

This follows from:

Φ(g) = \frac{D′_{SO}(g′)}{D′_{Sp}(g)} (g′ ⟷ g)

where D′ is the normalized Weyl denominator.

(3) The fact that Γ only agrees with theta-lifting for tempered representations is analogous to the fact that the sum of the representations in an arbitrary L-packet is not stable. In other words, Γ has a nice formula on standard modules, whereas theta-lifting is defined on irreducibles — only in the tempered case do these coincide.

(4) Since Φ is non-singular near the identity, Γ preserves the order [Barbasch-Vogan, JFA 1980] of an invariant eigendistribution.

(5) A stable virtual character on SO(n + 1, n) may be identified with a stable virtual character on {SO(p, q)}. The relationship with theta-lifting uses the latter interpretation.

(6) This lifting is similar to character lifting coming from endoscopic groups [Shelstad, Math. Annalen 1982], SO(n + 1, n) playing the role of the quasisplit inner form of S̃p(2n, ℝ). In endoscopy there is a map

f → f_{H}

from C_{c}^{∞}(G) to C_{c}^{∞}(H) (well defined on quotients of these spaces). This map satisfies a matching of orbital integrals, and is dual to a lifting of characters - character formulas on each Cartan subgroup are a consequence.

Conjecture. There is a map f_{Sp} → f_{SO}, defining a matching of orbital integrals, such that Γ(Θ)(f_{Sp}) = Θ(f_{SO}).

A plausible candidate for f_{Sp} → f_{SO} has been given by Rallis.

(7) In the case n = 1 this is related, but not identical, to Flicker’s lifting [Inventiones 1980] between GL(2) and its two-fold cover G̃L(2) (via the isomorphism P GL(2, ℝ) ≃ SO(2, 1)). Kazhdan-Patterson [Advances 1986] have conjectural generalizations of Flicker’s result to G̃L(n). The same methods used in the proof of the Main Theorem should yield a proof of some or all of these conjectures over ℝ.

Sketch of the proof of the Main Theorem.

(a) Θ′ satisfies Hirai’s matching conditions [Japanese Journal 1976] (using (2) above). Therefore Γ takes invariant eigendistributions to invariant eigendistributions.

(b) Discrete series: determined (among invariant eigendistributions) by the restriction to a compact Cartan.

(c) Induction: induced character formula as in endoscopic lifting [Shelstad, Compositio 1979].

(d) Therefore Γ takes standard modules to standard modules these span the virtual representations.