Advanced Studies in Pure Mathematics 26, 1998 Analysis on Homogeneous Spaces and Representations of Lie Groups pp. 1–19

Characters of Non-Linear Groups

Jeffrey Adams

§1. Introduction

Two of the primary methods of constructing automorphic forms are the Langlands program and Howe’s theory of dual pairs.

The Langlands program concerns a reductive linear group G defined over a number field. Associated to G is its dual group ^{L}G. The conjectural principle of functoriality says that a homomorphism ^{L}H → ^{L}G should provide a “transfer” of automorphic representations from H to those of G.

On the other hand Howe’s theory of dual pairs, the theta correspondence, starts with the oscillator representation of the non-linear metaplectic group Mp(2n), the two-fold cover of Sp(2n). Restricting this automorphic representation to a commuting pair of subgroups (G, G′) of Mp(2n) gives a relationship between the automorphic representations of G and G′.

This suggests a natural question: is the theta-correspondence in some sense “functorial”. As Langlands points out in [16]: “the connection between theta series and functoriality is quite delicate, and therefore quite fascinating … ”. Now G and G′ may be non-linear groups, and so even to define the notion of functoriality requires some work. In particular the L-groups of G and G′ are not defined. Nevertheless it is reasonable to ask that theta-lifting be given by some sort of data on the “dual” side. This can be done in some cases in which the nonlinearity of G and G′ do not play an essential role. Nevertheless a proper understanding of the relationship between theta-lifting (and its generalizations) and functoriality requires bringing the representation theory of non-linear groups into the Langlands program.

Some discussion of the relation of the theta-correspondence to functoriality may be found in [15], [21], and [2]. The case of U(3) has been discussed in great detail in [8].

Partially supported by NSF grant DMS–97–05872,

Received March 31, 1998

Let G = 𝔾(𝔽) be the 𝔽 points of an algebraic group 𝔾 defined over a local field 𝔽. Suppose G̃ is a non-linear covering group of G. A representation of G̃ is genuine if it does not factor to any proper quotient. Our approach is to relate:

G̃_{gen} = {irreducible genuine representations of G̃}

to the representations of some linear group H (which may or may not equal G). This would reduce questions about genuine representations of G̃ to those of H.

The (purely local) approach we shall take here is through character theory, analogous to endoscopic transfer and base change. The first example of this is GL(2) [6], which is related to the Shimura correspondence. We follow the roadmap, and some of the notation, of “the crux of the matter” ([12], §4), which gives more detail in the case of GL(n). What is known (to this author) in general is discussed in section 5.

This paper grows out of a talk at the Hayashibara Form, Okayama Japan, July 1997. I thank the organizers for their generous hospitality.

§2. Non-Linear Groups

Let 𝔽 be a local field of characteristic 0. Assume 𝔾 is a simple, simply connected algebraic group defined over 𝔽, and let G = 𝔾(𝔽). Let A be an abelian, locally compact group with trivial G-action. Then H^{2}(G, A) as defined by Moore [19] classifies the topological central extensions G̃ of G by A up to equivalence, i.e.

(2.1)  1 → A →^{ι} G̃ →^{p} G → 1

where ι, p are continuous, ι is closed and p is open, i.e. induces an isomorphism G̃/A ≃ G.

The following Theorem of Prasad and Rapinchuk is the culmination of work which has a number of contributors. It originates in the fundamental paper of Matsumoto [18], and uses results of Deligne [4].

Theorem 2.2 [20]. Suppose 𝔽 is non-archimedean and G is isotropic, i.e. contains a one-dimensional split torus. Then

(2.3)  H^{2}(G, A) ≃ Hom(μ(𝔽), A)

Here μ(𝔽) is the set of roots of unity in 𝔽. Let μ_{N}(𝔽) be the N^{th} roots of unity in 𝔽.

If 𝔽 = ℂ then H^{2}(G, A) = 1. If 𝔽 = ℝ let K be a maximal compact subgroup of G, and π_{1}(G) ≃ π_{1}(K) the ordinary topological fundamental group. This is isomorphic to 1, ℤ/2ℤ or ℤ, and H^{2}(G, A) ≃ Hom(π_{1}(G), A).

Definition 2.4. Assume μ_{N}(𝔽) = N. Let G[N] be the central extension

1 → μ_{N}(𝔽) → G[N] → G → 1

corresponding via (2.3) to the homomorphism ζ → ζ^{|μ(F)|/N} taking μ(𝔽) to μ_{N}(𝔽).

If 𝔽 = ℝ we define G[N] similarly if N = 2 and π_{1}(G) ≠ 1.

Definition 2.5. Let G̃ be a central extension of G by A as in (2.1). Let π be an irreducible admissible representation of G̃. Then π is genuine if the central character χ_{π} of π is an injection when restricted to A.

The representation theory of extensions of G reduces to that of the groups G[N]. First of all we may assume A is cyclic, and secondly that the homomorphism on the right hand side of (2.3) is surjective. Henceforth we assume |μ_{N}(𝔽)| = N, and write G̃ = G[N].

Let T be a Cartan subgroup of G, and let T̃ = p^{−1}(T). A key point is that T̃ is not necessarily abelian.

Definition 2.6. An element t of T̃ is said to be regular if p(g) is regular in T. Suppose g ∈ T̃ is a regular. Then

g is relevant if g ∈ Z(T̃)

where Z(T̃) denotes the center of T̃.

Let G̃_{gen} be the equivalence classes of irreducible genuine admissible representations of G̃. For π ∈ G̃_{gen} let Θ_{π} be its global character. For π a genuine virtual representation, i.e. a finite integral linear combination of irreducible representations π = ∑_{i} a_{i}π_{i} we define Θ_{π} = ∑ a_{i}Θ_{π_{i}} as usual. We identify Θ_{π} with the function on the regular semisimple elements which represents it.

For GL(n) the next Proposition may be found in [6] and [11].

Proposition 2.7. Suppose π ∈ G̃_{gen}. If g is not relevant then Θ_{π}(g) = 0.

Proof. If g ∈ T̃ is not in the center of T̃, then there exists h ∈ T̃ such that

(a)  hgh^{−1} ≠ g.

Projecting the left hand side gives

(b)  p(hgh^{−1}) = p(h)p(g)p(h^{−1}) = p(g)

since T is abelian. Therefore hgh^{−1} = zg for some z ∈ μ_{N}(𝔽) with z ≠ 1 by (a). Now Θ_{π} is conjugation invariant, so it takes the same value at g and hgh^{−1} = zg. This gives

Θ_{π}(g) = Θ_{π}(zg)
= χ_{π}(z)Θ_{π}(g).

Since π is genuine χ_{π}(z) ≠ 1, proving the Proposition.

Remark. The Proposition holds for any conjugation invariant function f which is genuine, i.e. f(zg) ≠ f(g) for all z ∈ μ_{N}(𝔽), z ≠ 1.

That this condition is highly non-trivial is illustrated by:

Proposition 2.8 [18]. Suppose 𝔽 is p-adic, G is split, and T is a split Cartan subgroup. Then T̃ is abelian if and only if G = Sp(2n) and N = 2 i.e. G̃ is the metaplectic group.

Remark. In some sense the the metaplectic group exists “because” of the oscillator representation. The proposition is an example of the fact that the oscillator representation is distinguished.

§3. General Lifting

Our discussion of lifting of characters for non-linear groups is modeled on the theory of endscopy and twisted endoscopy, including base change. In broad outline this is defined as follows.

The ingredients are:

(L.1) Reductive groups G, H, (L.2) a map

H_{0}/ ∼_{st}→^{t} G_{0}/ ∼

where: G_{0} is the strongly regular semisimple elements of G, ∼ is conjugacy by G,

∼_{st} is stable conjugacy, i.e. conjugacy by H(\overline{F}),

(L.3) Transfer factors: a map

Φ : H_{0} × G_{0} → ℂ

satisfying a number of conditions. In particular Φ(h, g) = 0 unless g = t(h), and

|Φ(h, g)| = |Δ(h)|/|Δ(g)|

where Δ is the usual Weyl denominator (with well defined absolute value).

Suppose π_{H} is a stable virtual character of H, with character Θ_{π_{H}}.

Definition 3.1.

(3.2)(a)  t_{∗}(Θ_{π_{H}})(g) = ∑_{{h|t(h)=g}/∼_{st}} Φ(h, g)Θ_{π_{H}} (h) (g ∈ G_{0}).

This is a finite sum, and defines a conjugation invariant function on G_{0}. This is conjectured to be the character of a virtual representation π_{G} of G: we say π_{G} is the lift of π_{H}, and write t_{∗}(π_{H}) = π_{G}, if t_{∗}(Θ_{π_{H}}) = Θ_{π_{G}}, i.e.

(3.2)(b)  Θ_{π_{G}}(g) = ∑_{{h|t(h)=g}/∼} Φ(h, g)Θ_{π_{H}} (h).

In the case of twisted endoscopy (e.g. base change) conjugacy is replaced by twisted conjugacy, and characters by twisted characters.

In the standard theory lifting (transfer) of orbital integrals is defined first, and lifting of characters is defined to be dual to this. Formula (3.2)(a) is then a consequence of the Weyl integration formula. It is possible, and sometimes convenient, to take (3.2)(a) as the definition, and prove a result on characters directly without the use of orbital integrals. One advantage of this approach is that the representation theory may suggest what to do (for example see Remark 5.13). The corresponding result on orbital integrals should then follow by similar arguments.

The transfer factors are a critical and difficult part of the theory. In the case of endoscopy and twisted endoscopy they contain deep arithmetic information ([17], [13]). It is necessary to define them carefully to insure the right hand side of (3.2) (which is a priori only a conjugation invariant function) is in fact a virtual character.

§4. Lifting for Non-Linear Groups

We return to the setting of section 2: 𝔽 is a local field of charactertistic zero, 𝔾 is a simple simply connected reductive group defined over 𝔽, G = 𝔾(𝔽) and G̃ = G[N].

We seek the following data:

(L.1) a linear reductive group ℍ defined over 𝔽, with 𝔽-points H, (L.2) a notion of stable conjugacy of strongly regular semisimple ele-

ments of G̃, and a map

H_{0}/ ∼_{st}→^{t} G_{0,relevant}/ ∼_{st}

The key point here is that the image should be the relevant semisimple elements of G̃_{0}, (cf. Proposition 2.7).

(L.3) Transfer factors

Φ : H_{0} × G̃_{0} → ℂ

satisfying Φ(h, g) = 0 unless g = t(h), and

|Φ(h, g)| = |Δ(h)|/|Δ(g)|

For π_{H} a stable virtual character of H define t_{∗} by (3.1)(a):

t_{∗}(Θ_{π_{H}})(g) = ∑_{{h|t(h)=g}/∼_{st}} Φ(h, g)Θ_{π_{H}} (h).

This is a stably conjugation invariant function on G̃_{0}. The hope is to define the data such that it is a genuine stable virtual character or 0, and every such virtual character arises this way.

Note that t_{∗} conjecturally involves stable virtual characters on both H and G̃. This is analogous to endoscopic transfer from the quasisplit inner form G_{qs} to G.

§5. Examples

Character theory as in Section 4 or related results are known in the following cases, which will be discussed in more detail. In each case G, G̃ and H are as in §4.

(1) G = H = GL(n, 𝔽)

(a) For n = 2 this is due to Flicker [6]. For general n there is a series of papers by Flicker, Kazhdan and Patterson [10], [11], [12]. The most complete results are for tempered representations.

(b) For 𝔽 = ℂ this operation preserves unitarity [29] (there are no covers in this case).

(c) For 𝔽 = ℝ unitarity is preserved as well [3]. In (b) and (c) the correspondence of unitary representations is computed explicitly.

(2) Unramified Representations. Let G̃ be cover of a split, simply connected group over a p-adic field 𝔽. Savin [25] constructs a split linear group H and an isomorphism of Iwahori Hecke algebras. This gives a candidate for H in this case. Huang proves that unitary is preserved [9].

(3) G = Sp(2n, 𝔽) and N = 2. Take H = SO(2n + 1, 𝔽) (split). If 𝔽 = ℝ or ℂ there is a bijection of genuine stable virtual characters between S̃p(2n, 𝔽) and stable virtual characters of SO(2n + 1, 𝔽) [1]. For 𝔽 p-adic and n = 1 the same result holds [26]. In the real case the corresponding result on orbital integrals is due to Renard [23].

(4) G = SL(n, 𝔽) for 𝔽 p-adic, and N = n. Take H = P GL(n), as predicted by (2) [25]. The character relations hold in this case, as can be derived from (1a).

(1) \underline{Kazhdan-Patterson Lifting for GL(n, 𝔽)} Let 𝔽 be a local field of charactertistic 0, G = GL(n, 𝔽). Fix N.

Since G is not simple some extra work is required to define an Nfold cover G̃ [11]. There is an additional integral parameter c coming from the center, and we write G̃ = GL(n, 𝔽)[N, c]. The restriction of GL(n, 𝔽)[N, c] to SL(n, 𝔽) is SL(n, 𝔽)[N]. The case n=2 is due to Flicker [6].

The center Z(G̃) of G̃ is a bit complicated ([11], Proposition 0.1.1): it is the inverse image of

(5.1)(a)  {xI |x^{n−1+2nc} ∈ 𝔽^{∗N}}.

Note that if (n−1+2nc, N) = 1 (greatest common divisor) this becomes

(5.1)(b)  {xI |x ∈ 𝔽^{∗N}}.

Let T be any Cartan subgroup of G.

Lemma 5.2 ([11], Proposition 0.1.4). Suppose g ∈ T̃ is regular. Then g is relevant if and only if g ∈ Z(G̃)p^{−1}(T^{N}).

If (n − 1 + 2nc, N) = 1 by (5.1)(b) the relevant set is therefore p^{−1}(T^{N}). This holds for example if n = 2 and c = 0. In general the center causes some some technical difficulties. Formally we may take c = −\frac{1}{2}; this is the approach of [24], provided |μ_{2N}(𝔽)| = 2N.

In any event the Lemma suggests taking H = G, and the N^{th} power map for the orbit correspondence. More precisely, let s be a map s : G → G̃ satisfying p ∘ s = Id. A simple observation [6], [12] is that the map

t_{0}(g) = s(g)^{N}

is independent of the choice of s and is well-defined on conjugacy classes. Together with Proposition 2.7 this suggests the ingredients of the lifting theory should be:

(L.1) H = G = GL(n, 𝔽), (L.2) Let t_{0}(g) = s(g)^{N}, where s : G → G̃ is any section. Then

t(g) = t_{0}(g)u(g) where u(g) is trivial if N is odd, and u : G → ±1 ∈ μ_{N}(𝔽) is a certain map ([11], Section 2) if N is even.

(L.3) Φ(h, g) = \{ \begin{matrix}\frac{|Δ(h)|}{|Δ(g)|} = \frac{|Δ(h)|}{|Δ(h^{N})|} & if p(g) = h^{N} \\ 0 & otherwise \end{matrix}

The need to modify t_{0} by u is a subtle and crucial point of the theory ([12], §2).

Assume (n − 1 + 2nc, N) = 1. Then t_{∗} is defined by (3.1)(a), with a constant c_{T} on the right hand side (which could be absorbed in the transfer factors) depending on the torus containing g.

In general some modification is necessary if Z(G̃) is not generated by t(Z(G)) and μ_{N}(𝔽), so an auxiliary choice of character of Z(G̃) is necessary. See ([12], Proposition 5.6) for details.

Conjecture 5.3 [10]. If π is an irreducible representation of G = GL(n, 𝔽) then t_{∗}(π) = 0, or t_{∗}(π) = ±π̃ for some irreducible genuine representation π̃ of G̃.

Theorem 5.4 [10]. The conjecture holds for π tempered.

In the case 𝔽 = ℂ, G = GL(n, ℂ) there are essentially no covers to consider (the derived group SL(n, ℂ) is simply connected). Nevertheless for any N t_{∗} may be defined as in (3.1)(a) taking GL(n, ℂ) to itself [29].

Theorem 5.6 [29]. If π is unitary then t_{∗}(π) = 0 or t_{∗}(π) = ±π̃ for π̃ an irreducible unitary representation of GL(n, ℂ).

Theorem 5.7 [3]. Let 𝔽 = ℝ, N = 2.

(1) If π is irreducible and unitary then t_{∗}(π) = 0 or t_{∗}(π) = ±π̃ for π̃ an irreducible unitary genuine representation of G̃.

(2) If π is one-dimensional then t_{∗}(π) is a minimal unitary genuine representation π̃ with infinitesimal character ρ/2 or zero.

(3) Let L be a theta-stable Levi factor in G, π_{L} a virtual character for L. Then (notation below) t_{∗} “commutes with the Euler characteristic of cohomological induction”:

t_{∗}(R̂_{L}^{G}(π_{L})) = RG̃ ; L̃(t_{∗}^{L}(π_{L}))

In (3), 𝔮 = 𝔩 ⊕ 𝔲 is a theta-stable parabolic subalgebra of 𝔤 = Lie(G)_{ℂ}. Let

R̂_{L}^{G}(π_{L}) = ∑(−1)^{k}R_{𝔮}^{k}(π_{L})

where R_{𝔮}^{k} is the derived functor in degree k, and π_{L} is a virtual character

of L. Taking L̃ = p^{−1}(L), R̂G̃ ; L̃ is defined similarly. Finally t_{∗}^{L} is Kazhdan-Patterson lifting defined for L, which is a product of copies GL(m, ℝ) and GL(m, ℂ).

Remark. In Theorems 5.6 and 5.7 t_{∗}(π) is computed explicitly. In particular there is an explicit condition for when it is 0.

Remark. Statement (3) is a cohomological version of the fact that, very generally in the setting of (3.1), t_{∗} commutes with ordinary parabolic induction: with the obvious notation

t_{∗}(Ind_{M}^{G}(π_{M})) = Ind_{M}^{G}(t_{∗}^{M}(π_{M}))

This follows from the induced character formula.

(2) \underline{Unramified Representations} Let 𝔽 be a p-adic field and fix a split, simply connected algebraic

group 𝔾 defined over the ring of integers R of 𝔽. Let G = 𝔾(𝔽) and G̃ = G[N] as in Section 2. Assume (p, N) = 1, so that the cover splits over G(R) and hence over an Iwahori subgroup I. Choose a splitting s : I → G̃. Fix a character χ of Ĩ = p^{−1}(I) such that χ|_{s(I)} = 1 and χ restricted to μ_{N}(𝔽) is injective. Consider the category R(G̃, χ) of smooth representations (π, V) of G̃ such that V is generated by

(5.8)(a)  V^{Ĩ,χ} = {v ∈ V | π(g)v = χ(g)v for all g ∈ Ĩ}.

Now let H(G̃, χ) be the space of compactly supported functions on G̃ satisfying

(5.8)(b)  f(xgy) = χ(xy)f(g) (x, y ∈ Ĩ, g ∈ G̃).

We call this the Iwahori Hecke algebra for G̃ (more precisely for the genuine representations of G̃ in which μ_{N}(𝔽) acts by χ.)

As in the linear case the map V → V^{Ĩ,χ} induces an equivalence of categories between R(G̃, χ) and the category of finite dimensional representations of H(G̃, χ).

A global correspondence of automorphic representations between a linear group H and a non-linear group G̃ should be given at the unramified places by a correspondence of unramified representations of H and G̃. It is natural to realize this correspondence via an isomorphism between H(G̃, χ) and the standard Iwahori Hecke algebra H(H). This is the case for GL(n) ([12], §3), and was considered in general by Savin in [25]. Huang [9] computed the induced correspondence of unramified representations and showed that unitarity is preserved.

We will only state a qualitative version of [25] here.

Theorem 5.9 [25]. Let G̃ = G[N]. For any χ there is a split linear group H defined over 𝔽 and an isomorphism

H(G̃, χ) ≃ H(H).

The group H is independent of χ and depends on N. With the exception of some cases with G = SL(n) or SO(4n + 2), H may be taken to be either G or the split form of the dual group ^{L}G, depending on N.

One consequence of this result is that it gives a candidate for the group H of Section 4 in these cases.

(3) \underline{The metaplectic group} Let 𝔽 be a local field of characteristic 0. Let G = Sp(2n, 𝔽) and

N = 2, so G̃ = S̃p(2n, 𝔽) is the metaplectic two-fold cover of Sp(2n, 𝔽).

The Cartan subgroups T̃ of S̃p(2n, 𝔽) are abelian. The map on conjugacy classes is not the squaring map as in example (2).

The ingredients are as follows.

(L.1) H = SO(2n + 1, 𝔽) (the split orthogonal group) (L.2) Suppose g ∈ Sp(2n, 𝔽)_{0}. Then we define t by the condition:

t(g) = h ∈ SO(2n + 1, 𝔽) if g, h have the same non-trivial (i.e. ≠ 1) eigenvalues. This defines a bijection

Sp(2n, 𝔽)_{0}/ ∼_{st} ⟷^{1−1} SO(2n + 1, 𝔽)_{0}/ ∼_{st}

If g ∈ S̃p(2n, 𝔽) define t(g) = t(p(g)).

(L.3) Fix a non-trivial character ψ : 𝔽 → S^{1}. Let ω(ψ) = ω = ω_{e} ⊕ ω_{o} be the corresponding oscillator representation. Let

Φ(g) = Θ_{ω_{e}}(g) − Θ_{ω_{o}}(g) (g ∈ S̃p(2n, 𝔽)_{0}).

Evidence that Φ(g) is a reasonable definition of the transfer factor is provided by:

Lemma 5.10.

|Φ(g)| = \frac{|Δ_{SO}(h)|}{|Δ_{Sp}(g)|}

where t(g) = h.

Here Δ_{SO} and Δ_{Sp} are the Weyl denominators for SO(2n + 1, 𝔽) and Sp(2n, 𝔽) respectively.

Proof. Recall |Δ(g)| = | ∏_{α∈Δ}(1 − α(g))|^{\frac{1}{2}} where Δ is the set of roots. Suppose g has eigenvalues {x_{i}}. The roots are then x_{i}^{2} and x_{i}/x_{j}. For SO(2n + 1, 𝔽) the roots evaluated at t(g) are x_{i} and x_{i}/x_{j}. In the quotient the x_{i}/x_{j} terms cancel to give

\frac{|Δ_{SO}(h)|}{|Δ_{Sp}(g)|} = \frac{| ∏(1 − x_{i})|^{\frac{1}{2}}}{| ∏(1 − x_{i}^{2})|^{\frac{1}{2}}}
= |∏(1 + x_{i})|^{− \frac{1}{2}}
= |det(1 + g)|^{− \frac{1}{2}} .

By a result of Howe for g̃ ∈ S̃p(2n, 𝔽), with g = p(g̃), |Θ_{ω}(g̃)| = |det(1 − g)|^{− \frac{1}{2}} . Let z ∈ S̃p(2n, 𝔽) be a central element satisfying p(z) = −I. The central characters of ω_{e} and ω_{o} have opposite sign on z. It follows that

|Φ(g̃)| = |Θ_{ω}(zg̃)| = |det(1 − (−g))|^{− \frac{1}{2}} ,

proving the result.

Definition 5.11. Let π_{H} be a stable virtual character of SO(2n + 1, 𝔽). Define

t_{∗}(Θ_{π_{H}})(g) = Φ(g)Θ_{π_{H}} (t(g))

for g ∈ S̃p(2n, 𝔽)_{0}. This is a genuine conjugation invariant function on S̃p(2n, 𝔽)_{0}.

Conjecture 5.12. t_{∗} is a bijection from

{stable virtual characters of SO(2n + 1, 𝔽)} to

{“stable” genuine virtual characters of S̃p(2n, 𝔽)}.

The notion of stable for S̃p(2n, 𝔽) requires definition. It reduces to S̃L(2, 𝔽), and should be a condition on each elliptic torus. (It is it not the same as restriction from G̃L(2, 𝔽).) Over ℝ it comes down to defining an action of the full Weyl group on a compact Cartan subgroup T̃_{c}.

Remark 5.13. Suppose one assumes the conjecture for some unknown transfer factor Φ. Let π_{H} = ℂ (the trivial representation); the conjecture gives

(5.14)(a)  t_{∗}(ℂ)(g) = Φ(g)

which says the transfer factor Φ(g) is equal to the character of the small virtual representation t_{∗}(ℂ). Note that the subtleties in the definition of the transfer factor are accounted for by the character of t_{∗}(ℂ).

The same idea in the general setting of (3.2)(b) gives

(5.14)(b)  t_{∗}(ℂ) = ∑_{{h|t(h)=g}/∼} Φ(h, g)

i.e. the transfer factor averaged over H should be a small virtual character of G. This might be useful in defining the transfer factors.

Theorem 5.15 [1]. The conjecture is true for 𝔽 = ℝ.

Theorem 5.16 [26]. The conjecture is true for 𝔽 p-adic, n = 1 and p ≠ 2.

(The assumption p ≠ 2 is probably not necessary.)

Theorem 5.17. The conjecture is true for the minimal principal series of S̃p(2n, 𝔽) and SO(2n + 1, 𝔽).

Proof. Let T be a split Cartan subgroup of Sp(2n, 𝔽) and let B ⊃ T be a Borel subgroup. Let B̃, T̃ be the inverse images of B and T in S̃p(2n, 𝔽); T̃ is abelian. For χ̃ a genuine character of T̃ the genuine induced representation π = π(χ̃) is defined as usual.

Similarly let B′ = T′N′ be a Borel subgroup of the split group SO(2n + 1, 𝔽). For χ′ a character of T′ let π′ = π′(χ′) be the corresponding induced representation.

The Weyl group W of T in Sp(2n, 𝔽) acts naturally on T̃ and by the induced character formula [5] for g ∈ T̃ ∩ G̃_{0}

(5.18)  Θ_{π}(g) = \frac{1}{|Δ_{Sp}(g)|} ∑_{W} χ̃(w · g)

Write g = (p(g), ϵ) in cocycle notation (ϵ = ±1). Let x_{1}^{±1}, …, x_{n}^{±1} be the eigenvalues of p(g). Fix a unitary (additive) character ψ of 𝔽, and let γ(x, ψ) be the corresponding gamma factor of Weil ([22], Appendix). Define a genuine additive character of T̃ by

(5.19)  \begin{aligned}χ̃_{0}(g) & = γ(∏x_{i}, ψ)ϵ \\ & = ∏_{i}γ(x_{i}, ψ)∏_{i<j}(x_{i}, x_{j})_{𝔽}ϵ \end{aligned}

where (,)_{𝔽} is the Hilbert symbol. A cocycle calculation shows that χ̃_{0}(w · g) = χ̃_{0}(g) for all w ∈ W. The character χ = χ̃_{0}^{−1}χ̃ factors to T and (5.18) becomes

(5.20)  Θ_{π}(g) = \frac{χ̃_{0}(g)}{|Δ_{Sp}(g)|} ∑_{W} χ(w · p(g))

Let h = t(g), so h and g have the same eigenvalues (h has eigenvalue 1 in addition). With the natural isomorphism ϕ : T′ ≃ T (take any such isomorphism which preserves eigenvalues) let χ′ = χ ∘ ϕ. Let W′ ≃ W be the Weyl group of T′ in SO(2n + 1, 𝔽). Then

(5.21)

Θ_{π}(g) = \frac{χ̃_{0}(g)}{|Δ_{Sp}(g)|} ∑_{W} χ(w · p(g))

= \frac{χ̃_{0}(g)}{|Δ_{Sp}(g)|} ∑_{W′} χ′(w′h)

= χ̃_{0}(g)\frac{|Δ_{SO}(h)|}{|Δ_{Sp}(g)|} \frac{1}{|Δ_{SO}(h)|} ∑_{W′} χ′(w′h)

= χ̃_{0}(g)\frac{|Δ_{SO}(h)|}{|Δ_{Sp}(g)|}Θ_{π′}(h).

It only remains to prove the factor

(5.22)  χ̃_{0}(g)\frac{|Δ_{SO}(h)|}{|Δ_{Sp}(g)|}

is equal to Φ(g). In absolute value this is Lemma 5.10. Suppose n = 1, p(g) = diag(x, \frac{1}{x}), g = (p(g), ϵ). The character of the oscillator representation has been computed in [26]. The result is:

(5.23)  Φ(g) = Θ_{ω_{e}}(g) − Θ_{ω_{o}}(g) = \frac{γ(x, ψ)ϵ}{|(1 + x)(1 + \frac{1}{x})|^{\frac{1}{2}}}

which equals (5.22) since χ̃_{0} = γ(χ, ψ)ϵ by (5.19).

The general case follows since the character of the oscillator representation representation is essentially diagonal, as is (5.22). More precisely by ([22], Proposition 3.8 and Corollary 5.6) Φ is diagonal with the exception of a cross term ∏_{i<j}(x_{i}, x_{j}) which is accounted for by (5.19). This completes the proof.

More generally it should follow easily that t_{∗} commutes with parabolic induction (this is true over ℝ and ℂ), which would reduce the conjecture to the discrete series.

(4) \underline{SL(n)} Let 𝔽 be a p-adic field satisfying |μ_{n}(𝔽)| = n and let S̃L(n) =

SL(n, 𝔽)[n].

The standard approach to representation theory of SL(n) is via restriction from GL(n). In general this is a difficult problem. For n = 2 this was the first case of endoscopy [14] and is highly non-trivial. For some results on the general case see [7], [28].

It is rather surprising that in contrast to the linear case the irreducible representations of S̃L(n) may be understood very simply in terms of those of the corresponding cover G̃L(n) of GL(n). In fact the restriction of an irreducible representation of G̃L(n) to S̃L(n) always has |𝔽^{∗}/𝔽^{∗n}| components which may be distinguished via their central characters. We sketch the situation briefly, details will appear elsewhere.

Let G̃L(n) = GL(n)[n, 0] (notation as above), which contains S̃L(n) as a normal subgroup. A standard construction in the theory of covering groups is the commutator. If ghg^{−1}h^{−1} = 1 then, for any inverse images g̃, h̃ of g, h, η = g̃h̃g̃^{−1}h̃^{−1} is in μ_{N}(𝔽), and we write {g, h} = η. The key point is the commutator formula ([11], proof of Proposition 0.1.1):

(5.24)  \begin{aligned}\{xI, g\} & = (x, det(g))_{n}^{n−1} \\ & = (x, det(g))^{−1} \end{aligned}

where (,)_{n} is the n^{th} norm residue symbol of 𝔽 [27].

Let D be the diagonal subgroup of GL(n),

H = SL(n)D = {g | det(g) ∈ 𝔽^{∗n}},

H̃ = p^{−1}(H). By (5.24) the center Z(H̃) of H̃ is D̃ = p^{−1}(D) and this gives

(5.25)  H̃ = S̃L(n)Z(H̃).

Therefore the representation theory of S̃L(n) reduces easily to that of H̃.

Note that H̃ is normal, and has index m = |𝔽^{∗}/𝔽^{∗n}|, in G̃L(n). Again from (5.24) we see that the centralizer of Z(H̃) in G̃L(n) is H̃. That is, if g ∈ G̃L(n), g ∉ H̃ then gzg^{−1} = z′ for some z, z′ ∈ Z(H̃), z ≠ z′, i.e.

(5.26)  gzg^{−1} = ηz

for some η ∈ μ_{N}(𝔽), η ≠ 1.

Now let σ be any genuine irreducible representation of H̃ with central character χ_{σ}. Let σ^{g} be the image of σ under the action of g on H̃. Then (5.24) implies

(5.27)  g ∉ H̃ ⇒ χ_{σ^{g}} ≠ χ_{σ}

and therefore σ^{g} ≄ σ.

Now let π be an irreducible genuine representation of G̃L(n), and suppose σ is a summand of π restricted to H̃. It follows from (5.27) and Clifford theory that

(5.28)  π|H̃ = ∑ σ^{g}

where the sum runs over g ∈ S = G̃L(n)/H̃.

The determinant induces an isomorphism S ≃ 𝔽^{∗}/𝔽^{∗n}. For \overline{x} ∈ 𝔽^{∗}/𝔽^{∗n} let x ∈ 𝔽^{∗} be a representative, and let z\overline{x} = (xI, ζ) be any element of D̃ satisfying p(z\overline{x}) = (xI, ζ).

Proposition 5.29. Let σ be an irreducible genuine representation of H̃ with central character χ_{σ}, and suppose σ is contained in the restriction of an irreducible representation π of G̃L(n).

Θ_{σ}(g) = \frac{1}{m} ∑_{\overline{x}∈S} χ_{σ}(z\overline{x})^{−1}Θ_{π}(z\overline{x}g) (g ∈ H̃_{0}).

(This is independent of the choices of z\overline{x}, \overline{x} ∈ S.)

Proof. Write π|H̃ = ∑_{y∈S} σ^{y}. Take the character of both sides at z_{x}g:

(5.30)  \begin{aligned}Θ_{π}(z_{x}g) & = ∑_{y} Θ_{σ^{y}}(z_{x}g) \\ & = ∑_{y} χ_{σ^{y}}(z_{x})Θ_{σ^{y}}(g). \end{aligned}

Multiply both sides by χ_{σ}(z_{x})^{−1}, sum over x, and use orthogonality of characters on S. The result follows.

This reduces the computation of irreducible characters of S̃L(n) to G̃L(n), and hence (via [10]) to GL(n) (at least for tempered representations). In fact there is a natural reduction to P GL(n), as we now sketch.

For ν_{0} ∈ 𝔽̂^{∗} let χ(ν_{0})(xI; ζ) = ν_{0}(x)ζγ(x^{n}, ψ). This is a genuine character of D̃, and every genuine character arises this way. Each χ_{σ} is equal to χ(ν_{0}) for some ν_{0}.

Suppose π is an irreducible representation of GL(n) whose central character satisfies χ_{π}(ζI) = 1 for all ζ ∈ μ_{n}(𝔽). Let π̃ be the lift of π to G̃L(n) via [10]. By the basic relationship between the central characters of π and π̃ [10], the central character of any summand of the restriction of π̃ to H̃ is of the form χ(ν_{0}), where ν_{0} satisfies

(5.31)  ν_{0}(x)^{n} = χ_{π}(xI).

The choices of ν_{0} satisfying (5.31) parametrize the summands of the restriction of π̃ to H̃.

Definition 5.32. For ν_{0} and π satisfying (5.31), let L(π, ν_{0}) be the summand of π̃|H̃ with central character χ(ν_{0}).

For β_{0} a character of 𝔽^{∗} let β(g) = β_{0}(det(g)). The relation with P GL(n) arises from the observation that πν^{−1} has trivial central character, i.e. factors to P GL(n). However the map L(π, ν_{0}) → πν^{−1} is not well defined. The Proposition implies

(5.33)  L(πβ, ν_{0}β_{0}) = L(π, ν_{0})

but only provided β(ζ) = 1 for all ζ ∈ μ_{n}(𝔽), i.e. β ∈ 𝔽̂^{∗n}. Let X = 𝔽̂^{∗}/𝔽̂^{∗n} ≃ μ_{n}(𝔽) .

This suggests a definition of stability for S̃L(n).

Definition 5.34.

L_{st}(π, ν_{0}) = ∑_{α∈X} L(πα, ν_{0}α_{0})

Such a virtual character is said to be stable.

Now (5.33) implies L_{st}(πβ, ν_{0}β) = L_{st}(π, ν_{0}) for all β. Hence

(5.35)  L_{st}(π, ν_{0}) → πν^{−1}

is a well–defined map from stable genuine virtual representations of H̃ to representations of GL(n) with trivial central character, or alternatively P GL(n). Tracing through the maps gives a character identity relating stable genuine virtual representations of S̃L(n) to irreducible representations of P GL(n) (again assuming the conjecture of [10], which holds for tempered representations).

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Math Dept, University of Maryland, College Park, MD 20742; jda@math.umd.edu, www.math.umd.edu/^{∼}jda