Characters of covering groups of SL(n)

Jeffrey Adams University of Maryland

May 5, 2000

Abstract

We study characters of an n-fold cover S̃L(n, 𝔽) of SL(n, 𝔽) over a non-archimedean local field. We compute the character of an irreducible representation of S̃L(n, 𝔽) in terms of the character of an irreducible representation of a cover G̃L(n, 𝔽) of GL(n, 𝔽). We define an analogue of L-packets for S̃L(n, 𝔽), such that the character of a linear combination of the representations in such a packet is computed in terms of the character of an irreducible representation of PGL(n, 𝔽). This is analogous to stable endoscopic lifting for linear groups. We also prove an inversion formula expressing the character of a genuine irreducible representation of S̃L(n, 𝔽) as a linear combination of virtual characters, each of which is lifted from PGL(n, 𝔽).

1 Introduction

Let 𝔾 be a reductive linear group defined over a local field 𝔽 of characteristic 0, and let G = 𝔾(𝔽). One of the ingredients of the local portion of the Langlands program for G is the study of the characters of admissible representations of G. These are used on one side of the trace formula, and provide information about automorphic representations of 𝔾 over a global field.

Important examples of automorphic representations involve reductive groups which are not linear, such as the oscillator representation of the metaplectic group, the two–fold cover of Sp(2n, 𝔽). We refer to a finite central extension G̃ of G which is not itself a linear group as a non-linear group. It would be interesting to understand the representation theory of such groups, and to extend the Langlands program to the study of their automorphic representations.

A representation π of G̃ is said to be genuine if it does not factor to any proper quotient of G̃. One approach to the representation theory of G̃ is to relate genuine representations of G̃ to representations of a linear group via character theory. There are a number of examples of this approach. See the references and [2] for a survey.

Now assume the cardinality of the n^{th} roots of unity μ_{n} = μ_{n}(𝔽) is n. We consider a central extension S̃L(n, 𝔽) of SL(n, 𝔽) by μ_{n} (cf. Section 2).

Arbitrary covers G̃L(n, 𝔽) of GL(n, 𝔽) have been studied extensively [5], [8], [7], [4]. Flicker, Kazhdan and Patterson relate character theory of G̃L(n, 𝔽) to that of GL(n, 𝔽). The group S̃L(n, 𝔽) is a subgroup of a corresponding group G̃L(n, 𝔽), and a natural approach is to study representations of S̃L(n, 𝔽) by restricting representations of G̃L(n, 𝔽). The corresponding problem for SL(n, 𝔽) and GL(n, 𝔽) is quite difficult [17], [12]. For example the case of n = 2 is the first example of endoscopy and is highly non–trivial [6].

Surprisingly the corresponding restriction problem for genuine representations of S̃L(n, 𝔽) is very easy, and character theory of S̃L(n, 𝔽) reduces to that of G̃L(n, 𝔽). Our first step is to write a formula (Theorem 3.1) for the character of an irreducible genuine representation π of S̃L(n, 𝔽) in terms of the character of an irreducible representation of G̃L(n, 𝔽) which contains π in its restriction.

Flicker, Kazhdan and Patterson have defined a lifting theory, conjecturally taking an irreducible unitary representation π of GL(n, 𝔽) to an irreducible genuine unitary representation t_{∗}(π) of G̃L(n, 𝔽) or 0 [7], [4], [8]. The character of t_{∗}(π) is computed in terms of the character of π. Together with Theorem 3.1, restriction from G̃L(n, 𝔽) to S̃L(n, 𝔽), this expresses the character of an irreducible constituent of t_{∗}(π) restricted to S̃L(n, 𝔽) in terms of characters of GL(n, 𝔽).

A constituent of t_{∗}(π) restricted to S̃L(n, 𝔽) is determined by a character ν of 𝔽^{∗} for which ν^{n} equals the central character of π. We write L(π, ν) for this constituent. For any character α, L(πα^{n}, να^{n}) ≈ L(π, ν); we sum over 𝔽̂^{∗}/𝔽̂^{∗n} ≈ μ̂_{n} and define (cf. Definition 5.3):

L_{st}(π, ν) = ∑_{α∈μ̂_{n}} L(πα, να).

Now πν^{−1} factors to PGL(n, 𝔽), and it turns out that the character of L_{st}(π, ν) may be computed in terms of the character of πν^{−1}. The main result is (Theorem 8.1):

Θ_{L_{st}(π,ν)}(g) = ∑_{\begin{matrix}h∈PGL(n,𝔽) \\ ϕ(h)=p(g) \end{matrix}} Δ_{μ}(h, g)Θ_{πν^{−1}}(h).

Here g is a regular semisimple element of G̃L(n, 𝔽), and we identify the character of a representation with a function on the regular semisimple elements. Also ϕ is the orbit correspondence: ϕ(g) = det(g^{−1})g^{n} ∈ SL(n, 𝔽) (Section 6), p is projection from S̃L(n, 𝔽) to SL(n, 𝔽), and Δ_{μ}(h, g) is a transfer factor (Section 7). The group PGL(n, 𝔽) is the one predicted by the Hecke algebra isomorphism of [13].

The set Π(π, ν) = {L(πα, να) | α ∈ μ̂_{n}} appearing here is analogous to an Lpacket for a linear group G. The character formula is is analogous to endoscopic lifting from the quasisplit form G_{qs} to G, and L_{st}(π, ν) is analogous to the stable lift of πν^{−1}, although since S̃L(n, 𝔽) is non-linear the notion of stable distribution is not defined.

An L-packet for SL(n, 𝔽) is conjecturally the set of constituents of the restriction of an irreducible representation of GL(n, 𝔽) to SL(n, 𝔽). The character of the sum of these representations is stable, i.e. invariant by conjugation by GL(n, 𝔽), and these sets satisfy other required properties of L–packets. It is interesting to note that Π(π, ν) is not the set of constituents of the restriction of a representation of G̃L(n, 𝔽). In particular (see the Remark following Theorem 8.1) Θ_{L_{st}(π,ν)} is typically not G̃L(n, 𝔽) conjugation invariant. It would be interesting to find an intrinsic characterization of the virtual characters L_{st}(π, ν).

In the theory of endoscopy for a linear group an irreducible representation π contained in an L-packet Π may be expressed as a linear combination of virtual characters, in the span of the elements of Π, each of which is stable, or lifted from a stable character on a smaller endoscopic group. This is known as inversion. By analogy with this we seek to write L(π, ν) as a linear combination of virtual representations, in the span of the elements of Π(π, ν), each of which is computed in terms of characters of a linear group. For ζ ∈ μ_{n} let

L_{ζ}(π, ν) = ∑_{α∈μ̂_{n}} α(ζ)L(πα, να).

We obtain an inversion formula (Theorem 9.1):

\begin{aligned}Θ_{L(π,ν)}(g) & = \frac{1}{n} ∑_{ζ∈μ_{n}} L_{ζ}(π, ν) \\ & = \frac{1}{n} ∑_{ζ∈μ_{n}} χ^{−1}(z_{ζ})Θ_{L_{st}(π,ν)}(z_{ζ}g). \end{aligned}

Here χ is the central character of L(π, ν) and z_{ζ} is an element of S̃L(n, 𝔽) with image ζI ∈ SL(n, 𝔽).

These results all hold as stated for 𝔽 = ℝ and n = 2, in which case they are equivalent to a special case of [3].

Similar results hold for certain other N-fold covers of S̃L(n, 𝔽). One would not expect the general N-fold cover to be amenable to these methods, as the case N = 1 makes abundantly clear.

The case of n = 2, worked out in detail, is the subject of the University of Maryland thesis of Jason Schultz [14]. This gives an intrinsic characterization of the local lift of Waldspurger [18]. In this case the set Π containing a genuine discrete series representation π consists of two elements π, π′ where π′ is the “Waldspurger involution” [18] applied to π. This goes back to the Shimura correspondence for modular forms of half–integral weight which is the origin of the theory of non-linear groups.

1.1 Desiderata

We consider covering groups:

1 → μ_{n} → G̃ →^{p} G → 1

with μ_{n} central in G̃ (cf. Section 2). We write χ_{π} for the central character of a representation π. We say a representation π of G̃ is genuine if π has a central character χ_{π} whose restriction to μ_{n} is injective. If π is not genuine then π factors to a representation of a cover of G with kernel a subgroup of μ_{n}. If ι : μ_{n} ↪ ℂ^{n} is an embedding we say π is of type ι if χ_{π}|_{μ_{n}} = ι.

An important role is played by the exact sequences

1 → μ_{n} → 𝔽^{∗} →^{n} 𝔽^{∗n} → 1  (1)
1 → 𝔽^{∗n} →^{ι} 𝔽^{∗} → 𝔽^{∗}/𝔽^{∗n} → 1  (2)

and their Pontriagin duals:

1 → 𝔽̂^{∗n} → 𝔽̂^{∗} →^{res} μ̂_{n} → 1  (3)
1 → { \atop 𝔽^{∗}/𝔽^{∗n}} → 𝔽̂^{∗} →^{res} 𝔽̂^{∗n} → 1  (4)

Suppose μ_{n} is in the kernel of λ ∈ 𝔽̂^{∗}. Then by (3) λ(x) = μ(x^{n}) for some character μ of 𝔽̂^{∗n}, which by (4) extends to τ ∈ 𝔽̂^{∗}. This gives the following well-known lemma which we use repeatedly:

Lemma 1.1 Let λ ∈ 𝔽̂^{∗}. Then λ = μ^{n} for some μ ∈ 𝔽̂^{∗} if and only if λ(ζ) = 1 for all ζ ∈ μ_{n}.

We identify the center Z of GL(n, 𝔽) with 𝔽^{∗} and the central character χ_{π} of a representation of GL(n, 𝔽) with an element of 𝔽̂^{∗}.

For α ∈ 𝔽̂^{∗} we write α for the character α ∘ det of of GL(n, 𝔽), and also for the character α ∘ p of G̃L(n, 𝔽). Note that for π a representation of GL(n, 𝔽) (with a central character)

χ_{πα} = χ_{π}α^{n}.  (5)

We write Θ_{π} for the global character of a representation π, considered as a function on the set of regular semisimple elements.

2 Group Structure

We continue with the notation of Section 1. We first define the group S̃L(n, 𝔽)(cf. [10], [9], [16]): this is a topological group which fits in an exact sequence:

1 → μ_{n} →^{ι} S̃L(n, 𝔽) →^{p} SL(n, 𝔽) → 1  (6)

with ι, p continuous, ι closed and p open. The classes of such extensions are parametrized by the group of (bilinear) Steinberg cocycles with values in μ_{n}. Let ( , )_{n} : 𝔽^{∗} × 𝔽^{∗} → μ_{n} denote the n^{th} norm residue symbol for 𝔽. For properties of ( , )_{n} see [15], ([7], §0.1). In particular ( , )_{n} is a perfect pairing on 𝔽^{∗}/𝔽^{∗n} and gives an isomorphism of 𝔽^{∗}/𝔽^{∗n} with { \atop 𝔽^{∗}/𝔽^{∗n}}. Each Steinberg cocycle is given by c(x, y) = (x, y)_{n}^{k} for some k. Write G[k] for the group defined by the cocycle (x, y)_{n}^{k}. Then G[k] and G[k′] are equivalent extensions if and only if k ≡ k′ mod (n).

The commutator subgroup G[k]_{c} of G[k] is a covering group of SL(n, 𝔽) with kernel a subgroup of μ_{n}. If G[k] is not perfect then G[k] = G[k]_{c}μ_{n} and the representations of G[k] of type ι are in bijection with the representations of G[k]_{c} of type ι|_{μ_{n}∩G[k]_{c}}. For this reason we assume G[k] is perfect, which holds if and only if gcd(k, n) = 1.

The map G[k] ∋ (g, ζ) → (g, ζ^{j}) ∈ G[kj] is a homomorphism, and is an isomorphism if gcd(j, n) = 1. In particular if gcd(k, n) = 1 then G[k] is isomorphic to G[1] (although not equivalent as an extension unless k ≡ 1 mod (n)). We let S̃L(n, 𝔽) = G[1]. Once and for all we fix an embedding

ι : μ_{n}(𝔽) ↪ ℂ^{∗}

and we identify μ_{n} with its image. Henceforth we assume all genuine representations are of type ι.

The Steinberg cocycle defines a cover G̃L(n, 𝔽) of GL(n, 𝔽) by [7], and S̃L(n, 𝔽) is a subgroup of G̃L(n, 𝔽) (we are taking c = 0 in the notation of [7]).

We write c( , ) for the cocycle defining G̃L(n, 𝔽). Then

G̃L(n, 𝔽) = {(g, ζ) | g ∈ GL(n, 𝔽), ζ ∈ μ_{n})}

with multiplication (g, ζ)(g′, ζ′) = (gg′, ζζ′c(g, g′)).

An essential role is played by the commutator. Suppose g and h are commuting elements of GL(n, 𝔽). Choose any inverse images g̃, h̃ of g, h in G̃L(n, 𝔽). Then η = g̃h̃g̃^{−1}h̃^{−1} ∈ μ_{n} is independent of the choices of g̃ and h̃. We write {g, h} = η.

An important property of the commutator is ([7], proof of Proposition 0.1.1):

{xI, g} = (x, det(g))_{n}^{−1}  (7)

2.1 Centers

Let

GL(n, 𝔽)_{+} = {g ∈ GL(n, 𝔽) | det(g) ∈ 𝔽^{∗n}}
= SL(n, 𝔽)Z.

Write H̃ for the inverse image in G̃L(n, 𝔽) of a subgroup H of GL(n, 𝔽). The following Lemma follows immediately from (7) and properties of the norm residue symbol.

Lemma 2.1 Let Z_{+} = {xI | x ∈ 𝔽^{∗n}}.

(1) The center of GL(n, 𝔽) is Z̃_{+},

(2) The center of G̃L(n, 𝔽)_{+} is Z̃,

(3) Cent_{G̃L(n,𝔽)}(G̃L(n, 𝔽)_{+}) = Z̃ and Cent_{G̃L(n,𝔽)}(Z̃) = G̃L(n, 𝔽)_{+}.

Thus Z̃ and G̃L(n, 𝔽)_{+} form a dual pair in the sense of Howe.

Therefore G̃L(n, 𝔽)_{+} = S̃L(n, 𝔽)Z̃, and Z̃ is the center of G̃L(n, 𝔽)_{+}. Consequently an irreducible representation of G̃L(n, 𝔽)_{+} restricts to an irreducible representation of S̃L(n, 𝔽), and every irreducible representation of S̃L(n, 𝔽) is obtained this way. For many purposes we may replace S̃L(n, 𝔽) by G̃L(n, 𝔽)_{+}. This is analogous to the corresponding situation for the linear groups. Note G̃L(n, 𝔽)/G̃L(n, 𝔽)_{+} ≈ GL(n, 𝔽)/GL(n, 𝔽)_{+} ≈ 𝔽^{∗}/𝔽^{∗n}.

The cocycle restricted to Z_{+} is trivial so Z̃_{+} ≈ 𝔽^{∗n} × μ_{n}. The cocycle restricted to Z is given by c(xI, yI) = ∏_{i<j}(x, y)_{n} = (x, y)_{n}^{n(n−1)/2}. This is equal to 1 if n is odd, or ±1 if n is even.

For later use we note there exists a (genuine) character μ of Z̃ satisfying

μ|Z̃_{+} = ι.  (8)

In fact we may take

μ(xI, ζ) = \begin{cases}ζ & n odd \\ γ(x, ψ)ζ & n even \end{cases}  (9)

Here ψ is a non-trivial additive character of 𝔽 and γ(x, ψ) ∈ {±1, ±i} is the Weil index ([11], appendix). In particular μ^{n} = 1 (n odd), and μ^{2n} = 1 (n even). We only use this explicit formula for (25).

Given μ, the genuine characters of Z̃ are in bijection with 𝔽̂^{∗}; given ν ∈ 𝔽̂^{∗} let

χ_{ν}(z) = μ(z)ν(x) (z ∈ Z̃, p(z) = xI)  (10)

i.e.

χ_{ν}(xI, ζ) = μ(xI, ζ)ν(x) = μ(xI, 1)ζν(x).  (11)

2.2 Cartan Subgroups

We define a Cartan subgroup of G̃L(n, 𝔽) or S̃L(n, 𝔽) to be the inverse image of a Cartan subgroup of the corresponding linear group. These groups are in general non-abelian, and an important role is played by the center. We say an element of a covering group is semisimple (respectively regular) if its image in the linear group is semisimple (resp. regular).

Lemma 2.2 Let T be a Cartan subgroup of GL(n) with inverse image T̃ in G̃L(n, 𝔽).

(1) The center of T̃ is p^{−1}(T^{n}),

(2) The center of T̃ ∩ S̃L(n, 𝔽) is p^{−1}(ZT^{n} ∩ SL(n, 𝔽)).

Proof. (1) is proved in ([4],§3), and (2) follows from this as well. We will sketch another proof of (2) in Section 3.

We say a regular semisimple element g ∈ T̃ is is relevant if it is contained in the center of T̃ [2]. It is a basic fact that if π is a genuine representation of G̃ then Θ_{π}(g) = 0 if g is not relevant [5], ([2], Proposition 2.7).

3 Restriction from G̃L(n, 𝔽) to S̃L(n, 𝔽)

We compute the character of an irreducible representation of S̃L(n, 𝔽) or G̃L(n, 𝔽)_{+} in terms of a character of G̃L(n, 𝔽) (Theorem 3.1). The main point is that Clifford theory for restriction of a genuine representation Π of G̃L(n, 𝔽) to G̃L(n, 𝔽)_{+} is very easy: each such representation restricts to a direct sum of |𝔽^{∗}/𝔽^{∗n}| distinct irreducible representations which are permuted by the action of G̃L(n, 𝔽)/G̃L(n, 𝔽)_{+} ≈ 𝔽^{∗}/𝔽^{∗n}. Furthermore the character of each summand may be computed in terms of the character of Π using Fourier inversion on Z̃/Z̃_{+} ≈ 𝔽^{∗}/𝔽^{∗n}.

Let π be a genuine representation of G̃L(n, 𝔽)_{+}. Write g : π → π^{g} for the action (by conjugation on G̃L(n, 𝔽)_{+}) of g ∈ G̃L(n, 𝔽) on representations of G̃L(n, 𝔽)_{+}. Assume π has a central character χ_{π}. We compute χ_{π^{g}}. Let z ∈ Z̃ with p(z) = xI. Then

χ_{π^{g}}(z) = χ_{π}(gzg^{−1})
= χ_{π}({p(g), xI}z)
= χ_{π}((x, det(g))_{n}z) (by (7))
= χ_{π}(z)(x, det(g))_{n} (since π is genuine).

(12)

By non-degeneracy of the symbol, if det(g) ∉ 𝔽^{∗n} there exists x such that (x, det(g))_{n} ≠ 1. Therefore if g ∉ G̃L(n, 𝔽)_{+}, χ_{π^{g}} ≠ χ_{π}, and a fortiori π^{g} ≉ π. Note the assumption π is genuine is essential; the corresponding result is demonstrably false for representations of G̃L(n, 𝔽) which factor to GL(n, 𝔽).

Let Π = Ind_{G̃L(n,𝔽)_{+}}^{G̃L(n,𝔽)}(π). By (12) and Clifford theory G̃L(n, 𝔽)/G̃L(n, 𝔽)_{+}

acts simply transitively on the set of constituents of Π restricted to G̃L(n, 𝔽)_{+}. For each x ∈ 𝔽^{∗}/𝔽^{∗n} choose g_{x} ∈ G̃L(n, 𝔽) with det(g) ≡ x mod (𝔽^{∗n}). Let π^{x} = π^{g_{x}}; the isomorphism class of π^{g_{x}} is independent of the choice of g_{x}. Thus:

Π|_{G̃L(n,𝔽)_{+}} = ∑_{x∈𝔽^{∗}/𝔽^{∗n}} π^{x}.  (13)

If π′ is a constituent of the restriction of Π to G̃L(n, 𝔽)_{+} then χ_{π′} (a character of Z̃) restricted to Z̃_{+} is equal to χ_{Π}. The set of extensions of χ_{Π} to Z̃ is in bijection with { \atop 𝔽^{∗}/𝔽^{∗n}}. By (13) the constituents of this restriction are in bijection with { \atop 𝔽^{∗}/𝔽^{∗n}}. This proves:

Proposition 3.1 Let Π be an irreducible genuine representation of G̃L(n, 𝔽). Let S be the set of extensions of χ_{Π} to Z̃_{+}, this set is in bijection with { \atop 𝔽^{∗}/𝔽^{∗n}}. For λ ∈ S let Π_{λ} be the λ eigenspace of Π.

For all λ, Π_{λ} is an irreducible representation of G̃L(n, 𝔽)_{+} and

Π|_{G̃L(n,𝔽)_{+}} = ∑_{λ∈S} Π_{λ}.

Fix an irreducible constituent π of this restriction. Then

Π|_{G̃L(n,𝔽)_{+}} = ∑_{x∈𝔽^{∗}/𝔽^{∗n}} π^{x}

and the central character of π^{x} is χ_{π}(·, x)_{n}.

Remark: A similar result holds for S̃L(n, 𝔽): Π|_{S̃L(n,𝔽)} = ∑_{x} π^{x} as above. However the π^{x} are not necessarily distinct: in some cases π ≈ π^{x} (this implies (x, ζ)_{n} = 1 for all ζ ∈ μ_{n}).

We strengthen this result using Fourier inversion on 𝔽^{∗}/𝔽^{∗n} to write Θ_{π} in terms of Θ_{Π}.

For z ∈ Z̃, z′ ∈ Z̃_{+}, χ_{π}(zz′)^{−1}Θ_{Π}(zz′g) = χ_{π}(z)Θ_{Π}(zg). Thus χ_{π}(z)^{−1}Θ_{Π}(zg) is well defined for z ∈ Z̃/Z̃_{+}. We compute

\begin{aligned}∑_{z∈Z̃/Z̃_{+}} χ_{π}(z)^{−1}Θ_{Π}(zg) & = ∑_{z∈Z̃/Z̃_{+}} χ_{π}(z)^{−1} ∑_{x∈𝔽^{∗}/𝔽^{∗n}} Θ_{π^{x}}(zg) (by (13)) \\ & = ∑_{z∈Z̃/Z̃_{+}} ∑_{x∈𝔽^{∗}/𝔽^{∗n}} χ_{π}(z)^{−1}χ_{π^{x}}(z)Θ_{π^{x}}(g). \end{aligned}

Now χ_{π}(z)^{−1}χ_{π^{x}}(z) factors to Z̃/Z̃_{+} ≈ 𝔽^{∗}/𝔽^{∗n}, and by orthogonality of characters the right hand side equals |𝔽^{∗}/𝔽^{∗n}|Θ_{π}(g). Explicitly, by (12)

χ_{π}(z)^{−1}χ_{π^{x}}(z) = χ_{π}(z)^{−1}χ_{π}(z)(y, x)_{n} = (y, x)_{n}

where p(z) = yI. As z runs over Z̃/Z̃_{+}, y runs over representatives of 𝔽^{∗}/𝔽^{∗n}, and this gives

∑_{y∈𝔽^{∗}/𝔽^{∗n}} ∑_{x∈𝔽^{∗}/𝔽^{∗n}}(y, x)_{n}Θ_{π^{x}} = |𝔽^{∗}/𝔽^{∗n}|Θ_{π}(g)

since ( , )_{n} is a perfect pairing. This proves:

Theorem 3.1 Let π be a genuine representation of G̃L(n, 𝔽)_{+} with a cen-

tral character. Let Π = Ind_{G̃L(n,𝔽)_{+}}^{G̃L(n,𝔽)}(π). This is a genuine representation of

G̃L(n, 𝔽) and is irreducible if π is irreducible. Assume Θ_{π} exists. Then for g ∈ G̃L(n, 𝔽)_{+} (a regular semisimple element)

\begin{aligned}Θ_{π}(g) & = \frac{1}{|𝔽^{∗}/𝔽^{∗n}|} ∑_{z∈Z̃/Z̃_{+}} χ_{π}(z)^{−1}Θ_{Π}(zg) \\ & = \frac{1}{|𝔽^{∗}/𝔽^{∗n}|} ∑_{x∈𝔽^{∗}/𝔽^{∗n}} χ_{π}(z_{x})^{−1}Θ_{Π}(z_{x}g). \end{aligned}  (14)

In the first sum z runs over any set of coset representatives of Z̃/Z̃_{+}. In the second the sum runs over any coset representatives of 𝔽^{∗}/𝔽^{∗n}, and for each x z_{x} ∈ Z̃ is any element satisfying p(z_{x}) = xI. Each term is independent of the choices.

Essentially the same result holds for genuine representations of S̃L(n, 𝔽). Let π be a genuine representation of S̃L(n, 𝔽), and extend π to a genuine representation π′ of G̃L(n, 𝔽)_{+} (cf. Section 2). Then (3.1) holds for g ∈ S̃L(n, 𝔽) and π replaced by π′. Each summand is independent of the choice of π′.

We can now sketch a proof of Lemma 2.2 (2).

Sketch of Proof. By the Theorem if π is a genuine representation of G̃L(n, 𝔽)_{+} then Θ_{π}(g) = 0 unless p(zg) ∈ T^{n} for some z by Lemma 2.2 (1), i.e. p(g) ∈ ZT^{n}. Conversely if g satisfies this condition then there exists a genuine representation for which Θ_{π}(g) ≠ 0, by the property that characters separate points. This proves the result for regular semisimple elements. For general elements apply a continuity argument. Alternatively apply the argument of Theorem 3.1 directly to an irreducible finite–dimensional genuine representation π of T̃, in which case Θ_{π} is defined for all g ∈ T̃.

4 Lifting from GL(n, 𝔽) to G̃L(n, 𝔽)

In this section we summarize results on lifting of characters from GL(n, 𝔽) to G̃L(n, 𝔽) [7],[4], [8].

We first define transfer factors in this setting. Recall the Weyl denominator for GL(n, 𝔽) is given by Δ(g) = ∏_{i<j} |x_{i} − x_{j}|\overline{𝔽}/|x_{i}x_{j}|\overline{𝔽}^{\frac{1}{2}} if g is a regular semisimple element with (distinct) eigenvalues x_{i} (in an algebraic closure \overline{𝔽} of 𝔽).

Definition 4.1 Suppose h ∈ GL(n, 𝔽), g ∈ G̃L(n, 𝔽) are regular semisimple elements satisfying h^{n} = p(g).

Let

τ(h, g) = gs(h)^{−n}u(h).  (15)

Here u(h) = ±1 ∈ μ_{n} is defined by ([8], §2) (we take u(h) = 1 if n is odd), and s : GL(n, 𝔽) → G̃L(n, 𝔽) is any section. Note that pτ(h, g) = 1 and we consider τ(h, g) to be an element of μ_{n}.

Let

Δ(h, g) = |n^{n}|_{𝔽}^{− \frac{1}{2}} τ(h, g)\frac{Δ(h)}{Δ(g)}.  (16)

Let π be a representation of GL(n) with central character χ_{π} satisfying χ_{π}(ζI) = 1 for all ζ ∈ μ_{n}. Suppose g is a regular semisimple element of G̃L(n, 𝔽), so p(g) is contained in a Cartan subgroup T of GL(n, 𝔽). Let

t_{∗}(Θ_{π})(g) = ∑_{\begin{matrix}h∈T \\ h^{n}=p(g) \end{matrix}} Δ(h, g)Θ_{π}(h).  (17)

This is a conjugation invariant function on the regular semisimple elements of G̃L(n, 𝔽).

This is a special case of ([4], 26.1), and we have written it in a different form. We use the notation of [4]. To see that (17) agrees with [4] first note that in our case the center Z̃_{+} of G̃L(n, 𝔽) is equal to s(Z^{n})μ_{n}, and it follows that the supplementary choice of ω̃ of [4] is unnecessary. The summand in [4] is over

{h ∈ T | h^{∗−1}g ∈ Z̃_{+}}/Z.

Given \overline{h} in this set, choose a representative h ∈ T, and write h^{∗}z = g for z ∈ Z̃_{+}. Equivalently the sum is over

A = {h ∈ T | (hz)^{n} = p(g) for some z ∈ Z}/Z.

On the other hand we have written the sum over

B = {h ∈ T | h^{n} = p(g)}.

There is an n − 1 surjective map from B to A given by h → \overline{h}. Finally if h^{n} = p(g) then h^{∗−1}g = s(h)^{−n}u(h)g = τ(g, h), and since this is an element of μ_{n}, ω̃(τ(g, h)) = τ(g, h). We have incorporated this term, together with the constant b of ([4], §24) (divided by n because of the difference between A and B) into the transfer factor Δ(h, g).

Flicker, Kazhdan and Patterson conjecture that for π an irreducible unitary representation t_{∗}(π) is either 0 or ± the character of a genuine irreducible unitary representation of G̃L(n, 𝔽). We refine this conjecture into two hypotheses for later use.

Hypothesis I Let π be an irreducible representation of GL(n, 𝔽) such that χ_{π}(ζI) = 1 for all ζ ∈ μ_{n}. We say Hypothesis I holds for π if t_{∗}(π) is 0 or ± the character of an irreducible representation of G̃L(n, 𝔽). If this holds we define the virtual representation t_{∗}(π) by t_{∗}(Θ_{π}) = Θ_{t_{∗}(π)}. Furthermore if t_{∗}(π) ≠ 0 define ϵ(π) = ±1 so that ϵ(π)t_{∗}(π) is a representation.

Hypothesis II Every genuine irreducible unitary representation of G̃L(n, 𝔽) is isomorphic to ϵ(π)t_{∗}(π) for some irreducible unitary representation π satisfying Hypothesis I.

Hypotheses I and II are true for n = 2 [5]. Hypothesis I is true if π is a discrete series representation, and t_{∗} is a bijection between a subset of the discrete series of GL(n, 𝔽) and the genuine discrete series of G̃L(n, 𝔽) ([4], §26). Hence Hypothesis II holds in the context of discrete series representations. For π a discrete series representation ϵ(π) = 1. If t_{∗}(π) is supercuspidal then π is supercuspidal, but not conversely.

Hypothesis I holds if π is tempered [4], with the caveat that this statement depends on ([4], Proposition 26.2), and in some cases there is a technical obstruction to this result holding as stated. In any event if π is tempered and satisfies Hypothesis I then t_{∗}(π) is tempered and ϵ(π) = 1. Subject to the preceding caveat Hypothesis II holds for tempered representations and t_{∗} is a bijection between the genuine irreducible tempered representations of G̃L(n, 𝔽) and a subset of the irreducible tempered representations of GL(n, 𝔽) ([4], Theorem 27.3).

Assuming Hypothesis II holds for tempered representations, then the Grothendieck group of genuine representations of GL(n, 𝔽) is spanned by the t_{∗}(π) for π satisfying Hypothesis I. Furthermore the non-zero t_{∗}(π) as π runs over all standard modules for GL(n, 𝔽) is a basis of the Grothendieck group of genuine representations of G̃L(n, 𝔽).

We are particularly interested in non-tempered π satisfying Hypothesis I. For example Hypothesis I holds for any character α satisfying α(ζ) = 1 for all ζ ∈ μ_{n}. In this case t_{∗}(α) is a singular unitary quotient of a minimal principal series with a one-dimensional space of Whittaker functionals [5],([7], Corollary I.3.6). Hypothesis I should hold for all characters α. For example for n = 2, −t_{∗}(α) is the supercuspidal constituent of the oscillator representation if α(−1) = −1 [5].

The central characters of π and t_{∗}(π) are related by

χ_{t_{∗}(π)}(x^{n}I, 1) = χ_{π}(x).  (18)

We also have for any α ∈ 𝔽̂^{∗}

t_{∗}(πα^{n}) = t_{∗}(π)α.  (19)

These follow immediately from (17).

5 Parameters for S̃L(n, 𝔽)

We put lifting from GL(n, 𝔽) to G̃L(n, 𝔽) together with restriction from G̃L(n, 𝔽) to S̃L(n, 𝔽) to obtain a character formula relating GL(n, 𝔽) and S̃L(n, 𝔽).

We first consider G̃L(n, 𝔽)_{+}. Suppose for the moment that Hypothesis II is true. We parametrize the genuine irreducible unitary representations of G̃L(n, 𝔽)_{+} as follows.

Fix a genuine irreducible unitary representation Π of G̃L(n, 𝔽). By Proposition 3.1 a constituent of the restriction of Π to G̃L(n, 𝔽)_{+} is determined by a character λ of Z̃ satisfying λ|Z̃_{+} = χ_{Π}, i.e.

λ(x^{n}, 1) = χ_{Π}(x^{n}, 1) (x ∈ 𝔽^{∗}).

By Hypothesis II there exists an irreducible representation π of GL(n, 𝔽), with χ_{π}(μ_{n}) = 1, such that t_{∗}(π) = ±Π. By (18)

χ_{Π}(x^{n}, 1) = χ_{π}(x)

so we have

λ(x^{n}, 1) = χ_{π}(x).  (20)

Fix a genuine character μ of Z̃ satisfying (8). Then the set of characters λ of Z̃ satisfying (20) is (cf. 10):

{χ_{ν} | ν^{n} = χ_{π}}.

Note that by Lemma 1.1 and (4) the set of such ν is parametrized by { \atop 𝔽^{∗}/𝔽^{∗n}}, and by Proposition 3.1 this parametrizes the constituents of Π|_{G̃L(n,𝔽)_{+}}.

This motivates the following definition.

Definition 5.1 Let X be the set of pairs (π, ν) where:

(1) π is an irreducible representation of GL(n, 𝔽), with central character χ_{π} satisfying χ_{π}(ζI) = 1 for all ζ ∈ μ_{n},

(2) ν is a character of 𝔽^{∗} satisfying ν^{n} = χ_{π}.

Let (π, ν) ∈ X, and assume Hypothesis I holds for π.

(3) Let L_{0}(π, ν) be the constituent of t_{∗}(π) restricted to G̃L(n, 𝔽)_{+} with central character χ_{ν} (cf. (10)).

(4) Let L(π, ν) be the restriction of L_{0}(π, ν) to S̃L(n, 𝔽).

Remark: L and L_{0} depend on the choice of μ satisfying (8).

By definition ϵ(π)L(π, ν) is the character of a representation. Assuming Hypothesis II every genuine irreducible unitary representation of G̃L(n, 𝔽)_{+} is isomorphic to ϵ(π)L(π, ν) for some (π, ν) ∈ X.

If (π, ν) ∈ X then by (5)

χ_{πν^{−1}} = χ_{π}ν^{−n} = 1,  (21)

i.e. πν^{−1} factors to a representation of PGL(n, 𝔽). If π is a representation of GL(n, 𝔽) with trivial central character let \overline{π} be the corresponding representation of PGL(n, 𝔽).

Definition 5.2 For (π, ν) ∈ X, let M(π, ν) be the irreducible representation \overline{πν^{−1}} of PGL(n, 𝔽).

Thus X is the graph of a correspondence between irreducible genuine representations of G̃L(n, 𝔽)_{+} or S̃L(n, 𝔽) and PGL(n, 𝔽). That is for π an irreducible representation of G̃L(n, 𝔽)_{+} or S̃L(n, 𝔽) and π′ an irreducible representation of PGL(n, 𝔽) we say π corresponds to π′ if there exists x = (π, ν) ∈ X, with π satisfying Hypothesis I, such that L_{0}(x) = π or L(x) = π, and M(x) = π′. Assuming Hypothesis II every genuine irreducible unitary representation of S̃L(n, 𝔽) is in the image of the correspondence.

Lemma 5.1

(1) If (π, ν) ∈ X then (πα, να) ∈ X for all α ∈ 𝔽̂^{∗}. Thus x = (π, ν) → αx = (πα, να) defines an action of 𝔽̂^{∗} on X.

For all α ∈ 𝔽̂^{∗} and x ∈ X,

(2) M(αx) = x,

(3) L_{0}(α^{n}x) = L_{0}(x)α,

(4) L(α^{n}x) = L(x).

Proof. (1) and (2) are immediate. By (19) t_{∗}(α^{n}π) = t_{∗}(π)α, and by (5) L_{0}(α^{n}x) and L_{0}(x)α have the same central character; (3) follows and (4) is an immediate consequence of (3).

Remark If β ∈ 𝔽̂^{∗} is non–trivial on μ_{n}, then β ∉ 𝔽̂^{∗}^{n}, and there is no elementary relationship between L_{0}(βx) and L_{0}(x).

Remark The action of 𝔽̂^{∗}/𝔽̂^{∗n} ≈ μ̂_{n} on genuine representations of S̃L(n, 𝔽) given by α : L(x) → L(αx) generalizes the “Waldspurger involution” for S̃L(2, 𝔽) [18].

We compute the set of representations of S̃L(n, 𝔽) corresponding to a given irreducible representation of PGL(n, 𝔽).

Fix x = (π, ν) ∈ X. If M(x′) = M(x) then x′ = αx for some α. By Lemma 5.1 (4) if α ∈ (𝔽̂^{∗})^{n} ≈ 𝔽̂^{∗n} then L(αx) = L(x). Therefore the irreducible representations of S̃L(n, 𝔽) corresponding to M(x) are the L(αx) for α ∈ 𝔽̂^{∗}/𝔽̂^{∗n} (not to be confused with { \atop 𝔽^{∗}/𝔽^{∗n}}), which by (3) is isomorphic to μ̂_{n}.

Definition 5.3 Let π be an irreducible representation of GL(n, 𝔽) with central character trivial on μ_{n}. Suppose Hypothesis I holds for πα for all α ∈ 𝔽̂^{∗}.

(1) For (π, ν) ∈ X

L_{st}(π, ν) = ∑_{α} L(πα, να)

where the sum runs over a set of representatives of 𝔽̂^{∗}/𝔽̂^{∗n} ≈ μ̂_{n}.

(2) Let π be an irreducible representation of PGL(n, 𝔽), and let π′ denote π pulled back to GL(n, 𝔽). Assume π′α satisfies Hypothesis I for all α ∈ 𝔽̂^{∗}. Define L_{st}(π) = L_{st}(π′, 1).

Remark L_{st}(π, ν) = L_{st}(πα, να) for all α, and in particular

L_{st}(π, ν) = L_{st}(πν^{−1}, 1) = L_{st}(πν^{−1}).

As discussed in the Introduction L_{st}(π), respectively Π(π, ν) = {L(πα, να) | α ∈ μ̂_{n}}, is our candidate for a “stable” virtual character, respectively L-packet, of S̃L(n, 𝔽). Note that the non-zero L(πα, να) in Π(π, ν) are distinct, and in fact have distinct central characters on S̃L(n, 𝔽). One could define a stable virtual character of S̃L(n, 𝔽) to be in the span of the L_{st}(π). It is not clear how to characterize the stable virtual characters intrinsically.

Not all L(πα, να) are necessarily non-zero. For example suppose π is the principal series defined by the character λ(diag(h_{1}, …, h_{n}))) = ∏ λ_{i}(h_{i}). This has central character trivial on μ_{n} if ∏_{i} λ_{i}(ζ) = 1 for all ζ ∈ μ_{n}. On the other hand t_{∗}(π) = 0 unless λ_{i}(ζ) = 1 for all i, ζ ∈ μ_{n}. Assume this holds. Then L(π, α) is a principal series of G̃L(n, 𝔽), and L(πα, να) = 0 for all α ≠ 1, so L(π, ν) = L_{st}(π, ν).

If π is a discrete series representation with χ_{π}(μ_{n}) = 1 π satisfies Hypothesis I, t_{∗}(πα) ≠ 0 for all α, and L(πα, να) ≠ 0 for all α. Therefore in this case |Π(π, α)| = n.

In the case n = 2 L(1, 1) is non-tempered, and is isomorphic to ω_{e}, the even half of the oscillator representation ω = ω_{e} ⊕ ω_{o} of S̃L(n, 𝔽) (ω depends on an additive character ψ, which is determined by μ). If α(−1) = −1 then L(α, α) = −ω_{o} is supercuspidal, and L_{st}(1) = ω_{e} − ω_{o}. See [5], [14] and [1].

6 Orbit Correspondence

For g ∈ GL(n, 𝔽) write \overline{g} for the image of g in PGL(n, 𝔽).

Definition 6.1 For h ∈ GL(n, 𝔽) let

ϕ(h) = det(h^{−1})h^{n} ∈ SL(n, 𝔽).

Then ϕ(zg) = ϕ(g) for all z ∈ Z, so ϕ factors to a map from PGL(n, 𝔽) to SL(n, 𝔽).

Thus GL(n, 𝔽) is the graph of a correspondence between PGL(n, 𝔽) and SL(n, 𝔽) via the maps the maps g → \overline{g} ∈ PGL(n, 𝔽) and g → ϕ(g) ∈ SL(n, 𝔽). The following Lemma is immediate.

Lemma 6.1

(1) For all h ∈ PGL(n, 𝔽), g ∈ GL(n, 𝔽), ϕ(\overline{g}h\overline{g}^{−1}) = gϕ(h)g^{−1}.

(2) If h is a regular semisimple element then ϕ(h) is relevant (cf. Lemma 2.2).

We also need the weak orbit correspondence. Suppose h ∈ GL(n, 𝔽), g ∈ SL(n, 𝔽) satisfy

h^{n} = zg (z ∈ Z).

Multiplying both sides by det(h^{−1}) shows this is equivalent to

ϕ(h) = det(h^{−1})zg (z ∈ Z).

Since ϕ(h) and g have determinant one this gives

det(h^{−1})z = ϕ(h)g^{−1} = ζI (ζ ∈ μ_{n}).  (22)

Definition 6.2 We say h ∈ PGL(n, 𝔽), g ∈ SL(n, 𝔽) weakly correspond, written h ↔^{weak} g, if for any (equivalently all) h′ ∈ GL(n, 𝔽) with \overline{h′} = h,

h^{′n} = zg (z ∈ Z).

Equivalently

g = ζϕ(h) (ζ ∈ μ_{n}).

If h ↔^{weak} g define ζ(h, g) ∈ μ_{n} by

g = ζ(h, g)ϕ(h).  (23)

We give an alternative description of the orbit correspondences in terms of roots and weights. This is not needed for what follows. Given a Cartan subgroup of GL(n, 𝔽), we identify the root and weight lattices of the corresponding Cartan subgroups of PGL(n, 𝔽) and SL(n, 𝔽).

Lemma 6.2 Let h, g be semisimple elements.

(1) h ↔^{weak} g if and only if α(h^{n}) = α(g) for all roots α,

(2) ϕ(h) = g if and only if (nλ)(h) = λ(g) for all weights λ.

Proof. Part (1) is immediate. For (2) we need to show for h ∈ GL(n, 𝔽), ζ ∈ μ_{n},

(nλ)(h) = λ(ζdet(h^{−1})h^{n}) for all weights λ ⇔ ζ = 1.

The subtlety is that λ(h) is not defined for arbitrary elements of GL(n, 𝔽). If h ∈ SL(n, 𝔽)Z = GL(n, 𝔽)_{+} then λ(h) is defined and this is immediate. It is enough to work over the algebraic closure \overline{𝔽}, in which case GL(n, \overline{𝔽})_{+} = GL(n, \overline{𝔽}), proving the result. Remark If g is in the split torus then |{h | ϕ(h) = g}| = n^{n−2}. In general the cardinality of the inverse image of a (regular semisimple) g ∈ SL(n, 𝔽) depends on the Cartan subgroup containing g.

7 Transfer Factors

We continue with the notation of §6. Fix a character μ of Z̃ satisfying (8). We define transfer factors Δ_{μ}(h, g) (Definition 7.2). These satisfy one of the standard requirements of transfer fators: |Δ_{μ}(h, g)| = |Δ(h)/Δ(g)| (25), up to a constant which is 1 for almost every residual characteristic.

Recall the definition of Δ (Definition 4.1).

Definition 7.1 Suppose h ∈ GL(n, 𝔽), g ∈ S̃L(n, 𝔽) satisfy

h^{n} = p(zg) (z ∈ Z̃).  (24)

Let

Δ_{μ}(h, g) = \frac{n^{2}}{|𝔽^{∗}/𝔽^{∗n}|}μ(z)^{−1}Δ(h, zg).

This is independent of the choice of z satisfying (24).

Lemma 7.1 For all λ ∈ 𝔽^{∗}

Δ_{μ}(λh, g) = Δ_{μ}(h, g).

Definition 7.2 Suppose h ∈ PGL(n, 𝔽), g ∈ S̃L(n, 𝔽) satisfy

h ↔^{weak} p(g).

Choose h′ ∈ GL(n, 𝔽) satisfying \overline{h′} = h. Let

Δ_{μ}(h, g) = Δ_{μ}(h′, g).

By the Lemma this is independent of the choice of h′.

Given h, g as in Definition 7.2, choose h′ ∈ GL(n, 𝔽) satisfying \overline{h′} = h, and choose z ∈ Z̃ with h^{′n} = p(zg). Recall τ is given by Definition 4.1, and |𝔽^{∗}/𝔽^{∗n}| = n^{2}/|n|_{𝔽} ([7], Lemma 0.3.2). This gives

Δ_{μ}(h, g) = \frac{n^{2}}{|𝔽^{∗}/𝔽^{∗n}|}μ(z)^{−1}Δ(h′, zg)

= |n|^{1−n/2}μ(z)^{−1}τ(h′, zg)\frac{Δ(h)}{Δ(g)}

= |n|^{1−n/2}μ(z)^{−1}zgs(h′)^{−n}u(h′)\frac{Δ(h)}{Δ(g)}.

This is independent of the choices.

Proof of the Lemma. Choose z ∈ Z̃ satisfying h^{n} = p(zg), and w ∈ Z̃ satisfying p(w) = λ^{n}I. Then (λh)^{n} = p(wzg). We need to show Δ_{μ}(h, g) = Δ_{μ}(λh, g), i.e.

μ(z)^{−1}zgs(h)^{−n}u(h) = μ(wz)^{−1}wzgs(λh)^{−n}u(λh).

After cancellations this is equivalent to

s(λh)^{n}u(λh) = μ(w)^{−1}ws(h)^{n}u(h).

By ([4], §4) s(λh)^{n}u(λh) = s(h)^{n}u(h)s_{0}(λ^{n}), where s_{0} is the distinguished section, i.e. s_{0}(g) = (g, 1). Inserting this we are reduced to showing s_{0}(λ^{n}) = μ(w)^{−1}w, which is precisely the fact that μ|Z̃_{+} = ι.

Remark If n = 2 or the residual characteristic of 𝔽 does not divide n then by (9)

(\frac{Δ_{μ}(h, g)}{Δ(h)/Δ(g)})^{N} = 1  (25)

with N = n (n odd) or N = 2n (n even). Remark If μ′ is another character satisfying (8), then μ′(z) = μ(z)(y, x)_{n} for some y, where p(z) = xI, and

\frac{Δ_{μ′}}{Δ_{μ}} (h, g) = \frac{μ}{μ′} (h) = (det(h), y)_{n}

(det(h) is a well defined element of 𝔽^{∗}/𝔽^{∗n}).

Although we will not need it we state the invariance property of Δ_{μ}. Supppose h ↔^{weak} g. For y ∈ G̃L(n, 𝔽) let y_{0} = \overline{p(y)} ∈ PGL(n, 𝔽).

Lemma 7.2

Δ_{μ}(y_{0}hy_{0}^{−1}, ygy^{−1}) = Δ_{μ}(h, g)(det(h)ζ(h, g), det(y))_{n}

Proof. A straightforward computation which is left to the reader.

8 Stable Character Formula

We state the character formula relating the character of an irreducible representation π of PGL(n, 𝔽) to the character of the virtual genuine representation L_{st}(π) of S̃L(n, 𝔽).

Fix μ as in (8), define L_{st} as in Definition 5.3, ϕ as in Section 6 and Δ_{μ} as in Section 7.

Theorem 8.1 (Main Theorem) Let π be an irreducible representation of PGL(n, 𝔽), for which L_{st}(π) is defined (Definition 5.3). Then for g a regular semisimple element of S̃L(n, 𝔽),

Θ_{L_{st}(π)}(g) = ∑_{\begin{matrix}h∈PGL(n,𝔽) \\ ϕ(h)=p(g) \end{matrix}} Δ_{μ}(h, g)Θ_{π}(h).  (26)

Recall the hypothesis on π is that t_{∗}(πα) is defined for all α ∈ 𝔽̂^{∗} (we have pulled π back to GL(n, 𝔽)).

Remark By Lemma 7.2 the right hand side of (26) is a priori S̃L(n, 𝔽) conjugation invariant. We do not need this, and it is a consequence of the Theorem. Note that Θ_{L_{st}(π)} is not necessarily invariant by conjugation by G̃L(n, 𝔽), since Δ_{μ} is only S̃L(n, 𝔽) conjugation invariant (Lemma 7.2).

Proof.

We first give a formula for Θ_{L(π,ν)}(g) for arbitrary (π, ν) ∈ X (with π satisfying Hypothesis I).

By Theorem 3.1 Θ_{L(π,ν)}(g) = ∑_{z∈Z̃/Z̃_{+}} χ_{ν}(z)^{−1}Θ_{t_{∗}(π)}(zg) (sum over any set of coset representatives). Inserting (17) gives

Θ_{L(π,ν)}(g) = \frac{1}{|𝔽^{∗}/𝔽^{∗n}|} ∑_{z∈Z̃/Z̃_{+}} ∑_{h^{n}=p(zg)} χ_{ν}(z)^{−1}Δ(h, zg)Θ_{π}(h).  (27)

Write the summand as follows:

χ_{ν}(z)^{−1}Δ(h, zg)Θ_{π}(h) = μ(z)^{−1}ν(z)^{−1}Δ(h, zg)Θ_{π}(h) (by (11))

= \frac{|𝔽^{∗}/𝔽^{∗n}|}{n^{2}}ν(z)^{−1}Δ_{μ}(h, g)Θ_{π}(h) (Definition 7.1)

= \frac{|𝔽^{∗}/𝔽^{∗n}|}{n^{2}}ν(h)ν(z)^{−1}Δ_{μ}(h, g)Θ_{πν^{−1}}(h).

By (21) and Lemma 7.1 Δ(h, g) and Θ_{πν^{−1}}(h) only depend on the image \overline{h} ∈ PGL(n, 𝔽) of h. By (22) and (23) ν(h)ν(z)^{−1} = ν(ϕ(h)^{−1}g) = ν(ζ(\overline{h}, g)). This gives

\frac{|𝔽^{∗}/𝔽^{∗n}|}{n^{2}}ν(ζ(\overline{h}, g))Δ_{μ}(\overline{h}, g)θ_{πν^{−1}}(\overline{h}).

Inserting this in (27) and changing the order of summation gives the following intermediate result.

Proposition 8.1

Θ_{L(π,ν)}(g) = \frac{1}{n} ∑_{h∈PGL(n,𝔽) h ↔^{weak}g} ν(ζ(h, g))Δ_{μ}(h, g)θ_{πν^{−1}}(h)

= \frac{1}{n} ∑_{ζ∈μ_{n}} ν(ζ) ∑_{\begin{matrix}h∈PGL(n,𝔽) \\ ϕ(h)=ζg \end{matrix}} Δ_{μ}(h, g)θ_{πν^{−1}}(h).

Replace (π, ν) with (πα, να). On the right hand side only the term ν(ζ(h, g)) is affected. Summing over α gives:

Θ_{L_{st}(π,ν)}(g) = \frac{1}{n} ∑_{ζ∈μ_{n}} ∑_{α∈μ̂_{n}} ν(ζ)α(ζ) ∑_{\begin{matrix}h∈PGL(n,𝔽) \\ ϕ(h)=ζg \end{matrix}} Δ_{μ}(h, g)θ_{πν^{−1}}(h).

By orthogonality of characters for μ_{n} this equals

∑_{\begin{matrix}h∈PGL(n,𝔽) \\ ϕ(h)=g \end{matrix}} Δ_{μ}(h, g)θ_{πν^{−1}}(h).

This completes the proof.

9 Inversion

We continue in the setting of the preceding section. Suppose πα satisfies Hypothesis I for all α. Definition 9.1 For ζ ∈ μ_{n} let

L_{ζ}(π, ν) = ∑_{α∈μ̂_{n}} α(ζ)L(πα, να).

This is a virtual character in which we allow rational coefficients, and L_{1}(π, ν) = L_{st}(π, ν).

By Fourier inversion on μ_{n}:

L(π, ν) = \frac{1}{n} ∑_{ζ∈μ_{n}} L_{ζ}(π, ν).  (28)

Recall the central character of L(πα, να) is χ_{αν}, i.e.

χ_{L(πα,να)}(z_{ζ}) = χ_{ν}(z_{ζ})α(ζ)

where p(z_{ζ}) = ζI. That is

α(ζ)Θ_{L(πα,να)}(g) = χ_{ν}^{−1}(z_{ζ})Θ_{L(πα,να)}(z_{ζ}g).

Inserting this into the definition gives Lemma 9.1 For all ζ ∈ μ_{n},

Θ_{L_{ζ}(π,ν)}(g) = χ_{ν}^{−1}(z_{ζ})Θ_{L_{st}(π,ν)}(z_{ζ}g)

for any choice of z_{ζ} satisfying p(z_{ζ}) = ζI.

Inserting this in (28) gives:

Theorem 9.1 (Inversion) Suppose (π, ν) ∈ X, and πα satisfies Hypothesis I for all α ∈ 𝔽̂^{∗}. Then

\begin{aligned}Θ_{L(π,ν)}(g) & = \frac{1}{n} ∑_{ζ∈μ_{n}} L_{ζ}(π, ν)(g) \\ & = \frac{1}{n} ∑_{ζ∈μ_{n}} χ_{ν}^{−1}(z_{ζ})Θ_{L_{st}(π,ν)}(z_{ζ}g). \end{aligned}  (29)

By Theorem 8.1 each term on the right hand side of (29) may be expressed in terms the character Θ_{πν^{−1}} of PGL(n, 𝔽). The resulting formula is Proposition 8.1.

We record the analogue of Theorem 8.1 for L_{ζ}(π, ν).

Θ_{L_{ζ}(π,ν)}(g) = ν(ζ)^{−1} ∑_{\begin{matrix}h ↔^{weak}g \\ ζ(h,g)=ζ^{−1} \end{matrix}} Δ_{μ}(h, g)Θ_{πν^{−1}}(h).

References

[1] J. Adams. Character of the oscillator representation. Israel J. Math., 98:229–252, 1997.

[2] J. Adams. Characters of non-linear groups. Proceedings of Conf. on Representation Theory and Harmonic Analysis, Okayama, Japan, 26:1–18, 2000.

[3] Jeffrey Adams. Lifting of characters on orthogonal and metaplectic groups. Duke Math. J., 92(1):129–178, 1998.

[4] Y. Flicker D. A. Kazhdan. Metaplectic correspondence. Inst. Hautes Études Sci. Publ. Math., 64:53–110, 1986.

[5] Y. Flicker. Automorphic forms on covering groups of GL(2). Invent. Math., 57:119–182, 1980.

[6] R.P. Langlands J.-P. Labesse. L-indistinguishability for SL(2). Canad. J. Math., 31:726–785, 1979.

[7] D. A. Kazhdan and S. J. Patterson. Metaplectic forms. Inst. Hautes Études Sci. Publ. Math., (59):35–142, 1984.

[8] D. A. Kazhdan and S. J. Patterson. Towards a generalized Shimura correspondence. Adv. in Math., 60(2):161–234, 1986.

[9] H. Matsumoto. Sur les sous groupes arithmétiques des groupes semi– simples. Ann. Sci. École Norm. Sup. (4), pages 1–62, 1969.

[10] C. Moore. Group extensions of p-adic and adelic linear groups. Inst. Hautes Études Sci. Publ. Math., (35):157–222, 1968.

[11] R. Ranga Rao. On some explicit formulas in the theory of the weil representation. Pacific J. Math., 157:335–371, 1993.

[12] A. Knapp S. Gelbart. L-indistinguishability and R groups for the special linear group. Adv. in Math., 43:101–121, 1982.

[13] G. Savin. Local Shimura correspondence. Math. Ann., 280:185–190, 1988.

[14] J. Schultz. Lifting of Characters on S̃L(2) and SO(2, 1) over a nonarchimedean local field. PhD thesis, University of Maryland, 1998. Thesis.

[15] J.-P. Serre. Local Fields. Number 67 in Graduate Texts in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York, 1979.

[16] R. Steinberg. Générateurs, relations et revêtements de groupes algébriques. Colloq. Théorie des Groupes Algébriques, Bruxelles (1962), pages 113–127, 1962.

[17] M. Tadić. Notes on representations of non-archimedean SL(n). Pacific J. Math., 152(2):375–396, 1992.

[18] J.-L. Waldspurger. Correspondances de Shimura et quaternions. Forum Math., 3:219–307, 1991.