ABSTRACT

Title of dissertation:  LIFTING OF CHARACTERS OF S̃L_{2}(F) AND SO_{1,2}(F) FOR F A NONARCHIMEDEAN LOCAL FIELD

Jason Paul Schultz, Doctor of Philosophy, 1998

Dissertation directed by: Professor Jeffrey D. Adams Department of Mathematics

We make a definition of “stable” virtual character of the two fold cover of SL_{2}(F) for F p–adic and establish a bijection between stable virtual characters π̃ of S̃L_{2}(F) and irreducible representations π′ of SO_{1,2}(F). This is achieved through character theory; the characters Θπ̃, Θ_{π′} of π̃ and π̃′ are related by

Θπ̃(g̃) = Γ_{−}(g̃)Θ_{π′}(τ(g)).

Here g̃ ∈ S̃L_{2}(F) and τ(g) is contained in the unique conjugacy class of elements of SO_{1,2}(F) having the same nontrivial eigenvalues as g̃. The transfer factor Γ_{−}(g̃) is the character of the difference of the two halves of the oscillator representation of S̃L_{2}(F). Included is a complete development of character formulas for the oscillator representation of S̃L_{2}(F).

This work was motivated by the results of Adams in [3] where a very similar character formula affords a bijection of stable virtual characters of S̃p(2n, R) and stable virtual characters of SO_{n+1,n}(R) with good properties.

LIFTING OF CHARACTERS OF

S̃L_{2}(F) AND SO_{1,2}(F) FOR F A

NONARCHIMEDEAN LOCAL FIELD

by

Jason Paul Schultz

Dissertation submitted to the Faculty of the Graduate School of the

University of Maryland, College Park in partial fulfillment

of the requirements for the degree of Doctor of Philosophy 1998

Advisory Committee:

Professor Jeffrey D. Adams, Chairman/Advisor Professor Rebecca A. Herb Professor Stephen S. Kudla Professor Ronald L. Lipsman Professor Oscar W. Greenberg, Physics

ACKNOWLEDGEMENTS

I am happy to have the chance to thank my advisor, Jeff Adams, for the outstanding education that he gave me. He was very patient, supportive, encouraging, and consistent. I am in debt to Jeff for all the time he spent working with me and answering my questions, repeatedly if necessary, in an effort to bring me to this point. I will very much miss our frequent discussions.

I would like to thank Jian–shu Li for introducing me to the field of representation theory and automorphic forms. I would also like to thank David Manderscheid and David Joyner for their input at various stages.

My mother and father, Carol Grosso Cole and Harvey Schultz and my stepmother and stepfather, Nanette Rainone and Roy Cole have helped me immeasurably. I wouldn’t be the person that I am without their love and influence and hope to always make them proud.

John Parkhurst and John Van Brunt both helped me through a very difficult time in different ways, and I am grateful to have had the honor of being influenced by both of them.

Finally, I would like to single out two friends who have influenced me profoundly. The first is Kirsty Eisenhart. I met her in my first year of graduate school and I would never have made it without her friendship. The second is my close friend and confidante, Amanda Lubell. She always knew what I meant, helped make Maryland home to me, and will always hold a special place in my heart.

TABLE OF CONTENTS

1 Introduction 1

2 Preliminaries 10 2.1 The Hilbert Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Quadratic Extensions of F . . . . . . . . . . . . . . . . . . . . . . 12 2.3 SO_{1,2}(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Covering Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Weil’s Gamma Factors . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Flicker’s Correspondence . . . . . . . . . . . . . . . . . . . . . . . 30

3 Representations of G̃L_{2}(F)_{+} 37 3.1 Restricting to G̃L_{2}(F)_{+} . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Inversion via the Center . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 The Character of the Oscillator Representation 46 4.1 Hyperbolic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Elliptic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Elliptic Set, Case I: g = h^{2} . . . . . . . . . . . . . . . . . . 51

4.2.2 Elliptic Set, Case II: g ≠ h^{2} . . . . . . . . . . . . . . . . . 54 4.3 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 The Character Formula 61 5.1 Hyperbolic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Elliptic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Elliptic Set, Case I: g = h^{2} . . . . . . . . . . . . . . . . . . 64 5.2.2 Elliptic Set, Case II: g ≠ h^{2} . . . . . . . . . . . . . . . . . 66

5.3 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.4 Pushing Down to SO_{1,2}(F): The Character Formula . . . . . . . . 69 5.5 Stable Representations of S̃L_{2}(F) . . . . . . . . . . . . . . . . . . 74

6 Properties of the Correspondence 77 6.1 Principal Series Representations . . . . . . . . . . . . . . . . . . . 77 6.2 One Dimensional Representations and the Dependence on ψ … 84 6.3 Special and Supercuspidal Representations . . . . . . . . . . . . . 89

6.3.1 Special Representations . . . . . . . . . . . . . . . . . . . 89 6.3.2 Supercuspidal Representations . . . . . . . . . . . . . . . . 90

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Bibliography 91

Chapter 1

Introduction

Overview

The study of correspondences between representations of Lie groups is an important theme in the theory of representations and automorphic forms. Lifting, or transfer, of the representation theory of one group to another has been the subject of extensive study. Langlands functoriality, roughly stated, says that given linear groups G and H, a homomorphism ^{L}H → ^{L}G of their L–groups gives a transfer of representations of H to representations of G. On the other hand, the theta correspondence of Howe [16] provides a way to relate the representation theories of a reductive dual pair G and G′ embedded in the metaplectic group. The groups G and G′ are sometimes non–linear.

Shimura [34] defined a correspondence between the space of cusp forms of half integral weight k/2 and the space of cusp forms of even integral weight k − 1. This correspondence preserves eigenvalues for the Hecke ring. Shimura’s correspondence is based on L–functions and suggests a correspondence between automorphic forms on S̃L_{2} and P GL_{2}. Subsequently, Shintani [35] and Niwa [27] discovered this correspondence using the oscillator (or Weil) representation.

Interpreting the Shimura correspondence in representation theoretic terms then became an important problem. A systematic study of the dual pair correspondence for the reductive dual pair (S̃L_{2}, P GL_{2}) was made by Rallis and Schiffman [30] and was later parameterized by Manderscheid [24]. Gelbart initiated the study of the Shimura correspondence via group theory in [9]; this was continued with Piatetski–Shapiro in [10] and [11]. Finally, in [39], [40] and [41], Waldspurger gave a very deep study of the Shimura correspondence and automorphic forms on S̃L_{2}. An excellent survey of this work was made by Piatetski–Shapiro in [28].

Using character theory, Flicker [6] has defined a correspondence between m– fold covering groups G̃L_{2} and GL_{2}. This correspondence has been generalized by Kazhdan and Patterson in [19] and [20] and by Flicker and Kazhdan in [7] in the case of G̃L_{r} and GL_{r}.

Our work is motivated by a correspondence of Adams in [3] using character theory. Suppose F is a local field of characteristic 0. Let S̃p(2n, F) be the two-fold cover of Sp(2n, F). Fix an additive character ψ of F and let ω(ψ) = ω_{e}(ψ) ⊕ ω_{o}(ψ) be the oscillator representation of S̃p(2n, F) attached to ψ. If π is a representation, let Θ_{π} be its character (see Chapter 2). We consider the character as a function on the regular semisimple elements, or equivalently, conjugacy classes. Let Γ_{−}^{ψ} be the difference of the characters of the two halves of the oscillator representation, i.e.,

Γ_{−}^{ψ} = Θ_{ω_{e}(ψ)} − Θ_{ω_{o}(ψ)}.

Now assume F = R. If g ∈ Sp(2n, R), then we say that h ∈ SO_{n+1,n}(R) and g correspond via the orbit correspondence if g and h have the same non–trivial eigenvalues (see Adams [3]). We write t(g) = h for the orbit correspondence. If g ∈ S̃p(2n, R) then write t(g) = t(pr(g)) (here pr is the projection S̃p(2n, R) → Sp(2n, R)). Adams [3] (see also [1]) has given the definition of “stable” genuine invariant eigendistribution on S̃p(2n, R) and has shown that the map t_{∗} given by

t_{∗}(Θ)(g) = Γ_{−}^{ψ}(g)Θ(t(g))

is a bijection between

{stable invariant eigendistributions of SO_{n+1,n}(R)}

and

{stable genuine invariant eigendistributions of S̃p(2n, R)}.

This bijection has good properties. For example, t_{∗}(Θ) is a virtual character if and only if Θ is, t_{∗}(Θ) is tempered if and only if Θ is, t_{∗} takes stable discrete series representations of SO_{n+1,n}(R) to the stable genuine discrete series representations of S̃p(2n, R), t_{∗} commutes with parabolic induction, and if Θ is a tempered virtual character, then t_{∗}(Θ) is the normalized theta–lift of Θ. See Adams in [3] for the proof.

Essential to the proof of Adams’ lifting are Hirai’s matching conditions. These are special for the field F = R. Our problem considers the analogue of this correspondence in the case of n = 1 with F p–adic. We now give a detailed summary of our methods and results.

Let π be an irreducible representation of GL_{2}(F). If π has trivial central character, then π factors to a representation of P GL_{2}(F) ≃ SO_{1,2}(F). Let p be the map from GL_{2}(F) to SO_{1,2}(F). Then define π′, a representation of SO_{1,2}(F), by pulling back π via p, i.e., their characters satisfy

Θ_{π}(g) = Θ_{π′}(p(g)).

Assume g ∈ SL_{2}(F). Let V be a vector space equipped with a quadratic form of signature (1, 2). For g regular semisimple let τ(g) be the unique conjugacy class of elements with the same nontrivial eigenvalues as g. Then

p(g) ∈ τ(g^{2}).

(For details, see Section 2.3.) Our character identities relate character values at g̃ ∈ S̃L_{2}(F) satisfying pr(g̃) = g and character values at τ(g) ⊆ SO_{1,2}(F).

The key idea is then the following. Start with an irreducible representation π of GL_{2}(F) whose central character χ_{π} satisfies χ_{π}(−I) = 1 (here I is the 2 × 2 identity matrix) and lift it via Flicker’s theory [6] to π̃ = Lift_{F}(π) of G̃L_{2}(F). (The lifting is actually defined for arbitrary covers of GL_{2}(F) but we only use the case of 2–fold covers. See Section 2.7 for a full discussion of Flicker’s correspondence.) Then restrict π̃ to G̃L_{2}(F)_{+} = {g̃ ∈ G̃L_{2}(F) : det pr(g̃) ∈ F^{×2}}. Let σ̃ be an irreducible summand of this restriction. By Clifford Theory (see Theorem 3.8), we have

π̃|_{G̃L_{2}(F)_{+}} ≃ ∑_{x∈F^{×}/F^{×2}} σ̃^{x}.

In particular, the summands are parameterized by x ∈ F^{×}/F^{×2}. If G is any group, we let Z(G) be its center. Since

G̃L_{2}(F)_{+} = S̃L_{2}(F) · Z(G̃L_{2}(F)_{+}),

there is no further reducibility when we restrict to S̃L_{2}(F). We are able to recover the character of σ̃^{x} by using an “inversion formula”:

Θ_{σ^{x}}(g̃) = \frac{1}{|F^{×}/F^{×2}|} ∑_{ξ∈F^{×}/F^{×2}} χ_{σ^{x}}(z_{ξ})^{−1}Θπ̃(z_{ξ}g̃),  g̃ ∈ G̃L_{2}(F)_{+}.

This is given in Theorem 3.12. Note that this yields character formulas for all irreducible representations of S̃L_{2}(F), and that both Theorems 3.8 and 3.12 are valid for any genuine representation π̃ of G̃L_{2}(F).

Flicker’s correspondence is defined by a character identity, giving a relationship between the character of π̃ = Lift_{F}(π) on G̃L_{2}(F) and the character of π on GL_{2}(F). Flicker’s formula is only nonzero on the “relevant elements”, that is, elements g̃ such that pr(g̃) is a square. In particular, Flicker [6] points out that:

characters of genuine representations of G̃L_{2}(F) vanish off of the set {(g^{2}; ε)}.

Using Flicker’s identity together with the inversion formula, we are able to relate the character of an individual σ̃^{x} with the character of π which is defined on all elements. The problem is then to relate the character of an individual σ̃^{x} with the character of a representation π′ of SO_{1,2}(F). It turns out that this can’t be done because the resulting character formula has two terms. However, if we “stabilize” the formulas (see below) we are able to achieve this.

Here and throughout, if ν_{0} is a character of F^{×}, then the letter ν will always be reserved for the one–dimensional representation ν = ν_{0} ∘ det of GL_{2}(F).

Given a representation π on GL_{2}(F) such that χ_{π}(−I) = 1, there exists a character ν_{0} of F^{×} such that

χ_{π}(xI) = ν_{0}(x^{2}).

Let

χ_{ν_{0}}(xI; ε) = ν_{0}(x)γ_{F}(x, ψ)ε.

Note that this depends on the choice of the additive character ψ. The γ_{F}(x, ψ) are the gamma–factors introduced by Weil; see Section 2.5. Each genuine character of the center of G̃L_{2}(F)_{+} is of this form. In particular, there exists ν_{0} such that χ_{σ^{x}} = χ_{ν_{0}}.

Definition 1.1. Fix a character ν_{0} of F^{×} satisfying χ_{π}(xI) = ν_{0}(x^{2}). Let ρ̃ be the constituent of π̃|_{G̃L_{2}(F)_{+}} with central character χ_{ν_{0}}. We define L(π, ν_{0}) to be the restriction of ρ̃ to S̃L_{2}(F).

This central character condition is consistent with Flicker’s character condition in Definition 2.63.

Our first main result is stated in Theorem 5.30. We repeat it here.

Theorem 1.2. Suppose that π is an irreducible representation of GL_{2}(F) satisfying χ_{π}(−I) = 1. Suppose that ν_{0} is a character of F^{×} satisfying ν_{0}(x^{2}) = χ_{π}(xI). Let g̃ be a regular semisimple element of S̃L_{2}(F) and let g = pr(g̃). Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}^{ψ}(g̃)Θ_{(πν^{−1})′}(τ(g)) + ν_{0}(−1)Γ_{+}^{ψ}(g̃)Θ_{(πν^{−1})′}(τ(−g))] .

In the above statement, Γ_{+}^{ψ} and Γ_{−}^{ψ} are respectively the sum and difference of the two halves of the oscillator representation attached to ψ. We remark that in Chapter 4, we have developed formulas for the characters of the even and odd halves of the oscillator representation and therefore obtain the character of their sum and difference. See Theorem 4.40 and Theorem 4.41.

If β_{0} is a character of F^{×} which satisfies β_{0}(−1) = 1, then

L(π, ν_{0}) = L(πβ, ν_{0}β_{0}).

In fact, we have shown that L(π, ν_{0}) = L(σ, λ_{0}) if and only if there is a character β_{0} satisfying σ = πβ, λ_{0} = ν_{0}β_{0}, and β_{0}(−1) = 1.

Theorem 1.2 suggests that we have a map

L(π, ν_{0}) ↦ (πν^{−1})′.

However, the map is not one–to–one. Indeed, if β is such that β_{0}(−1) = −1, then L(π, ν_{0}) and L(πβ, ν_{0}β_{0}) are generally non–zero, are inequivalent because they have different central characters, and map to the same representation (πν^{−1})′. To remedy this situation, we “stabilize” our formulas as follows.

Definition 1.3. Let β_{0} satisfy β_{0}(−1) = −1. Let π be an irreducible representation of GL_{2}(F) satisfying χ_{π}(−I) = 1. Let ν_{0} be a character of F^{×} such that χ_{π}(xI) = ν_{0}(x^{2}). Define

L_{st}(π, ν_{0}) = L(π, ν_{0}) + L(πβ, ν_{0}β_{0}).

This is independent of the choice of β.

Let Gr_{st}(S̃L_{2}(F)) be the linear span of all virtual characters L_{st}(π, ν_{0}) where π is an irreducible representation of GL_{2}(F) satisfying χ_{π}(−I) = 1 and ν_{0} is a character of F^{×} satisfying χ_{π}(xI) = ν_{0}(x^{2}). If ρ ∈ Gr_{st}(S̃L_{2}(F)), then we say that ρ is a stable virtual representation of S̃L_{2}(F).

After making this definition, we obtain the following formula.

Theorem 1.4. Continue with the notation in Theorem 1.2. Then

Θ_{L_{st}(π,ν_{0})}(g̃) = Γ_{−}^{ψ}(g̃)Θ_{(πν^{−1})′}(τ(g)).

The formula defines a bijection

\begin{matrix}Gr_{st}(S̃L_{2}(F)) & ↔ Gr(SO_{1,2}(F)) \\ L_{st}(π, ν_{0}) & ↔ (πν^{−1})′. \end{matrix}

[Here, Gr(SO_{1,2}(F)) is the space of virtual representations of SO_{1,2}(F).] The bijection satisfies the following properties. A precise statement appears in Theorem 6.27.

• principal series representations of SO_{1,2}(F) are in bijection with genuine principal series representations of S̃L_{2}(F), • the one dimensional representations of SO_{1,2}(F) are in bijection with the

finite set

{Γ_{−}^{ψ} : ψ ∈ F̂},

• a special representation of SO_{1,2}(F) corresponds with the sum of an odd oscillator representation and a special representation of S̃L_{2}(F),

• an irreducible supercuspidal representation of SO_{1,2}(F) maps to the sum of two supercuspidal representations of S̃L_{2}(F).

Connection to Work of Waldspurger

Let σ̃ be a discrete series representation of S̃L_{2}(F) and let ψ be an additive character of F. In [41], Waldspurger has defined an involution σ̃ ↦ σ̃^{W} as follows:

σ̃ ⟶^{Θ(ψ)} π ⟶^{JL} π′ ⟶^{Θ′(ψ^{−1})} σ̃^{W} .

In the above diagram, π is an irreducible admissible representation of P GL_{2}(F) and JL denotes the Jacquet–Langlands correspondence (see [12]). The maps Θ(ψ), Θ′(ψ) are Waldspurger’s correspondences (see [41]).

We have essentially defined a stable “packet”

L_{st}(π, ν_{0}) = {L(π, ν_{0}), L(πβ, ν_{0}β_{0})}.

On the other hand, Waldspurger has proved the following theorem:

Theorem 1.5 ([41],[11]). Let σ̃ be a discrete series representation of S̃L_{2}(F) and suppose ψ is a nontrivial additive character of F such that σ̃ has a ψ– Whittaker model (see [10]). Then:

1. There is an irreducible admissible representation π of P GL_{2}(F) such that Θ(ψ)^{−1}(π) = σ̃.

2. The “near equivalence set” NE(σ̃) defined by

NE(σ̃) = {Θ(ψ_{ξ})^{−1}(π ⊗ q_{ξ}) : ξ ∈ F^{×}/F^{×2}}

has exactly two elements. (Here q_{ξ} is the quadratic character associated to the extension F(\sqrt{ξ}) of F.)

3. Suppose that

NE(σ̃) = {Θ(ψ)^{−1}(π), Θ(ψ_{ξ})^{−1}(π ⊗ q_{ξ})}.

Then

NE(σ̃) = {σ̃, σ̃^{W} }.

Our “packets” are the same as the “near equivalence sets” of Waldspurger. In particular, we claim that if σ̃ = L(π, ν_{0}) is a discrete series representation, then

{L(π, ν_{0}), L(πβ, ν_{0}β_{0})} = {σ̃, σ̃^{W} }.

Thus our stable packets are the character–theory analogue of near equivalence.

Preliminaries

In this chapter we will explain various constructions and techniques that we will assume and use throughout.

In Section 1, we state the definition and properties of the quadratic Hilbert symbol. In Section 2, we explain how to embed the nonzero elements of a quadratic extension in GL_{2}(F) and give information about squares in quadratic extensions. We define the group SO_{1,2}(F) in Section 3. The basic theory of covering groups is given in Section 4 and the groups S̃L_{2}(F), G̃L_{2}(F) and G̃L_{2}(F)_{+} are defined. Section 5 is concerned with the definitions and properties of the gamma factors γ_{F}(x, ψ) introduced by Weil. We summarize the basic representation theory we will need, including a statement of the existence of global characters, in Section 6. Finally, in Section 7, we elucidate Flicker’s correspondence in the special case n = 2. This is used heavily in the derivation of our character formulas. Flicker’s character formula is given as it appears in Flicker–Kazhdan [7]. We have also included Flicker’s character formulas in the cases of principal series representations and the oscillator representation.

2.1 The Hilbert Symbol

Let F be a local field. We recall briefly the definition and properties of the Hilbert symbol attached to F. An elegant account of this material is found in Serre [32] and [33]. Definition 2.1. Let a, b be elements of F. We define

(a, b)_{F} = \begin{cases}1 & if ax^{2} + by^{2} = z^{2} has a nonzero solution (x, y, z) ∈ F^{3} \\ −1 & otherwise. \end{cases}

The number (a, b)_{F} is called the Hilbert symbol of a and b relative to F. Proposition 2.2. The following properties of the Hilbert symbol are true for all a and b in F^{×}.

1. (a, b)_{F} = (b, a)_{F},

2. (a, b^{2})_{F} = 1,

3. (a, −a)_{F} = 1,

4. (a, a)_{F} = (−1, a)_{F},

5. (a, 1 − a)_{F} = 1,

6. (a, b)_{F} = (a + b, −ab)_{F}. Proposition 2.3. The Hilbert symbol is a nondegenerate, symmetric bilinear form on the F_{2}–vector space F^{×}/F^{×2} × F^{×}/F^{×2}. (Here F_{2} = {0, 1}.) In particular, the following properties hold.

1. If (a, b)_{F} = 1 for all a ∈ F^{×}/F^{×2}, then b is a square.

2. We have that (aa′, b)_{F} = (a, b)_{F}(a′, b)_{F} for all a, a′, b ∈ F^{×}/F^{×2}. Proof. See Serre [32].

2.2 Quadratic Extensions of F

Let E be the quadratic extension F(δ) of F with δ^{2} = Δ ∈ F. We fix the ordered basis {1, δ} for E as a vector space over F. If z = x + yδ ∈ E, the norm of z relative to this extension, written N_{E/F}(z) = N(z), is defined to be N(z) = x^{2} − y^{2}Δ. The norm is a multiplicative map, i.e., N(zw) = N(z)N(w). If z = x + yδ ∈ E, we write \overline{z} = x − yδ for the conjugate of z in E. Note that N(z) = z\overline{z}. We have that (a, b)_{F} = 1 if and only if a is a norm in F(\sqrt{b}) and that N(E^{×}) has index two in F^{×}. Let E^{1} denote the group of elements of norm one in E. Let φ(z) = z/\overline{z}. Hilbert’s Theorem 90 (see Jacobson, [17]) states that z ∈ E^{1} if and only if there exists w ∈ E^{×} such that z = φ(w) = w/\overline{w}.

Let z = x + yδ ∈ E and a ∈ F^{×}. The matrix of left multiplication of z on E in the ordered basis {1, aδ} is

(\begin{matrix}x & yΔa \\ y/a & x \end{matrix}) .

This gives an embedding ι_{a} : E^{×} ↪ GL_{2}(F). We write ι_{a}(E^{×}) = T_{a,Δ}, ι = ι_{1} and T_{1,Δ} = T_{Δ}. Any embedding of E^{×} in GL_{2}(F) is conjugate to ι. Via ι, the determinant corresponds to the norm, so that N = det ∘ι. In other words,

N(x + y\sqrt{Δ}) = det (\begin{matrix}x & yΔ \\ y & x \end{matrix}) .

In a similar way, we obtain an embedding ι_{a}(E^{1}) ↪ SL_{2}(F) and abuse notation by writing ι_{a}(E^{1}) = T_{a,Δ} when the context is clear. Any such embedding extends to an embedding ι of E^{×} in GL_{2}(F).

We use the following proposition repeatedly when we apply the inversion formula (see Theorem 3.12 below). Proposition 2.4. Suppose z ∈ E^{1}. Then there are exactly two values of ξ in F^{×}/F^{×2} such that ξz ∈ E^{×2}. Proof. We claim that if ξ ∈ E^{×2} ∩ F^{×}, then ξ ∈ F^{×2} ∪ ΔF^{×2}. Indeed, write ξ = (a + bδ)^{2}, so that ξ = (a^{2} + b^{2}Δ) + 2abδ. Now ξ ∈ F^{×} implies that 2ab = 0, i.e., a = 0 or b = 0, so that ξ = b^{2}Δ or a^{2}, respectively.

First assume that z ∈ E^{1} ∩ E^{×2}. If ξz ∈ E^{×2} it follows that ξ ∈ E^{×2}. Modulo squares in F^{×}, there are exactly two values of ξ, namely 1 and Δ.

Now assume that z ∉ E^{×2}. By Hilbert’s Theorem 90 there exists w ∈ E^{×} such that z = w/\overline{w}. Multiplying by w/w gives

z = \frac{1}{Nw}w^{2}.

Putting ξ_{0} = Nw we have that ξ_{0}z = w^{2} ∈ E^{×2}. But ξz = ξξ_{0}^{−1}w^{2} ∈ E^{×2} implies that ξξ_{0}^{−1} is a square in E^{×}. Modulo squares in F^{×}, we have that ξξ_{0}^{−1} = 1 or Δ and there are exactly two values of ξ, namely ξ_{0} and ξ_{0}Δ.

2.3 SO_{1,2}(F)

Let F be a nonarchimedean local field, let M_{2}(F) be the vector space of all two–by–two matrices with entries in F, and let V be the subspace of M_{2}(F) of trace–zero matrices. So

V = \{(\begin{matrix}a & b \\ c & −a \end{matrix}) : a, b, c ∈ F\} .

Let

A = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ,  (2.5)

and define σ : M_{2}(F) → M_{2}(F) by

σ(x) = AxA^{−1}.  (2.6)

Note that if x ∈ GL_{2}(F),

(σ(x))^{t} = (det x)x^{−1}.

This implies that for g invertible,

g^{t}σ(g) = det(g)I.

In addition σ has the property

σ(xy) = σ(x)σ(y) for all x, y ∈ M_{2}(F).

Now define a form B(·, ·) on V by

B(x, y) = trace(σ(x)y^{t}).  (2.7)

We have a map p : GL_{2}(F) → GL(V) defined by

p(g)x = gxg^{−1}.

This satisfies

B(p(g)x, p(g)y) = trace(σ(gxg^{−1})(gyg^{−1})^{t})
= trace(σ(g)σ(x)σ(g^{−1})(g^{−1})^{t}y^{t}g^{t})
= trace(g^{t}σ(g)σ(x)σ(g^{−1})(g^{−1})^{t}y^{t})
= trace(det(g)σ(x) det(g^{−1})y^{t})
= trace(σ(x)y^{t})
= B(x, y).

So p(g) ∈ O(V).

A basis of V consists of the elements

X = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  Y = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) ,  and  H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

We compute

\begin{aligned}B(X, X) & = 0 \\ B(X, Y) & = −1 \\ B(X, H) & = 0 \\ B(Y, Y) & = 0 \\ B(Y, H) & = 0 \\ B(H, H) & = −2. \end{aligned}

Therefore the matrix of B(·, ·) with respect to the ordered basis {X, Y, H} is

[B] = (\begin{matrix}& −1 \\ −1 \\ & & −2 \end{matrix}) .

If g = (\begin{matrix}a & b \\ c & d \end{matrix}) then

p(g)X = \frac{1}{det g} [(a^{2})X + (−ac)H + (−c^{2})Y]
p(g)Y = \frac{1}{det g} [(−b^{2})X + (bd)H + (d^{2})Y]
p(g)H = \frac{1}{det g} [(−2ab)X + (ad + bc)H + (2cd)Y].

Conseqently, with respect to the ordered basis {X, Y, H}, the map p is

(\begin{matrix}a & b \\ c & d \end{matrix}) ↦ \frac{1}{ad − bc} (\begin{matrix}a^{2} & −b^{2} & −2ab \\ −c^{2} & d^{2} & 2cd \\ −ac & bd & ad + bc \end{matrix}) .

In particular, the map p satisfies

diag(x, y) ↦ diag(x/y, y/x, 1).  (2.8)

Also note that

[diag(x, 1/x, 1)]^{t}[B][diag(x, 1/x, 1)] = [B]

so that diag(x, 1/x, 1) ∈ SO(V).

Now let E be the quadratic extension F(δ) of F with δ^{2} = Δ ∈ F. Let ι be the corresponding embedding of E^{×} in GL_{2}(F). Another basis of V consists of the elements

H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ,  X′ = (\begin{matrix}0 & −Δ \\ 1 & 0 \end{matrix}) ,  and  Y′ = (\begin{matrix}0 & Δ \\ 1 & 0 \end{matrix}) .

We compute

\begin{aligned}B(X′, X′) & = 2Δ \\ B(X′, Y′) & = 0 \\ B(X′, H) & = 0 \\ B(Y′, Y′) & = −2Δ \\ B(Y′, H) & = 0 \\ B(H, H) & = −2. \end{aligned}

Therefore the matrix of B(·, ·) with respect to the ordered basis {H, X′, Y′} is

[B′] = diag(−2, 2Δ, −2Δ).  (2.9)

Letting g = (\begin{matrix}a & bΔ \\ b & a \end{matrix}) we have

p(g)H = \frac{1}{det g} [(a^{2} + b^{2}Δ)H + (2ab)X′]
p(g)X′ = \frac{1}{det g} [(2abΔ)H + (a^{2} + b^{2}Δ)X′]
p(g)Y′ = Y′.

(2.10)

We have shown that with respect to the ordered basis {H, X′, Y′}, if g ∈ ι(E^{×}) then

g ↦ (\begin{matrix}g^{2}/(det g) \\ & 1 \end{matrix}) .

Also note that

[diag(g, 1)]^{t}[B′][diag(g, 1)] = [B′].

In particular diag(g, 1) is conjugate to an element of O(V). So if det g = 1, diag(g, 1) is conjugate to a matrix in SO(V). Definition 2.11. Let g ∈ GL_{2}(F). We let τ(g) ⊆ SO_{1,2}(F) be the unique conjugacy class of elements having the same nontrivial eigenvalues as those of g. Proposition 2.12. If g ∈ SL_{2}(F) then p(g) ∈ τ(g^{2}). Remark. Fix a Cartan subgroup T of SL_{2}(F). Then with respect to an appropriate basis of V,

τ(g) = (\begin{matrix}g \\ & 1 \end{matrix}) , g ∈ T.

This follows from the preceding computations.

In what follows, we do not distinguish between the conjugacy class τ(g) and elements of the conjugacy class. Thus if π is a representation with character Θ_{π}, we write Θ_{π}(τ(g)) for Θ_{π}(h) where h is any element of τ(g). Our main formulas, given in Chapter 5, relate characters of S̃L_{2}(F) on an element (g; ε) in S̃L_{2}(F) to a character of SO_{1,2}(F) on any element of τ(g). Proposition 2.13. Let E = F(\sqrt{Δ}). Let ι be an embedding of E^{×} in GL_{2}(F) and φ(z) = z/\overline{z}. The map defined by ι′(z) = τ(ι(z)), z ∈ E^{1}, has image in SO(V). Moreover, the diagram

E^{×} ⟶^{φ} E^{1}
ι↓ ↓_{ι′}
GL(2) ⟶^{p} SO(V)

commutes.

Proof. The first part follows from Proposition 2.12. By (2.10), if z = x+yδ ∈ E^{×} then

pι(z) = (\begin{matrix}ι(z^{2})/Nz \\ & 1 \end{matrix})

and

ι′φ(z) = ι′(z/\overline{z}) = ι′(z^{2}/Nz) = (\begin{matrix}ι(z^{2})/Nz \\ & 1 \end{matrix})

as desired.

2.4 Covering Groups

Let G be a locally compact topological group. We first explain how to define n–fold covers of G. See Moore, [25] and [26] for more details. Let μ_{n} denote the group of n–th roots of unity in F. The group G acts trivially on μ_{n}. Definition 2.14. A Borel 2–cocycle c is a Borel measurable map

c : G × G → μ_{n} which satisfies

c(g_{1}g_{2}, g_{3})c(g_{1}, g_{2}) = c(g_{1}, g_{2}g_{3})c(g_{2}, g_{3})

and

c(1, g) = c(g, 1) = 1

for all g, g_{i} in G. The cocycle c is said to be trivial if there is a map s : G → μ_{n} which satisfies c(g, g′)s(gg′) = s(g)s(g′). Otherwise the cocycle is said to be nontrivial. Definition 2.15. Let c be a Borel 2–cocycle. The n–fold covering group G̃ of G with associated cocycle c is defined to be

G̃ = {(g; ε) : g ∈ G, ε ∈ μ_{n}}

with group law

(g; ε)(g′; ε′) = (gg′; εε′c(g, g′)).

The projection map pr : G̃ → G is pr(g; ε) = g. The group G̃ is a group extension of G in the sense that the sequence

1 ⟶ μ_{n} ⟶ G̃ ⟶^{pr} G ⟶ 1

is exact. This extension is central because μ_{n} is contained in the center of G via the map ε → (1; ε). Definition 2.16. Suppose that G is abelian. Let g, h be elements of G and g̃ ∈ pr^{−1}(g), h̃ ∈ pr^{−1}(h). We define the commutator of g and h, written {g, h}, to be

{g, h} = g̃h̃g̃^{−1}h̃^{−1}.  (2.17)

It is clear that {g, h} is independent of the choices of g̃ and h̃.

We now define the covering groups that we will use. We will construct two– fold covers of SL_{2}(F) and GL_{2}(F) by explicitly specifying the cocycle. We use the construction given in Gelbart [9], chapter 2.

If g = (\begin{matrix}a & b \\ c & d \end{matrix}) is any 2 × 2 matrix with entries in F, define a map x by

x(g) = \begin{cases}c & if c ≠ 0 \\ d & otherwise. \end{cases}  (2.18)

Assume that g_{1}, g_{2} ∈ SL_{2}(F). Define a map c_{α} by

c_{α}(g_{1}, g_{2}) = (x(g_{1}), x(g_{2}))_{F}(−x(g_{1})x(g_{2}), x(g_{1}g_{2}))_{F}.  (2.19)

The following proposition is proved by Kubota [22]. Proposition 2.20. The map c_{α} given in (2.19) is a nontrivial Borel 2–cocycle on SL_{2}(F).

Thus we have defined a two–fold covering group S̃L_{2}(F) of SL_{2}(F) associated to the cocycle c_{α}. All nontrivial two–fold covers of SL_{2}(F) are isomorphic to S̃L_{2}(F) (see Gelbart [9], pp. 14–15 for the proof).

We extend c_{α} to GL_{2}(F), and, following Gelbart [9] define similarly a covering group G̃L_{2}(F) of GL_{2}(F). This will describe a two–fold cover of GL_{2}(F) which is a semidirect product of S̃L_{2}(F) and F^{×}.

Let g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(F). Put

q(g) = (\begin{matrix}a & b \\ c/ det g & d/ det g \end{matrix}) .  (2.21)

We have that q(g) ∈ SL_{2}(F). In addition, if y ∈ F^{×}, define

g^{y} = [diag(1, y)]^{−1}g[diag(1, y)]  (2.22)

and

v(y, g) = \begin{cases}1 & if c ≠ 0 \\ (y, d)_{F} & otherwise. \end{cases}  (2.23)

A Borel cocycle c_{β} on GL_{2}(F) is

c_{β}(g_{1}, g_{2}) = c_{α}(q(g_{1})^{det g_{2}}, q(g_{2}))v(det(g_{2}), q(g_{1})).  (2.24)

Note that c_{β} restricted to SL_{2}(F) is equal to c_{α}.

Kazhdan and Patterson [19] have a convenient simplification of these formulas. In the same notation as above, the cocycle c_{β} on GL_{2}(F) may be written

c_{β}(g_{1}, g_{2}) = (\frac{x(g_{1}g_{2})}{x(g_{1})}, \frac{x(g_{1}g_{2})}{x(g_{2})})_{F} (det(g_{1}), \frac{x(g_{1}g_{2})}{x(g_{1})})_{F} .  (2.25)

Restricting this to SL_{2}(F) we get the simple formula

c_{α}(g_{1}, g_{2}) = (\frac{x(g_{1}g_{2})}{x(g_{1})}, \frac{x(g_{1}g_{2})}{x(g_{2})})_{F} .  (2.26)

The following proposition is mentioned in Gelbart [9] and is proved in Kubota [23]. Proposition 2.27. Let y ∈ F^{×}. The map

(g; ε) ↦ (g^{y}; εv(y, g))

is an automorphism of S̃L_{2}(F). It therefore determines a semidirect product of S̃L_{2}(F) and F^{×}. This semidirect product is isomorphic to the covering group G̃L_{2}(F) defined by (2.24).

Instead of c_{β}, it will be necessary to use a more convenient, but equivalent, cocycle c in defining G̃L_{2}(F). Define s : GL_{2}(F) → {±1} by

s(g) = \begin{cases}(c, d/ det g)_{F} & if cd ≠ 0 and ord(c) is odd, \\ 1 & otherwise. \end{cases}  (2.28)

Then define c by

c(g_{1}, g_{2}) = c_{β}(g_{1}, g_{2})s(g_{1})s(g_{2})s(g_{1}g_{2})^{−1}.

This is the cocycle we will use to do all of our calculations. (Recall that if x ∈ F and ϖ is a uniformizer then x = ϖ^{ord(x)}u for some unit u.) Definition 2.29. If g ∈ GL_{2}(F) then g^{y} is always given by (2.22). If (g; ε) ∈ G̃L_{2}(F), then (g; ε)^{y} always denotes the element

(g; ε)^{y} = (diag(1, y); 1)^{−1} (g; ε) (diag(1, y); 1) .  (2.30)

Definition 2.31. Let G̃L_{2}(F)_{+} be the subgroup of G̃L_{2}(F) defined by

G̃L_{2}(F)_{+} = {(g; ε) : det g ∈ F^{×2}}.  (2.32)

If G is any group, we write Z(G) for the center of G. The following proposition computes the centers of the various covering groups we have mentioned thus far. Let I denote the 2 × 2 identity matrix. Proposition 2.33.

\begin{aligned}Z(S̃L_{2}(F)) & = \{(±I; ±1)\} \\ Z(G̃L_{2}(F)) & = \{(y^{2}I; ±1) : y ∈ F^{×}\} \\ Z(G̃L_{2}(F)_{+}) & = \{(yI; ±1) : y ∈ F^{×}\}. \end{aligned}

Proof. Write G̃ for any of the three groups S̃L_{2}(F), G̃L_{2}(F), G̃L_{2}(F)_{+} and G for pr(G̃). Clearly, if z̃ = (z; ε) ∈ Z(G̃), then z ∈ Z(G). So we may assume that z = yI.

If g ∈ G, then

c(z, g) = (y, x(g))_{F}s(z)s(g)s(zg)^{−1},

while

c(g, z) = (y, x(g))_{F} · (y, det g)_{F}s(g)s(z)s(gz)^{−1}.

By definition, s(z) = s(yI) = 1 so that

s(z)s(g)s(zg)^{−1} = s(g)s(z)s(zg)^{−1} = s(g)s(z)s(gz)^{−1}.

Therefore, the commutator {z, g} equals

z̃g̃z̃^{−1}g̃^{−1} = (y, det g)_{F}.  (2.34)

Hence z̃ ∈ Z(G̃) if and only if (y, det g)_{F} = 1 for all g ∈ G.

If G̃ = G̃L_{2}(F), then (y, det g)_{F} = 1 for all g ∈ G implies that y is a square, by the nondegeneracy of the Hilbert symbol. Therefore,

Z(G̃L_{2}(F)) = {(y^{2}I; ±1) : y ∈ F^{×}}.

If G̃ = G̃L_{2}(F)_{+} then det g is a square for all g ∈ G. The commutator is 1 for any y. Therefore,

Z(G̃L_{2}(F)_{+}) = {(yI; ±1) : y ∈ F^{×}}.

Finally, if G̃ = S̃L_{2}(F), then det g = 1 for all g ∈ G. Thus the commutator {z, g} = 1 for all g. The element z̃ = (z, ε) will be in the center only if det z = y^{2} = 1, i.e. y = ±1. This shows that Z(S̃L_{2}(F)) = {(±I; ±1)}.

We end this section by doing some calculations which will simplify the formulas in Chapter 4.

Lemma 2.35. If a, b, c, and d are in F^{×}, then

c(diag(a, b), diag(c, d)) = (a, d)_{F}.  (2.36)

Proof. By (2.25), we have

c_{β}(diag(a, b), diag(c, d)) = (d, b)_{F}(ab, d)_{F}
= (ab^{2}, d)_{F}
= (a, d)_{F}.

Since s is trivial on diagonal matrices, the result follows. Lemma 2.37. For ξ ∈ F^{×}, g ∈ GL_{2}(F),

(−1, ξ)_{F}s(ξg)c(ξI, g) = s(g)c_{β}(ξI, −g).  (2.38)

Proof. We have that

\begin{aligned}(−1, ξ)_{F}s(ξg)c(ξI, g) & = (−1, ξ)_{F}s(ξg)c_{β}(ξI, g)s(ξI)s(g)s(ξg)^{−1} \\ & = s(g)(−1, ξ)_{F}c_{β}(ξI, g) \end{aligned}

since s is trivial on the diagonal matrix ξI. But

(−1, ξ)_{F}c_{β}(ξI, g) = (−1, ξ)_{F}(x(g), ξ)_{F}
= (ξ, −x(g))_{F}
= c_{β}(ξI, −g)

so the result follows.

2.5 Weil’s Gamma Factors

Let ψ be a nontrivial additive character of F. If a ∈ F^{×}, the character ψ_{a} of F is given by

ψ_{a}(x) = ψ(ax).  (2.39)

Definition 2.40. Let γ_{F}(ψ) be the Weil index of the map x ↦ ψ(x^{2}) (see [42]). Define γ_{F}(x, ψ) by

γ_{F}(x, ψ) = \frac{γ_{F}(ψ_{x})}{γ_{F}(ψ)}.  (2.41)

From this definition, it immediately follows that γ_{F}(1, ψ) = 1.

The next proposition gives the basic properties of the gamma–factors just defined. This is stated in Rao, [31]. Proposition 2.42. The following formulas are valid for all x and y in F^{×}.

1. γ_{F}(xy, ψ) = γ_{F}(x, ψ)γ_{F}(y, ψ)(x, y)_{F}.

2. γ_{F}(x^{2}y, ψ) = γ_{F}(y, ψ). Hence γ_{F}(x^{2}, ψ) = 1 and γ_{F}(x^{−1}, ψ) = γ_{F}(x, ψ).

3. γ_{F}(x, ψ_{ξ}) = (x, ξ)_{F}γ_{F}(x, ψ). Hence γ_{F}(x, ψ_{ξ^{2}}) = γ_{F}(x, ψ).

4. {γ_{F}(x, ψ)}^{2} = (−1, x)_{F} = (x, x)_{F}. Hence γ_{F}(x, ψ)^{−1} = γ_{F}(x, ψ)(−1, x)_{F} and {γ_{F}(x, ψ)}^{4} = 1.

Proof. The formulas (1) and (2) are proved in Weil [42]. The other formulas follow from this. To prove (3), note that γ_{F}(x, ψ_{ξ}) = γ_{F}(ψ_{xξ})/γ_{F}(ψ_{ξ}). Thus

γ_{F}(x, ψ_{ξ})γ_{F}(ξ, ψ) = (γ_{F}(ψ_{xξ})/γ_{F}(ψ_{ξ})) (γ_{F}(ψ_{ξ})/γ_{F}(ψ))
= γ_{F}(xξ, ψ)
= γ_{F}(x, ψ)γ_{F}(ξ, ψ)(x, ξ)_{F}.

Cancelling γ_{F}(ξ, ψ) gives (3). To prove (4), we use (1) and (2) to write

γ_{F}(x, ψ)^{2} = γ_{F}(x^{2}, ψ)(x, x)_{F} = (x, x)_{F}.

2.6 Representation Theory

We state some basic definitions and theorems in the theory of representations of p–adic groups which we will need later. The basic references for this section are Silberger [36] and Gelbart [8]. Throughout this section, G is a locally compact p–adic group. Definition 2.43. Let Z be the center of G. If there is a character χ such that π(z)v = χ(z)v for all z ∈ Z then we call π a χ–representation and say that χ is the central character of π.

We remark that by Schur’s Lemma, if π is irreducible then π has a central character. In general, if π has a central character then we denote it by χ_{π}. Definition 2.44. Let n be a positive integer. Let F be a field which contains μ_{n}, the n–th roots of unity. Then a character χ of F^{×} is said to be n–even if χ(μ_{n}) = {1}. If n = 2 then we drop n from the notation and say that χ is even. If n = 2 and χ is not even then we say that χ is odd. Definition 2.45. A χ–representation of G is called n–even if χ is an n–even character. If n = 2, we drop n from the notation and say simply that the representation is even. Definition 2.46. Let H be a subgroup of G and let π be a representation of G on a complex vector space V. We write π|_{H} for the restriction of the homomorphism π of G to H. Definition 2.47. Let G be a group and H be a closed subgroup. Let δ_{H} denote the modular function for H. Let σ be a representation of H on a complex vector space V. Consider the space F (G, V) of all locally constant functions f : G → V satsifying f(hg) = δ_{H}(h)^{1/2}σ(h)f(g) for all h ∈ H, g ∈ G. The representation π of G on the space F (G, V) is called the representation induced by σ on G if

π(g)(f)(x) = f(xg)

for all x, g ∈ G. We write

π = Ind_{H}^{G}(σ).

Definition 2.48. Let G be a group and G̃ be a two–fold cover of G. Let π̃ be an irreducible representation of G̃ on a complex vector space V. Then π̃ is called genuine if π̃ does not factor to G. Definition 2.49. Let G be a group and H a closed subgroup of G. Let π be a representation of H on a complex vector space V. For any x ∈ G, the representation π^{x} is the representation of x^{−1}Hx on V given by

π^{x}(x^{−1}hx) = π(h), h ∈ H.

Definition 2.50. Let V be a complex vector space. A representation π : G → Aut(V) is said to be admissible if

1. The stabilizer in G of each v in V is an open subgroup of G, and

2. For every compact open subgroup K of G,

V^{K} = {v ∈ V : π(k)v = v for all k ∈ K}

is finite–dimensional.

Definition 2.51 (Principal series, GL_{2}(F)). Let G = GL_{2}(F). Let B be the subgroup

B = \{(\begin{matrix}x & z \\ 0 & y \end{matrix}) : x, y ∈ F^{×}, z ∈ F\} .

Let η_{1} and η_{2} be two characters of F^{×}. Let σ be the character of B defined by

σ (\begin{matrix}x & z \\ 0 & y \end{matrix}) = η_{1}(x)η_{2}(y).

Define the principal series representation π(η_{1}, η_{2}) of GL_{2}(F) to be

π(η_{1}, η_{2}) = Ind_{B}^{G}σ.

Proposition 2.52. A principal series π(η_{1}, η_{2}) is irreducible except when η(x) = η_{1}η_{2}^{−1}(x) = |x| or |x|^{−1}. If η(x) = |x|^{−1} then π(η_{1}, η_{2}) contains a one–dimensional invariant subspace and the representation induced on the resulting factor space is irreducible. If η(x) = |x|, then π(η_{1}, η_{2}) contains an irreducible invariant subspace of codimension one. Moreover, π(η_{1}, η_{2}) and π(λ_{1}, λ_{2}) are equivalent if and only if (η_{1}, η_{2}) = (λ_{1}, λ_{2}) or (λ_{2}, λ_{1}).

Proof. See Gelbart [8]. Definition 2.53 (Special representation). If π(η_{1}, η_{2}) is reducible, the resulting infinite dimensional subquotients of π(η_{1}, η_{2}) are called special representations and are denoted σ(η_{1}, η_{2}). Definition 2.54. We call a principal series strongly even if both μ_{1} and μ_{2} are even characters. Definition 2.55 (Supercuspidal representation). Suppose that π is an admissible representation of G = GL_{2}(F) on V. Let N be the subgroup

N = \{(\begin{matrix}1 & z \\ 0 & 1 \end{matrix}) : z ∈ F\}

of G. Then π is supercuspidal if for all v ∈ V,

∫_{U} π(n)v dn = 0

for some compact open subgroup U of N. Proposition 2.56. Suppose that π is an irreducible admissible representation of GL_{2}(F). If π is not supercuspidal, then it is a subrepresentation of a principal series representation. In particular, if it is infinite dimensional then it is an irreducible principal series or a special representation. Conversely, if π is supercuspidal, then it is not equivalent to a subquotient of any principal series representation.

Proof. See Gelbart [8].

We now give a brief explanation of principal series representations of G̃L_{2}(F). Let A = {diag(x, 1/x) : x ∈ F^{×}}. Hence Ã ≃ B̃/N. Let B_{0} denote the subgroup of B whose diagonal entries have even v–adic order. (Recall that if x ∈ F and ϖ is a uniformizer then x = ϖ^{ord_{v}(x)}u for some unit u.) Then Ã_{0} = B̃_{0}/N ≃ A_{0}×Z/2Z is a maximal abelian subgroup of finite index in Ã. Moreover, Ã^{2} ⊆ Ã_{0}. Let η̃ = (η_{1}, η_{2}) be a genuine one–dimensional representation of Ã^{2} and extend this to Ã_{0} arbitrarily. Then extend η̃ to B̃_{0} by requiring that η̃ be trivial on N. Any genuine representation of G̃L_{2}(F) which is not supercuspidal is a subquotient of an induced representation IndG̃L_{2}(F) ; B̃_{0}(τ̃) where τ̃ = IndÃ ; Ã_{0} η̃. (For the proof, see Gelbart [9].) We write π̃(η_{1}, η_{2}) = IndG̃L_{2}(F) ; B̃_{0}(τ̃). Flicker [6] points out that this depends only on the restriction of the pair (η_{1}, η_{2}) to Ã^{2}.

We end this section by recalling the theorem on the existence of a global character of an irreducible representation of a group G. Definition 2.57. Let G be a locally compact group and Z be the center of G. Let π be an irreducible admissible representation of G with central character χ. Let f be any function satisfying the property f(zg) = χ^{−1}(z)f(g) for all z ∈ Z, g ∈ G. Then define

π(f) = ∫_{G/Z} π(g)f(g) dg.

Here dg is a Haar measure on G/Z.

Recall that the regular set (see Harish–Chandra [13]) is open and dense in G and its complement in G has measure zero.

The following theorem states that every admissible representation of a group G has a character defined on the regular set. The theorem holds for all of the groups that we will consider. It holds in particular for G any reductive algebraic group; it also holds for both of the covering groups S̃L_{2}(F) and G̃L_{2}(F). The proof is due to Harish–Chandra [14]. See also Flicker [6], p. 140 and Flicker– Kazhdan [7], p. 68. Theorem 2.58. The operator π(f) is of finite rank. Let trace(π(f)) denote the trace of this operator. Then there is conjugation–invariant function Θ_{π}, defined on the regular set of G, which satisfies

trace(π(f)) = ∫_{G/Z} f(g)Θ_{π}(g) dg.

The function Θ_{π} is called the character of π. Equivalent irreducible representations have the same character and conversely. If χ is the central character of π, then Θ_{π} transforms by χ on Z.

Note that if π̃ is genuine, then Θπ̃(g; ε) = εΘπ̃(g; 1).

2.7 Flicker’s Correspondence

In this section, let G = GL_{2}(F) and let G̃ be an n–fold cover of G. In [6], Flicker has defined a correspondence between genuine irreducible admissible representations π̃ of G̃ and admissible representations π of G. We use Flicker’s character formula extensively when we derive our character formulas.

Let g be regular with λ_{1} and λ_{2} the distinct eigenvalues of g. The map D is defined on regular g by

D(g) = |\frac{(λ_{1} − λ_{2})^{2}}{λ_{1}λ_{2}} |^{1/2}  (2.59)

and satisfies

D(ζg) = D(g),  ζ ∈ F^{×}.  (2.60)

Flicker and Kazhdan [7], p. 93, have defined a constant b and showed that b = n/(d|n^{r}/d|^{1/2}). In their notation, we have taken r = 2 and c = 0, so that d = gcd(n, r − 1 + 2rc) = 1. Thus

b = n/|n|_{p}.

Kazhdan and Patterson have shown that

|F^{×}/F^{×n}| = n^{2}/|n|_{p}

(see Kazhdan and Patterson [19], Lemma 0.3.2 or Flicker and Kazhdan [7], Lemma 3.24.2). The constant b becomes

b = \frac{|F^{×}/F^{×n}|}{n}.

Flicker’s correspondence for n–fold covers appears in Definition 5.0.1 of [6] and is repeated here. In our statement, we have corrected the character formula defining this correspondence to conform with the correspondence of Flicker and Kazhdan in [7].

Definition 2.61. Suppose π̃ is a genuine irreducible admissible representation of an n–fold cover of GL_{2}(F) with central character χ̃ and that π is an admissible representation of GL_{2}(F) with central character χ. Assume that χ and χ̃ satisfy the relationship

χ̃(x^{n}I; 1) = χ(xI)

for all x ∈ F^{×}. Then π̃ and π correspond via Flicker’s correspondence if whenever g^{n} is regular the following relationship holds:

D(g^{n})Θπ̃(g^{n}; s(g^{n})^{−1}) = \begin{cases}b D(g)Θ_{π}(g), & if g is elliptic \\ b ∑_{ζ∈μ_{n}} D(g_{ζ})Θ_{π}(g_{ζ}), & otherwise. \end{cases}

In the above formula, g_{ζ} = diag(ζx, y) if g is conjugate to diag(x, y). Remark. For g hyperbolic, s(g^{n})^{−1} = 1.

We are primarily interested in the case n = 2, so Flicker’s correspondence is stated in this special case. When we quote Flicker’s corresondence, we always mean the correspondence in this specific case. The constant b is

b = \frac{|F^{×}/F^{×2}|}{2} = 2/|2|_{p}.  (2.62)

This is 2 if p ≠ 2, and a power of 2 if p = 2. Definition 2.63 (Flicker’s Correspondence). Let π̃ be a genuine irreducible admissible representation of G̃L_{2}(F) with central character χ̃ and let π be an admissible representation of GL_{2}(F) with central character χ. Suppose that χ and χ̃ satisfy the relationship

χ̃(x^{2}I; 1) = χ(xI)  (2.64)

for all x ∈ F^{×}. Then we say that π̃ and π correspond via Flicker’s correspondence if whenever g^{2} is regular the following relationship holds:

D(g^{2})Θπ̃(g^{2}; s(g^{2})^{−1}) = \begin{cases}b D(g)Θ_{π}(g), & if g is elliptic, \\ b (D(g)Θ_{π}(g) + D(g′)Θ_{π}(g′)) & if g is hyperbolic. \end{cases}

In the above formula, if g is conjugate to the hyperbolic element diag(x, y) then

g′ = diag(−x, y).  (2.65)

Suppose π and π̃ correspond via Flicker’s correspondence. Then we say that π̃ is Flicker’s lift of π and we write

π̃ = Lift_{F}(π).  (2.66)

Flicker has shown the following theorem about this correspondence (see Theorem 5.2 and Corollary 5.2.1 of Flicker [6]). Flicker’s theorem is stated in the case of n = 2. Theorem 2.67. The correspondence π̃ = Lift_{F}(π) is a bijection between irreducible admissible representations π of GL_{2}(F) satisfying χ_{π}(−I) = 1 and genuine irreducible admissible representations π̃ of G̃L_{2}(F). If π is a supercuspidal representation of GL_{2}(F) with even central character, then there is a supercuspidal representation π̃ of G̃L_{2}(F) such that π̃ = Lift_{F}(π). If π is a special representation σ(μ_{1}, μ_{2}) of GL_{2}(F) with both μ_{i} odd, then Lift_{F}(π) is equivalent to an odd oscillator representation of G̃L_{2}(F). If π is an even one–dimensional representation of GL_{2}(F) then Lift_{F}(π) is an even oscillator representation of G̃L_{2}(F). (See Proposition 2.70 below.) Remark. An odd oscillator representation of G̃L_{2}(F) is supercuspidal. This is an example of a supercuspidal representation occuring in Flicker’s lifting which does not come from a supercuspidal representation of GL_{2}(F). This phenomenon is further illustrated by the following. Let π = π(μ_{1}, μ_{2}) be a reducible principal series of GL_{2}(F) such that μ_{1} and μ_{2} are both odd, i.e., π is even but not strongly even. Then Flicker’s lifting of π is zero by Flicker’s formula for principal series (see Proposition 2.68 below). On the other hand, π is the sum of a special representation and an odd one–dimensional representation. The special representation lifts to an odd oscillator representation and the odd one–dimensional lifts to minus that odd oscillator representation (Proposition 2.70 below). This gives a different way of seeing why the principal series π lifts to 0.

Flicker gives examples of special cases of this correspondence. He computes the correspondence for principal series and for the oscillator representations. We state these in the case n = 2. Flicker uses the full power of the trace formula to get the correspondence on supercuspidal representations. Proposition 2.68. Let η_{1} and η_{2} be even characters of F^{×}. Let μ_{1} and μ_{2} be two characters of F^{×2} which satisfy μ_{i}(x^{2}) = η_{i}(x) for all x and extend these to characters of F^{×} arbitrarily. We have the character relation

Θ_{π̃(μ_{1},μ_{2})}(g^{2}; 1) = \begin{cases}b \frac{D(g)}{D(g^{2})}Θ_{π(η_{1},η_{2})}(g) & if g ∼ diag(x, y) \\ 0 & otherwise. \end{cases}  (2.69)

(In the above statement g ∼ h means that g is conjugate via GL_{2}(F) to h.) Proposition 2.70. Let μ_{0} be a character of F^{×}. Let ω(μ) be the oscillator representation attached to μ on G̃L_{2}(F). Then

Θ_{ω(μ)}(g^{2}; s(g^{2})^{−1}) = b μ_{0}(−1)μ(g)\frac{D(g)}{D(g^{2})}

for g elliptic and Θ_{ω(μ)} (g^{2}; 1) = b [μ_{0}(− det g)D(g) + μ_{0}(det g′)D(g′)]/D(g^{2}) if g = diag(x, y) is hyperbolic [with g′ defined by (2.65)]. Corollary 2.71. If μ is even, then

D(g^{2})Θ_{ω(μ)}(g^{2}; s(g^{2})^{−1}) = \begin{cases}b D(g)μ(g) & if g is elliptic \\ b (D(g)μ(g) + D(g′)μ(g′)) & otherwise \end{cases}

while if μ is odd,

D(g^{2})Θ_{ω(μ)}(g^{2}; s(g^{2})^{−1}) = \begin{cases}−b D(g)μ(g) & if g is elliptic \\ −b (D(g)μ(g) + D(g′)μ(g′)) & otherwise. \end{cases}

= \begin{cases}b D(g)Θ_{σ}(g) & if g is elliptic \\ b (D(g)Θ_{σ}(g) + D(g′)Θ_{σ}(g′)) & otherwise. \end{cases}

Here g′ is defined in (2.65) and σ is a special representation σ(μ| · |^{−1/2}, μ| · |^{1/2}) as in Definition 2.53.

The next proposition shows that characters of genuine admissible representations vanish off of elements of the form (g^{2}; ε). This property, combined with Proposition 2.4 shows that when we apply the inversion formula (see Theorem 3.12 below) to Flicker’s lifting, there will be exactly two nonzero terms which appear in the sum. Proposition 2.72. Let π̃ be a genuine admissible representation of G̃L_{2}(F). Suppose that y ∈ G̃L_{2}(F) is regular and is not of the form (g^{2}; ε). Then Θπ̃(y) = 0.

Proof. By Flicker [6], pp. 127–128, such y have the following property:

there exists h ∈ G̃L_{2}(F) such that h^{−1}yh = y · (1; −1).

Since π̃ is genuine, we have

Θπ̃(y) = Θπ̃(h^{−1}yh) = Θπ̃(y(1; −1)) = −Θπ̃(y).

The commutator can be computed explicitly on an elliptic Cartan T_{Δ}. Proposition 2.73. Choose an embedding ι : E^{×} = F(\sqrt{Δ}) → T_{Δ}. Let g = ι(z) and h = ι(w). Then

\{g, h\} = (z, \overline{w})_{E}.

Proof. See Flicker [6], pp. 127–128 and Blondel [4], p. 14. Remark. In the above proposition, (·, ·)_{E} is the Hilbert symbol of the quadratic extension E = F(\sqrt{Δ}) of F. See Serre [33]. It satisfies the property

(α, β)_{E} = (Nα, β)_{F}, α ∈ E^{×}, β ∈ F^{×}.

Representations of G̃L_{2}(F)_{+}

We would like to derive a formula relating characters of S̃L_{2}(F) and characters of SO_{1,2}(F). The first step is to restrict representations of G̃L_{2}(F) to G̃L_{2}(F)_{+}; this is done in Theorem 3.8. The unexpected result says that the restriction is a sum of representations parameterized by F^{×}/F^{×2}. We then derive an “inversion formula” which computes the character of an individual summand of this restriction if the character of the original representation is known. Finally, we explicitly parameterize summands appearing in the restriction of a representation to G̃L_{2}(F)_{+}.

3.1 Restricting to G̃L_{2}(F)_{+}

We address the problem of what happens when we restrict a genuine irreducible representation π̃ of G̃L_{2}(F) to the subgroup G̃L_{2}(F)_{+}.

Recall that G̃L_{2}(F)_{+} = {(h; ε) : det h ∈ F^{×2}}. Write

V = F^{×}/F^{×2} = Z_{2} × ⋯ × Z_{2} = ⟨α_{1}⟩ × ⋯ × ⟨α_{n}⟩.

The Hilbert symbol (·, ·)_{F} is a nondegenerate bilinear form on this F_{2} vector space (see Proposition 2.3). If W is a subspace of V, the orthogonal complement of W is

W^{⊥} = {v ∈ V : (w, v)_{F} = 1 for all w ∈ W }.  (3.1)

Write

H̃_{k} = G̃L_{2}(F)_{α_{1},... ,α_{k}} = ⟨{(h; ε) : det h ∈ F^{×2} ∪ ⋃_{i=1}^{k} α_{i}F^{×2}}⟩  (3.2)

for k = 0, …, n. Note that H̃_{0} = G̃L_{2}(F)_{+} and H̃_{n} = G̃L_{2}(F). We have the chain of inclusions

H̃_{0} ⊆ H̃_{1} ⊆ ⋯ ⊆ H̃_{k} ⊆ ⋯ ⊆ H̃_{n}

with H̃_{k}/H̃_{k−1} of order 2 for all k = 1, …, n. Proposition 3.3. Let H̃ be a subgroup of G̃L_{2}(F). Then Z(H̃), the center of H̃, equals

Z(H̃) = {(h; ε) : h ∈ Z(H) and (h, det g)_{F} = 1 for all g ∈ H}.  (3.4)

In particular, we have

\begin{aligned}Z(H̃_{k}) & = \{(diag(y, y); ±1) : (α_{i}, y)_{F} = 1 for all i = 1, …, k\} \\ & = \{(diag(y, y); ±1) : y ∈ ⟨α_{1}, …, α_{k}⟩^{⊥}\} \end{aligned}

for all k = 0, …, n.

Proof. The proof of this proposition is very similar to that of Proposition 2.33.

It is clear that if (h; ε) ∈ Z(H̃_{k}) then h = diag(y, y) for some y ∈ F^{×}. The condition (y, det g)_{F} = 1 [see (2.34)] for all g ∈ H_{k} means that (y, α_{i})_{F} = 1 for all i = 1, …, k. This is true if and only if y ∈ ⟨α_{i}⟩^{⊥} for all i = 1, …, k.

Proposition 3.5. For some β ∈ ⟨α_{1}, …, α_{k−1}⟩^{⊥}, we have that (α_{k}, β)_{F} = −1.

Proof. Assume not. This implies that if β ∈ ⟨α_{1}, …, α_{k−1}⟩^{⊥}, then (β, α_{k})_{F} = 1, that is, β ∈ ⟨α_{k}⟩^{⊥}. So ⟨α_{1}, …, α_{k−1}⟩^{⊥} ⊆ ⟨α_{k}⟩^{⊥}. Taking ^{⊥} of both sides, we get that ⟨α_{k}⟩ ⊆ ⟨α_{1}, …, α_{k−1}⟩. This contradicts the fact that {α_{1}, …, α_{k}} is a linearly independent set.

Let I be the 2 × 2 identity matrix. Proposition 3.6. Let H̃ be a subgroup of G̃L_{2}(F) such that S̃L_{2}(F) ⊆ H̃ and Z(H̃) ⊆ {(xI; ε)}. Let σ̃ be a genuine irreducible representation of H̃ with central character χσ̃. Then

χ_{σ^{y}}(zI; ε) = (y, z)_{F} · χσ̃(zI; ε).

Proof. This follows from the computation

(zI; ε)^{y} = (y, z)_{F}(zI; ε)

(see Definition 2.29) and the fact that χσ̃ is genuine. Remark. Recall that Z(H̃_{k−1}) = {(diag(y, y), ±1) : y ∈ ⟨α_{1}, …, α_{k−1}⟩^{⊥}}. Proposition 3.6 says that (diag(1, α_{k}), 1) acts nontrivially on Z(H̃_{k−1}). Proposition 3.7 (Restriction principle). Let H ⊆ G be a subgroup of G having index two. Let y be a representative for the nontrivial coset of G/H. Let ω be the nontrivial character of G/H (so ω(h) = 1, ω(yh) = −1 for h ∈ H). Let π be an irreducible representation of G.

1. Suppose π ⊗ ω ≠ π. Then σ = π|_{H} is irreducible. Furthermore σ^{y} = σ and Ind_{H}^{G}(σ) = π ⊕ (π ⊗ ω).

2. Suppose π ⊗ ω = π. Then π|_{H} = σ_{1} ⊕ σ_{2} is reducible. Here each σ_{i} is irreducible. Furthermore σ_{1} ≠ σ_{1}^{y} ≃ σ_{2} and Ind_{H}^{G}(σ_{1}) = Ind_{H}^{G}(σ_{2}) are irreducible.

Proof. See Bröcker and tom Dieck, [5].

The following theorem achieves the goal of this section. It states that the restriction of a genuine irreducible representation of G̃L_{2}(F) is a sum of representations parameterized by F^{×}/F^{×2}. Theorem 3.8. Let π̃ be a genuine irreducible representation of G̃L_{2}(F). Let σ̃ be an irreducible constituent of π̃|_{G̃L_{2}(F)_{+}}. Then

π̃|_{G̃L_{2}(F)_{+}} = ∑_{α∈F^{×}/F^{×2}} σ̃^{α}.

Hence the irreducible constituents of π̃|_{G̃L_{2}(F)_{+}} are parameterized by α ∈ F^{×}/F^{×2}.

Proof. Fix k ≥ 1. Let ρ̃ be an irreducible representation of H̃_{k}. Let σ̃ be an irreducible constituent of ρ̃H̃_{k−1}. We may assume by Proposition 3.5 (and the remark after Proposition 3.6) that there exists β having the properties (α_{k}, β)_{F} = −1 and (diag(β, β); ε) ∈ Z(H̃_{k−1}). By Proposition 3.6, the central characters χ_{σ^{α_{k}}} and χσ̃ are of opposite sign on (diag(β, β); ε) in Z(H̃_{k−1}). Therefore σ̃^{α_{k}} ≠ σ̃. Since (diag(1, α_{k}); ε) is in H̃_{k} but not in H̃_{k−1}, the restriction principle gives that

ρ̃|H̃_{k−1} = σ̃_{1} ⊕ σ̃_{2}

with each σ̃_{i} irreducible. The proof now follows by restriction in stages.

Remark. A statement of this result appears in Gelbart and Piatetski–Shapiro [11] on p. 101.

3.2 Inversion via the Center

In light of Theorem 3.8, we now consider the problem of finding a formula for the character of the individual constituents σ̃^{α} appearing in the restriction of an irreducible genuine representation π̃ of G̃L_{2}(F) to G̃L_{2}(F)_{+}. We will show that Θ_{σ^{α}} can be written in terms of χ_{σ^{α}} and Θπ̃.

To begin, we let π̃ be a genuine irreducible representation of G̃L_{2}(F). By Theorem 3.8 we may write π̃|_{G̃L_{2}(F)_{+}} = ∑_{α} σ̃^{α}, where each σ̃^{α} is a genuine irreducible representation of G̃L_{2}(F)_{+}. The group F^{×}/F^{×2} acts transitively on the set {σ̃^{α}}.

Fix a set of coset representatives {α} for F^{×}/F^{×2}. For each α we have that

χπ̃(x^{2}I; ε) = χ_{σ^{α}}(x^{2}I; ε).  (3.9)

By Proposition 3.6 we also have

χ_{(σ^{α})^{a}}(xI; ε) = (a, x)_{F}χ_{σ^{α}}(xI; 1)  (3.10)

for every α.

Let z_{α} be any element satisfying

pr(z_{α}) = αI.  (3.11)

Each z_{α} is in the center of G̃L_{2}(F)_{+}. Theorem 3.12 (Inversion Formula). Let π̃ be a genuine, irreducible representation of G̃L_{2}(F) and write

π̃|_{G̃L_{2}(F)_{+}} = ∑_{α∈F^{×}/F^{×2}} σ̃^{α}.

Then for any α ∈ F^{×}/F^{×2},

Θ_{σ^{α}}(g̃) = \frac{1}{|F^{×}/F^{×2}|} ∑_{ξ∈F^{×}/F^{×2}} χ_{σ^{α}}(z_{ξ})^{−1}Θπ̃(z_{ξ}g̃).  (3.13)

Here we abuse notation and write ξ for the coset ξF^{×2} as well as an element of that coset.

Proof. First we show that the inversion formula is well defined. In particular, we claim that the formula is independent of the choice of coset representatives for F^{×}/F^{×2}. Fix ξ and let ξ′ = y^{2}ξ, so that ξF^{×2} = ξ′F^{×2}. By (2.35), (y^{2}ξI; 1) = (y^{2}I; 1)(ξI; 1). It follows that

\begin{aligned}χ_{σ^{α}}(z_{ξ′})^{−1}Θπ̃(z_{ξ′} g̃) & = χ_{σ^{α}}(y^{2}I; 1)^{−1}χ_{σ^{α}}(z_{ξ})^{−1}Θπ̃((y^{2}I; 1)z_{ξ}g̃) \\ & = χ_{σ^{α}}(y^{2}I; 1)^{−1}χ_{σ^{α}}(z_{ξ})^{−1}χπ̃(y^{2}I; 1)Θπ̃(z_{ξ}g̃). \end{aligned}

From (3.9), we see

\begin{aligned}χ_{σ^{α}}(z_{ξ′})^{−1}Θπ̃(z_{ξ′} g̃) & = χ_{σ^{α}}(y^{2}I; 1)^{−1}χ_{σ^{α}}(z_{ξ})^{−1}χ_{σ^{α}}(y^{2}I; 1)Θπ̃(z_{ξ}g̃) \\ & = χ_{σ^{α}}(z_{ξ})^{−1}Θπ̃(z_{ξ}g̃). \end{aligned}

This proves the claim.

Now we derive the inversion formula. Let z be in the center of G̃L_{2}(F)_{+}. Then

Θπ̃(zg̃) = ∑_{α} χ_{σ^{α}}(z)Θ_{σ^{α}}(g̃).

For any constants c_{ξ}, we have

\begin{aligned}∑_{ξ} c_{ξ}Θπ̃(z_{ξ}g̃) & = ∑_{ξ} c_{ξ} ∑_{α} χ_{σ^{α}}(z_{ξ})Θ_{σ^{α}}(g̃) \\ & = ∑_{α} Θ_{σ^{α}}(g̃) ∑_{ξ} c_{ξ}χ_{σ^{α}}(z_{ξ}). \end{aligned}

Fix α_{0} and let c_{ξ} = χ_{σ^{α_{0}}}(z_{ξ})^{−1}. Substitute in to get

∑_{ξ} χ_{σ^{α_{0}}}(z_{ξ})^{−1}Θπ̃(z_{ξ}g̃) = ∑_{α} Θ_{σ^{α}}(g̃) ∑_{ξ} χ_{σ^{α_{0}}}(z_{ξ})^{−1}χ_{σ^{α}}(z_{ξ}).

By transitivity of the action of F^{×}/F^{×2} on the σ̃^{α}’s, we may pick a so that (σ̃^{α_{0}})^{a} = σ̃^{α}. (This a depends on α.) By (3.6),

∑_{ξ} χ_{σ^{α_{0}}}(z_{ξ})^{−1}Θπ̃(z_{ξ}g̃) = ∑_{α} Θ_{σ^{α}}(g̃) ∑_{ξ}(a, ξ)_{F}.

But

∑_{ξ}(a, ξ)_{F} = \begin{cases}|F^{×}/F^{×2}| & if a is a square \\ 0 & otherwise. \end{cases}

Hence

Θ_{σ^{α_{0}}}(g̃) = \frac{1}{|F^{×}/F^{×2}|} ∑_{ξ} χ_{σ^{α_{0}}}(z_{ξ})^{−1}Θπ̃(z_{ξ}g̃).

This completes the proof, since α_{0} was arbitrary.

3.3 Parameterization

Our goal in this section is to parameterize the representations which appear in the restriction to G̃L_{2}(F)_{+} of a Flicker lifting. In order to do this, some preparation and notation is required.

Fix, once and for all, a nontrivial character ψ of F.

A two–fold cover of F^{×} is given by the cocycle c(x, y) = (x, y)_{F}. Proposition 3.14. Let γ_{F}(x, ψ) be Weil’s gamma factor as defined in Definition 2.40. The map

λ̃(x; ε) = γ_{F}(x, ψ)ε

is a genuine character of the two–fold cover of F^{×}.

Proof. We have

λ̃((x; ε)(x′; ε′)) = λ̃(xx′; εε′(x, x′)_{F})
= γ_{F}(xx′, ψ)εε′(x, x′)_{F}
= γ_{F}(x, ψ)γ_{F}(x′, ψ)(x, x′)_{F}εε′(x, x′)_{F}
= γ_{F}(x, ψ)εγ_{F}(x′, ψ)ε′
= λ̃(x; ε)λ̃(x′; ε′);

this establishes the proposition. Definition 3.15. Suppose that ν_{0} is any character of F^{×}. Define a character of Z(G̃L_{2}(F)_{+}) by the formula

χ_{ν_{0}}(xI; ε) = ν_{0}(x)γ_{F}(x, ψ)ε.  (3.16)

It is clear that χ_{ν_{0}} is a genuine character of Z(G̃L_{2}(F)_{+}). Since the quotient of two genuine characters is not genuine, i.e., a character of F^{×}, it is also obvious that each genuine character of Z(G̃L_{2}(F)_{+}) is of the form χ_{ν_{0}} for some character ν_{0} of F^{×}.

Now let π be an irreducible admissible representation of GL_{2}(F) and let π̃ = Lift_{F}(π). Recall that the central characters of π̃ and π are related by (2.64):

χπ̃(x^{2}I; 1) = χ_{π}(xI).

In particular, χ_{π}(−I) = 1. The central character of any representation appearing in the restriction of π̃ to G̃L_{2}(F)_{+} also satisfies (2.64), see (3.9). Definition 3.17. Let π̃ = Lift_{F}(π). Let ν_{0} be a character of F^{×} which satisfies

χ_{π}(xI) = ν_{0}(x^{2}).  (3.18)

If π and ν_{0} satisfy (3.18), we define L(π, ν_{0}) to be the constituent of π̃|_{G̃L_{2}(F)_{+}} with central character χ_{ν_{0}}. Remark. Since L(π, ν_{0}) has central character χ_{ν_{0}}, by (3.9) it follows that

χπ̃(x^{2}; 1) = ν_{0}(x^{2})γ_{F}(x^{2}, ψ) = ν_{0}(x^{2}).

By (2.64) this is χ_{π}(xI). This explains condition (3.18).

Proposition 3.19. Fix ν_{0}. Let x ∈ F^{×}. Put q_{x} = (x, ·)_{F}. Then

L(π, ν_{0})^{x} = L(π, ν_{0}q_{x}).

(See Definition 2.49.) Hence

π̃|_{G̃L_{2}(F)_{+}} = ∑_{x∈F^{×}/F^{×2}} L(π, ν_{0}q_{x}).

Proof. This follows from the way we are parameterizing summands in the restriction and the “central character trick” [see (3.10)]:

χ_{ν_{0}}^{x}(yI; ε) = (x, y)_{F}χ_{ν_{0}}(yI; ε)
= q_{x}(y)ν_{0}(y)γ_{F}(y, ψ)ε
= ν_{0}(y)γ_{F}(y, ψ_{x})ε
= χ_{ν_{0}q_{x}}(yI; ε).

Remark. We have that G̃L_{2}(F)_{+} ≃ S̃L_{2}(F) · Z(G̃L_{2}(F)_{+}). Hence there is no further reducibility of the restriction of π̃ to S̃L_{2}(F), and the irreducible constituents of π̃|_{S̃L_{2}(F)} are parameterized by F^{×}/F^{×2}. This result is surprising and its analogue for the linear group GL_{2}(F) is false (see Gelbart and Knapp, [21] and Tadic, [37]). In general, understanding the representation theory of SL_{2}(F) by restricting representations of GL_{2}(F) is quite difficult; see Labesse–Langlands [18].

We will restrict L(π, ν_{0}) to S̃L_{2}(F) in Chapter 5. The condition on (π, ν_{0}) and (σ, λ_{0}) needeed to guarantee L(π, ν_{0}) = L(σ, λ_{0}) as representations of S̃L_{2}(F) is nontrivial and is given in Lemma 5.34.

The Character of the Oscillator Representation

Let ω(ψ) = ω_{e}(ψ) ⊕ ω_{o}(ψ) be the oscillator representation of S̃L_{2}(F) attached to ψ. (See Gelbart and Piatetski–Shapiro in [10], p. 150, for a clear exposition of the definition of ω(ψ).) In this chapter, we explicitly compute the characters of ω_{e}(ψ) and ω_{0}(ψ) and get formulas for the sums and differences of these. The derivation uses the inversion formula and Flicker’s formula for the character of the oscillator representation (see Proposition 2.70). The idea motivates the derivation of the general character identities in the next chapter. In fact, we could have proved these general identities first and obtained the formulas in this chapter as a consequence. In the end, we felt it would be useful to have both computations and have included the explicit derivation of the character of the oscillator representation as a firm starting point. The fact that these identities are a special case of the general identities is given in Corollary 5.19.

For a general reference for the character of the oscillator representation, see Howe [15]. Additional information in the real case is given in Torasso [38] and Adams [2]. For the p–adic case see Prasad [29]. We remark that Prasad’s formula is given in terms of the norm one element z = w/\overline{w} corresponding to an element of SL_{2}(F).

We require some notation. Let m = |F^{×}/F^{×2}|. Let D_{SO} and D_{SL} be the Weyl denominators for SO_{1,2}(F) and SL_{2}(F) respectively. So if x, y ∈ F^{×},

D_{SO}(diag(x, 1/x, 1)) = |(1 − x)(1 − 1/x)|^{1/2},

and

D_{SL}(diag(x, 1/x)) = |(1 − x^{2})(1 − 1/x^{2})|^{1/2}.

We have the following lemmas. Lemma 4.1. Suppose that g ∈ SL_{2}(F). Then

\frac{D_{SO}(τ(±g))}{D_{SL}(g)} = | det(1 ± g)|^{−1/2}.

Proof. Let x ∈ F^{×} and put g = diag(x, 1/x). We have

\frac{D_{SO}(τ(±g))}{D_{SL}(g)} = \frac{|(1 ∓ x)(1 ∓ 1/x)|^{1/2}}{|(1 − x^{2})(1 − 1/x^{2})|^{1/2}} = |(1±x)(1±1/x)|^{−1/2} = | det(1±g)|^{−1/2}.

Let E be a the quadratic extension F(\sqrt{Δ}). Let ι be an embedding of E^{×} in GL_{2}(F) and write ι(E^{1}) = S_{Δ} ⊂ SL_{2}(F).

Now suppose that g ∈ S_{Δ}. Write g = ι(z) with z ∈ E^{1}. Then

\frac{D_{SO}(τ(±g))}{D_{SL}(g)} = \frac{|(1 ∓ z/\overline{z})(1 ∓ \overline{z}/z)|^{1/2}}{|(1 − z^{2}/\overline{z}^{2})(1 − \overline{z}^{2}/z^{2})|^{1/2}}
= |(1 ± z/\overline{z})(1 ± \overline{z}/z)|^{−1/2}
= | det(1 ± g)|^{−1/2}.

Since any element of SL_{2}(F) is conjugate to an element of one of the above two forms, the result follows.

Definition 4.2. For a ∈ F^{×}, define

Φ_{a}(g) = | det(1 + g/a)|^{−1/2}  (4.3)

We abuse notation and write Φ_{±}(g) in the case that a = ±1.

Notation 4.4. If ν_{0} is a character of F^{×}, we always reserve the letter ν to signify the one–dimensional representation of GL_{2}(F) given by ν = ν_{0} ∘ det.

Let μ_{0} be a character of F^{×} and suppose that ω(μ) is an oscillator representation of G̃L_{2}(F). (See Gelbart and Piatetski–Shapiro in [10], p. 151.) Let ν_{0} be another character of F^{×} and let ω(μ)(ν_{0}) be the summand of ω(μ)|_{G̃L_{2}(F)_{+}} which has central character

χ_{ν_{0}}(xI; ε) = ν_{0}(x)γ_{F}(x, ψ)ε.

4.1 Hyperbolic Set

We begin this section with some identities involving the map D defined by (2.59). We have that

\frac{D(diag(x, y))}{D(diag(x^{2}, y^{2}))} = |(1 + y/x)(1 + x/y)|^{−1/2}

and

\frac{D(diag(−x, y))}{D(diag(x^{2}, y^{2}))} = |(1 − y/x)(1 − x/y)|^{−1/2}.

Taking y = 1 and g = diag(x, 1/x) we obtain

\frac{D(diag(x, 1))}{D(diag(x^{2}, 1))} = |(1 + x)(1 + 1/x)|^{−1/2} = | det(1 + g)|^{−1/2} = Φ_{+}(g) (4.5)

and

\frac{D(diag(−x, 1))}{D(diag(x^{2}, y^{2}))} = |(1 − x)(1 − 1/x)|^{−1/2} = | det(1 − g)|^{−1/2} = Φ_{−}(g). (4.6)

Let g = diag(x, 1/x) and let g̃ = (g; ε). By the inversion formula, we have that

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{1}{m} ∑_{α∈F^{×}/F^{×2}} χ_{ν_{0}}(z_{α})^{−1}Θ_{ω(μ)} (z_{α}g̃) .

By Proposition 2.72, all but one of the terms in the sum is zero, i.e.,

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{ε}{m}(ξ, x)_{F}χ_{ν_{0}}(ξI; 1)^{−1}Θ_{ω(μ)}(diag(ξx, ξ/x); 1)  (4.7)

for ξ such that ξ ≡ x mod squares. This is a ξ which makes diag(ξx, ξ/x) a square; without loss of generality we take ξ = x.

We now make the following observations. By definition of χ_{ν_{0}} and properties of gamma factors (see Proposition 2.42) we have that (recall ξ = x)

(ξ, x)_{F}χ_{ν_{0}}(ξI; 1)^{−1} = (−1, x)_{F}χ_{ν_{0}}(x^{−1}I; (x, x)_{F})
= γ_{F}(x^{−1}, ψ)ν_{0}(x^{−1})
= γ_{F}(x, ψ)ν_{0}(x)^{−1}.

By Flicker’s formula for the character of the oscillator representation (see Proposition 2.70), we have that

\begin{aligned}Θ_{ω(μ)} (diag(ξx, ξ/x); 1) & = Θ_{ω(μ)} (diag(x^{2}, 1); 1) \\ & = b [μ_{0}(−x)\frac{D(diag(x, 1))}{D(diag(x^{2}, 1))} + μ_{0}(x)\frac{D(diag(−x, 1))}{D(diag(x^{2}, 1))}] . \end{aligned}

By (4.5) and (4.6), this becomes

Θ_{ω(μ)} (diag(ξx, ξ/x); 1) = bμ_{0}(x) [\frac{μ_{0}(−1)}{| det(1 + g)|^{1/2}} + \frac{1}{| det(1 − g)|^{1/2}} ] .

Incorporating these observations in (4.7) we obtain the following proposition. Proposition 4.8. Let g = diag(x, 1/x) and g̃ = (g; ε). Suppose that μ_{0} and ν_{0} are characters of F^{×}. Then

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{bεμ_{0}(x)ν_{0}(x)^{−1}γ_{F}(x, ψ)}{m} [\frac{μ_{0}(−1)}{| det(1 + g)|^{1/2}} + \frac{1}{| det(1 − g)|^{1/2}} ] .

(4.9)

Taking μ_{0} = ν_{0} in (4.9), and hence μ = ν, we get

Θ_{ω(ν)(ν_{0})}(g̃) = \frac{bεγ_{F}(x, ψ)}{m} [\frac{ν_{0}(−1)}{| det(1 + g)|^{1/2}} + \frac{1}{| det(1 − g)|^{1/2}} ] .  (4.10)

We apply (4.10) twice, once to a character ν satisfying ν_{0}(−1) = 1 and once to a character ν′ satisfying ν′_{0}(−1) = −1. Adding and subtracting the results we obtain

Θ_{ω(ν)(ν_{0})+ω(ν′)(ν′_{0})}(g̃) = \frac{2bεγ_{F}(x, ψ)}{m| det(1 − g)|^{1/2}}

and

Θ_{ω(ν)(ν_{0})−ω(ν′)(ν′_{0})}(g̃) = \frac{2bεγ_{F}(x, ψ)}{m| det(1 + g)|^{1/2}} .

But by (2.62), b = |F^{×}/F^{×2}|/2, so that 2b/m = 1. We have shown the following proposition. Proposition 4.11. Let g = diag(x, 1/x) and g̃ = (g; ε). Suppose that ν_{0} and ν′_{0} are characters of F^{×} that satisfy ν_{0}(−1) = 1 and ν′_{0}(−1) = −1. Then

Θ_{ω(ν)(ν_{0})±ω(ν′)(ν′_{0})}(g̃) = \frac{γ_{F}(x, ψ)ε}{| det(1 ∓ g)|^{1/2}} .

This is consistent with the calculation of the character of the oscillator representation given by Howe [15].

4.2 Elliptic Set

Let E = F(δ) be a quadratic extension of F, δ^{2} = Δ ∈ F. Fix an embedding ι of E^{×} in GL_{2}(F) and write S_{Δ} = ι(E^{1}) (see Section 2.2). Set δ′ = ι(δ).

Suppose g is an elliptic element of SL_{2}(F). Without loss of generality, g ∈ S_{Δ} for some Δ. By the inversion formula, we have that

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{1}{m} ∑_{α∈F^{×}/F^{×2}} χ_{ν_{0}}(z_{α})^{−1}Θ_{ω(μ)} (z_{α}g̃) .

The analysis is divided into two cases, the case of g a square and g not a square.

4.2.1 Elliptic Set, Case I: g = h^{2}

Suppose that g is the square of an element h in GL_{2}(F). Write ι(z) = g, ι(w) = h and g̃ = (g; ε). Since det g = 1, it follows that det h = Nw = ±1.

We will need the following calculations involving the function D from (2.59). Lemma 4.12. Suppose that g = h^{2}. Then

\frac{D(h)}{D(g)} = | det(1 + g/ det h)|^{−1/2} = Φ_{det h}(g).  (4.13)

Proof. Note that

\frac{D(h)}{D(g)} = |\frac{(w − \overline{w})^{2}/w\overline{w}}{(z − \overline{z})^{2}/z\overline{z}}|^{1/2}

= |\frac{(1 − w/\overline{w})(1 − \overline{w}/w)}{(1 − z/\overline{z})(1 − \overline{z}/z)}|^{1/2}

= |(1 + w/\overline{w})(1 + \overline{w}/w)|^{−1/2}

= | det(1 + ι(w^{2}/Nw))|^{−1/2} = | det(1 + g/ det h)|^{−1/2}

= Φ_{det h}(g).

Lemma 4.14. Suppose that g = h^{2}. Then

\frac{D(δ′h)}{D(g)} = | det(1 − g/ det h)|^{−1/2} = Φ_{− det h}(g).  (4.15)

Proof. We have

\frac{D(δ′h)}{D(g)} = |\frac{(δw − \overline{δw})^{2}/δw\overline{δw}}{(z − \overline{z})^{2}/z\overline{z}}|^{1/2}

= |\frac{(1 − δw/\overline{δw})(1 − \overline{δw}/δw)}{(1 − z/\overline{z})(1 − \overline{z}/z)}|^{1/2}

= |\frac{(1 + w/\overline{w})(1 + \overline{w}/w)}{(1 − z/\overline{z})(1 − \overline{z}/z)}|^{1/2}

= |(1 − w/\overline{w})(1 − \overline{w}/w)|^{−1/2} = | det(1 − ι(w^{2}/Nw))|^{−1/2} so that

\frac{D(δ′h)}{D(g)} = | det(1 − g/ det h)|^{−1/2} = Φ_{− det h}(g).

By Proposition 2.4, ξg is a square if and only if ξ = 1 or Δ. [In particular, Δg = (δ′h)^{2}.]

The inversion formula becomes

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{1}{m} [Θ_{ω(μ)}(g̃) + χ_{ν_{0}}(ΔI; 1)^{−1}Θ_{ω(μ)}((ΔI; 1)g̃)]  (4.16)

by Proposition 2.72. By Flicker’s formula for the character of the oscillator representation, we have that

Θ_{ω(μ)}(g̃) = εs(g)Θ_{ω(μ)}(g; s(g)^{−1})

= bεs(g)μ_{0}(−1)μ(h)\frac{D(h)}{D(g)}

= bεs(g)μ_{0}(− det h)\frac{D(h)}{D(g)}.

By (4.13) this becomes

Θ_{ω(μ)}(g̃) = \frac{bεs(g)μ_{0}(− det h)}{| det(1 + g/ det h)|^{1/2}} .  (4.17)

We now consider the term χ_{ν_{0}}(ΔI; 1)^{−1}Θ_{ω(μ)}((ΔI; 1)g̃). By definition, we have

χ_{ν_{0}}(ΔI; 1)^{−1} = ν_{0}(Δ)^{−1}γ_{F}(Δ, ψ)(−1, Δ)_{F}.  (4.18)

Flicker’s formula for the character of the oscillator representation yields

Θ_{ω(μ)}((ΔI; 1)g̃) = εs(Δg)c(ΔI, g)Θ_{ω(μ)}(Δg; s(Δg)^{−1})

= bεs(Δg)c(ΔI, g)μ_{0}(−1)μ(δ′h)\frac{D(δ′h)}{D(Δg)}

= bεs(Δg)c(ΔI, g)μ_{0}(Δ det h)\frac{D(δ′h)}{D(Δg)}.

Equation (2.60) implies that the right–hand side equals

bεs(Δg)c(ΔI, g)μ_{0}(Δ det h)\frac{D(δ′h)}{D(g)}.

Hence, by (4.15),

Θ_{ω(μ)}((ΔI; 1)g̃) = \frac{bεs(Δg)c(ΔI, g)μ_{0}(Δ det h)}{| det(1 − g/ det h)|^{1/2}}.  (4.19)

Putting (4.18) and (4.19) together we obtain

χ_{ν_{0}}(ΔI; 1)^{−1}Θ_{ω(μ)}((ΔI; 1)g̃) =

\frac{bε(−1, Δ)_{F}s(Δg)c(ΔI, g)μ_{0}(Δ det h)ν_{0}(Δ)^{−1}γ_{F}(Δ, ψ)}{| det(1 − g/ det h)|^{1/2}}.

We condense the result using (2.38) and obtain

χ_{ν_{0}}(ΔI; 1)^{−1}Θ_{ω(μ)}((ΔI; 1)g̃) =

\frac{bεs(g)c_{β}(ΔI, −g)μ_{0}(Δ det h)ν_{0}(Δ)^{−1}γ_{F}(Δ, ψ)}{| det(1 − g/ det h)|^{1/2}}.

(4.20)

Plugging (4.17) and (4.20) into (4.16), the following proposition follows. Proposition 4.21. Suppose g = h^{2} ∈ S_{Δ} is regular elliptic and g̃ = (g; ε). Suppose that μ_{0} and ν_{0} are characters of F^{×}. Then

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{bεs(g)}{m} [\frac{μ_{0}(− det h)}{| det(1 + g/ det h)|^{1/2}}

+ \frac{c_{β}(ΔI, −g)γ_{F}(Δ, ψ)μ_{0}(Δ det h)ν_{0}(Δ)^{−1}}{| det(1 − g/ det h)|^{1/2}} ] .

(4.22)

Taking μ_{0} = ν_{0} in (4.22), and hence μ = ν, we get

Θ_{ω(ν)(ν_{0})}(g̃) = \frac{bεs(g)}{m} [\frac{ν_{0}(− det h)}{| det(1 + g/ det h)|^{1/2}}

+ \frac{c_{β}(ΔI, −g)γ_{F}(Δ, ψ)ν_{0}(det h)}{| det(1 − g/ det h)|^{1/2}}] .

Hence we have that

Θ_{ω(ν)(ν_{0})}(g̃) = \begin{cases}\frac{bεs(g)}{m} & [\frac{ν_{0}(−1)}{| det(1 + g)|^{1/2}} + \frac{c_{β}(ΔI, −g)γ_{F}(Δ, ψ)}{| det(1 − g)|^{1/2}}] \\ \frac{bεs(g)}{m} & [\frac{1}{| det(1 − g)|^{1/2}} + \frac{ν_{0}(−1)c_{β}(ΔI, −g)γ_{F}(Δ, ψ)}{| det(1 + g)|^{1/2}}] \end{cases} (4.23)

according as det h = ±1.

We apply this twice, once to a character ν satisfying ν_{0}(−1) = 1 and once to a character ν′ satisfying ν′_{0}(−1) = −1. Adding and subtracting the results and noticing that 2b/m = 1, the following proposition follows. Proposition 4.24. Suppose that g = h^{2} ∈ S_{Δ} is regular elliptic and g̃ = (g; ε). Suppose that ν_{0} and ν′_{0} are characters of F^{×} that satisfy ν_{0}(−1) = 1 and ν′_{0}(−1) = −1. Then

Θ_{ω(ν)(ν_{0})+ω(ν′)(ν′_{0})}(g̃) = \begin{cases}\frac{εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)}{| det(1 − g)|^{1/2}}, & det h = 1 \\ \frac{εs(g)}{| det(1 − g)|^{1/2}} , & det h = −1. \end{cases}

and

Θ_{ω(ν)(ν_{0})−ω(ν′)(ν′_{0})}(g̃) = \begin{cases}\frac{εs(g)}{| det(1 + g)|^{1/2}} , & det h = 1 \\ \frac{εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)}{| det(1 + g)|^{1/2}}, & det h = −1. \end{cases}

4.2.2 Elliptic Set, Case II: g ≠ h^{2}

Assume now that g is not the square of an element in GL_{2}(F).

Put ι(z) = g. By Proposition 2.4, we may write ξ_{0}z = w^{2} with ξ_{0} = Nw. Let h = ι(w) and g̃ = (g; ε). Note that ξ_{0}Δz = (δw)^{2} and if ξz is a square, then ξ = ξ_{0} or ξ_{0}Δ.

We begin again with some computations involving D. Lemma 4.25. Suppose that ξ_{0}g = h^{2} with ξ_{0} = det h. Then

\frac{D(h)}{D(ξ_{0}g)} = | det(1 + g)|^{−1/2} = Φ_{+}(g).  (4.26)

Proof. Note that

\frac{D(h)}{D(ξ_{0}g)} = \frac{D(h)}{D(g)}

= |\frac{(w − \overline{w})^{2}/w\overline{w}}{(z − \overline{z})^{2}/z\overline{z}}|^{1/2}

= |\frac{(1 − w/\overline{w})(1 − \overline{w}/w)}{(1 − z/\overline{z})(1 − \overline{z}/z)}|^{1/2}

= |\frac{(1 − w/\overline{w})(1 − \overline{w}/w)}{(1 − ξ_{0}z/\overline{ξ_{0}z})(1 − \overline{ξ_{0}z}/ξ_{0}z)}|^{1/2}

= |(1 + w/\overline{w})(1 + \overline{w}/w)|^{−1/2} = | det(1 + ι(w^{2}/Nw))|^{−1/2}.

Since ι(w^{2}/Nw) = h^{2}/ det h = g, we get

\frac{D(h)}{D(ξ_{0}g)} = | det(1 + g)|^{−1/2} = Φ_{+}(g).

Lemma 4.27. Suppose that ξ_{0}g = h^{2} with ξ_{0} = det h. Then

\frac{D(δ′h)}{D(ξ_{0}Δg)} = | det(1 − g)|^{−1/2} = Φ_{−}(g).  (4.28)

Proof. We have

\frac{D(δ′h)}{D(ξ_{0}Δg)} = \frac{D(δ′h)}{D(g)}.

Hence,

\frac{D(δ′h)}{D(ξ_{0}Δg)} = |\frac{(δw − \overline{δw})^{2}/δw\overline{δw}}{(z − \overline{z})^{2}/z\overline{z}}|^{1/2}

= |\frac{(1 − δw/\overline{δw})(1 − \overline{δw}/δw)}{(1 − z/\overline{z})(1 − \overline{z}/z)}|^{1/2}

= |\frac{(1 + w/\overline{w})(1 + \overline{w}/w)}{(1 − z/\overline{z})(1 − \overline{z}/z)}|^{1/2}

= |\frac{(1 + w/\overline{w})(1 + \overline{w}/w)}{(1 − ξ_{0}z/\overline{ξ_{0}z})(1 − \overline{ξ_{0}z}/ξ_{0}z)}|^{1/2}

= |(1 − w/\overline{w})(1 − \overline{w}/w)|^{−1/2} = | det(1 − ι(w^{2}/Nw))|^{−1/2}.

Since ι(w^{2}/Nw) = h^{2}/ det h = g, we get

\frac{D(δ′h)}{D(ξ_{0}Δg)} = | det(1 − g)|^{−1/2} = Φ_{−}(g).

By Proposition 2.72, the inversion formula becomes

\begin{aligned}Θ_{ω(μ)(ν_{0})}(g̃) = \frac{1}{m} & [χ_{ν_{0}}(ξ_{0}I; 1)^{−1}Θ_{ω(μ)}((ξ_{0}I; 1)g̃) \\ & + χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θ_{ω(μ)}((ξ_{0}ΔI; 1)g̃)] \end{aligned}  (4.29)

First we calculate

χ_{ν_{0}}(ξ_{0}I; 1)^{−1}Θ_{ω(μ)}((ξ_{0}I; 1)g̃).

By definition, we have that

χ_{ν_{0}}(ξ_{0}I; 1)^{−1} = ν_{0}(ξ_{0})^{−1}γ_{F}(ξ_{0}, ψ)(−1, ξ_{0})_{F}.  (4.30)

By Flicker’s formula for the character of the oscillator representation, we have that

Θ_{ω(μ)}((ξ_{0}I; 1)g̃) = εs(ξ_{0}g)c(ξ_{0}I, g)Θ_{ω(μ)}(ξ_{0}g; s(ξ_{0}g)^{−1})

= bεs(ξ_{0}g)c(ξ_{0}I, g)μ_{0}(−1)μ(h)\frac{D(h)}{D(ξ_{0}g)}

= bεs(ξ_{0}g)c(ξ_{0}I, g)μ_{0}(−ξ_{0})\frac{D(h)}{D(ξ_{0}g)}.

By (4.26) this becomes

Θ_{ω(μ)}((ξ_{0}I; 1)g̃) = \frac{bεs(ξ_{0}g)c(ξ_{0}I, g)μ_{0}(−ξ_{0})}{| det(1 + g)|^{1/2}}.  (4.31)

Patching (4.30) and (4.31) together, we get

\begin{aligned}χ_{ν_{0}}(ξ_{0}I; 1)^{−1} & Θ_{ω(μ)}((ξ_{0}I; 1)g̃) = \\ & \frac{bεγ_{F}(ξ_{0}, ψ)(−1, ξ_{0})_{F}s(ξ_{0}g)c(ξ_{0}I, g)μ_{0}(−ξ_{0})ν_{0}(ξ_{0})^{−1}}{| det(1 + g)|^{1/2}} . \end{aligned}

We condense this using (2.38) and obtain

\begin{aligned}χ_{ν_{0}}(ξ_{0}I; 1)^{−1} & Θ_{ω(μ)}((ξ_{0}I; 1)g̃) = \\ & \frac{bεγ_{F}(ξ_{0}, ψ)s(g)c_{β}(ξ_{0}I, −g)μ_{0}(−ξ_{0})ν_{0}(ξ_{0})^{−1}}{| det(1 + g)|^{1/2}} . \end{aligned}  (4.32)

Next we calculate

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θ_{ω(μ)}((ξ_{0}ΔI; 1)g̃).

By definition, we have that

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1} = ν_{0}(ξ_{0}Δ)^{−1}γ_{F}(ξ_{0}Δ, ψ)(−1, ξ_{0}Δ)_{F}.  (4.33)

By Flicker’s formula for the character of the oscillator representation, we have that

Θ_{ω(μ)}((ξ_{0}ΔI; 1)g̃) = εs(ξ_{0}Δg)c(ξ_{0}ΔI, g)Θ_{ω(μ)}(ξ_{0}Δg; s(ξ_{0}Δg)^{−1})

= bεs(ξ_{0}Δg)c(ξ_{0}ΔI, g)μ_{0}(−1)μ(ξ_{0}δ′h)\frac{D(δ′h)}{D(ξ_{0}Δg)}

= bεs(ξ_{0}Δg)c(ξ_{0}ΔI, g)μ_{0}(ξ_{0}Δ)\frac{D(δ′h)}{D(ξ_{0}Δg)}.

By (4.28) this becomes

Θ_{ω(μ)}((ξ_{0}ΔI; 1)g̃) = \frac{bεs(ξ_{0}Δg)c(ξ_{0}ΔI, g)μ_{0}(ξ_{0}Δ)}{| det(1 − g)|^{1/2}}.  (4.34)

Patching (4.33) and (4.34) together, we get

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θ_{ω(μ)}((ξ_{0}ΔI; 1)g̃) =

\frac{bεγ_{F}(ξ_{0}Δ, ψ)(−1, ξ_{0}Δ)_{F}s(ξ_{0}Δg)c(ξ_{0}ΔI, g)μ_{0}(ξ_{0}Δ)ν_{0}(ξ_{0}Δ)^{−1}}{| det(1 − g)|^{1/2}} .

We condense this using (2.38) and obtain

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θ_{ω(μ)}((ξ_{0}ΔI; 1)g̃) =

\frac{bεγ_{F}(ξ_{0}Δ, ψ)s(g)c_{β}(ξ_{0}ΔI, −g)μ_{0}(ξ_{0}Δ)ν_{0}(ξ_{0}Δ)^{−1}}{| det(1 − g)|^{1/2}} .

(4.35)

Rewriting (4.29) using (4.32) and (4.35) results in the following proposition. Proposition 4.36. Suppose that g ∈ S_{Δ} is regular elliptic and is not a square. Let g̃ = (g; ε). Write ξ_{0}g = h^{2} with ξ_{0} = det h. Suppose that μ_{0} and ν_{0} are characters of F^{×}. Then

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{bεs(g)}{m} [\frac{γ_{F}(ξ_{0}, ψ)c_{β}(ξ_{0}I, −g)μ_{0}(−ξ_{0})ν_{0}(ξ_{0})^{−1}}{| det(1 + g)|^{1/2}}

+ \frac{γ_{F}(ξ_{0}Δ, ψ)c_{β}(ξ_{0}ΔI, −g)μ_{0}(ξ_{0}Δ)ν_{0}(ξ_{0}Δ)^{−1}}{| det(1 − g)|^{1/2}} ] .

(4.37)

Taking μ_{0} = ν_{0} in (4.37), and hence μ = ν, we get

Θ_{ω(ν)(ν_{0})}(g̃) = \frac{bεs(g)}{m} [\frac{γ_{F}(ξ_{0}, ψ)c_{β}(ξ_{0}I, −g)ν_{0}(−1)}{| det(1 + g)|^{1/2}} + \frac{γ_{F}(ξ_{0}Δ, ψ)c_{β}(ξ_{0}ΔI, −g)}{| det(1 − g)|^{1/2}}] .

(4.38)

We apply this twice, once to a character ν satisfying ν_{0}(−1) = 1 and once to a character ν′ satisfying ν′_{0}(−1) = −1. Adding and subtracting the results and noticing that 2b/m = 1, the following proposition follows. Proposition 4.39. Suppose that g ∈ S_{Δ} is regular elliptic and is not a square. Write ξ_{0}g = h^{2} with ξ_{0} = det h. Suppose that ν_{0} and ν′_{0} are characters of F^{×} that satisfy ν_{0}(−1) = 1 and ν′_{0}(−1) = −1. Then

Θ_{ω(ν)(ν_{0})+ω(ν′)(ν′_{0})}(g̃) = \frac{εs(g)c_{β}(ξ_{0}ΔI, −g)γ_{F}(ξ_{0}Δ, ψ)}{| det(1 − g)|^{1/2}}

and

Θ_{ω(ν)(ν_{0})−ω(ν′)(ν′_{0})}(g̃) = \frac{εs(g)c_{β}(ξ_{0}I, −g)γ_{F}(ξ_{0}, ψ)}{| det(1 + g)|^{1/2}}.

4.3 Recapitulation

We collect the results of this chapter in two theorems. Theorem 4.40. Let ω(μ)(ν_{0}) be the oscillator representation of G̃L_{2}(F)_{+} with central character χ_{ν_{0}} where

χ_{ν_{0}}(xI; ε) = ν_{0}(x)γ_{F}(x, ψ)ε.

1. Let g = diag(x, 1/x) and g̃ = (g; ε). Then

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{bεμ_{0}(x)ν_{0}(x)^{−1}γ_{F}(x, ψ)}{m} [\frac{μ_{0}(−1)}{| det(1 + g)|^{1/2}} + \frac{1}{| det(1 − g)|^{1/2}} ] .

2. Suppose g = h^{2} ∈ S_{Δ} is regular elliptic and g̃ = (g; ε). Then

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{bεs(g)}{m} [\frac{μ_{0}(− det h)}{| det(1 + g/ det h)|^{1/2}}

+ \frac{c_{β}(ΔI, −g)γ_{F}(Δ, ψ)μ_{0}(Δ det h)ν_{0}(Δ)^{−1}}{| det(1 − g/ det h)|^{1/2}} ] .

3. Suppose that g ∈ S_{Δ} is regular elliptic and is not a square. Let g̃ = (g; ε).

Write ξ_{0}g = h^{2} with ξ_{0} = det h. Then

Θ_{ω(μ)(ν_{0})}(g̃) = \frac{bεs(g)}{m} [\frac{γ_{F}(ξ_{0}, ψ)c_{β}(ξ_{0}I, −g)μ_{0}(−ξ_{0})ν_{0}(ξ_{0})^{−1}}{| det(1 + g)|^{1/2}}

+ \frac{γ_{F}(ξ_{0}Δ, ψ)c_{β}(ξ_{0}ΔI, −g)μ_{0}(ξ_{0}Δ)ν_{0}(ξ_{0}Δ)^{−1}}{| det(1 − g)|^{1/2}} ] .

Suppose that ν_{0} and ν′_{0} are characters of F^{×} that satisfy ν_{0}(−1) = 1 and ν′_{0}(−1) = −1. Put g̃ = (g; ε). Define

Γ_{±}(g̃) = Θ_{ω(ν)(ν_{0})±ω(ν′)(ν′_{0})}(g̃).

Theorem 4.41.

Γ_{+}(g̃) = \begin{cases}\frac{γ_{F}(x, ψ)ε}{| det(1 − g)|^{1/2}} , & if g = diag(x, 1/x) \\ \frac{εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)}{| det(1 − g)|^{1/2}}, & if g = h^{2} ∈ S_{Δ} and det h = 1 \\ \frac{εs(g)}{| det(1 − g)|^{1/2}} , & if g = h^{2} ∈ S_{Δ} and det h = −1 \\ \frac{εs(g)γ_{F}(ξ_{0}Δ, ψ)c_{β}(ξ_{0}ΔI, −g)}{| det(1 − g)|^{1/2}}, & if g ∈ S_{Δ}, ξ_{0}g = h^{2} and det h = ξ_{0} \end{cases}

and

Γ_{−}(g̃) = \begin{cases}\frac{γ_{F}(x, ψ)ε}{| det(1 + g)|^{1/2}} , & if g = diag(x, 1/x) \\ \frac{εs(g)}{| det(1 + g)|^{1/2}} , & if g = h^{2} ∈ S_{Δ} and det h = +1 \\ \frac{εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)}{| det(1 + g)|^{1/2}}, & if g = h^{2} ∈ S_{Δ} and det h = −1 \\ \frac{εs(g)γ_{F}(ξ_{0}, ψ)c_{β}(ξ_{0}I, −g)}{| det(1 + g)|^{1/2}}, & if g ∈ S_{Δ}, ξ_{0}g = h^{2} and det h = ξ_{0}. \end{cases}

Remark. We have proved in Lemma 4.1 that for all g ∈ SL_{2}(F),

| det(1 ± g)|^{1/2} = \frac{D_{SO}(τ(±g))}{D_{SL}(g)}.

The Character Formula

Let π be an irreducible representation of GL_{2}(F) satisfying χ_{π}(−I) = 1 and π̃ = Lift_{F}(π) be Flicker’s lifting of π to G̃L_{2}(F). In Chapter 3 we let L(π, ν_{0}) be the constituent of π̃|_{G̃L_{2}(F)_{+}} having central character χ_{ν_{0}}(xI; ε) = ν_{0}(x)γ_{F}(x, ψ)ε.

We restrict L(π, ν_{0}) to S̃L_{2}(F) and also write L(π, ν_{0}) for this restriction. Throughout the rest of this chapter and beyond, whenever we write L(π, ν_{0}) we intend the representation of S̃L_{2}(F). The representation π and the character ν_{0} are related by the equation χ_{π}(xI) = ν_{0}(x^{2}), see (3.18). The representation πν^{−1} has trivial central character and factors to a representation (πν^{−1})′ of SO_{1,2}(F).

This chapter contains the main results of this thesis. We derive a character formula relating the representations L(π, ν_{0}) of S̃L_{2}(F) and (πν^{−1})′ of SO_{1,2}(F). This character formula suggests a map L(π, ν_{0}) ↦ (πν^{−1})′ from irreducible representations of S̃L_{2}(F) to irreducible representations of SO_{1,2}(F). The map is surjective but not injective.

In order to obtain a bijection, we “stabilize” the formulas; we get a character formula relating “stable” representations L_{st}(π, ν_{0}) of S̃L_{2}(F) and representations (πν^{−1})′ of SO_{1,2}(F) through the following remarkably simple formula:

Θ_{L_{st}(π,ν_{0})}(g̃) = Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)).

The derivation of these formulas requires the inversion formula and Flicker’s character identity. The first step is to get a character formula between representations L(π, ν_{0}) and πν^{−1}. This is the bulk of the work and is contained in Sections 1–3. After this, we push over to SO_{1,2}(F) and then stabilize the correspondence.

5.1 Hyperbolic Set

Let g = diag(x, 1/x) and let g̃ = (g; ε). By the inversion formula, we have that

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{m} ∑_{α∈F^{×}/F^{×2}} χ_{ν_{0}}(z_{α})^{−1}Θπ̃ (z_{α}g̃) .

By Proposition 2.72, all but one of the terms in the sum is zero, i.e.,

Θ_{L(μ,ν_{0})}(g̃) = \frac{ε}{m}(ξ, x)_{F}χ_{ν_{0}}(ξI; 1)^{−1}Θπ̃(diag(ξx, ξ/x); 1)  (5.1)

for ξ such that ξ ≡ x mod squares. This is a ξ which makes diag(ξx, ξ/x) a square; without loss of generality we take ξ = x.

By definition of χ_{ν_{0}} and properties of gamma factors (see Proposition 2.42) we have that (recall ξ = x)

(ξ, x)_{F}χ_{ν_{0}}(ξI; 1)^{−1} = (−1, x)_{F}χ_{ν_{0}}(x^{−1}I; (x, x)_{F})
= γ_{F}(x^{−1}, ψ)ν_{0}(x^{−1})
= γ_{F}(x, ψ)ν_{0}(x)^{−1}.

We put

h = diag(x, 1)  and  h′ = diag(−x, 1).

By Flicker’s formula on the hyperbolic set (see Definiton 2.63), we have that

\begin{aligned}Θπ̃ (diag(ξx, ξ/x); 1) & = Θπ̃ (h^{2}; 1) \\ & = b [\frac{D(h)}{D(h^{2})}Θ_{π}(h) + \frac{D(h′)}{D(h^{2})}Θ_{π}(h′)] . \end{aligned}

[We remark here that if g is hyperbolic, then s(g^{2})^{−1} = 1.] By (4.5) and (4.6), this becomes

Θπ̃ (diag(ξx, ξ/x); 1) = b [Φ_{+}(g)Θ_{π}(h) + Φ_{−}(g)Θ_{π}(h′)] .

Incorporating these observations in (5.1) we obtain

Θ_{L(π,ν_{0})}(g̃) = \frac{bεν_{0}(x)^{−1}γ_{F}(x, ψ)}{m} [Φ_{+}(g)Θ_{π}(h) + Φ_{−}(g)Θ_{π}(h′)] .  (5.2)

By Theorem 4.41, we have that Γ_{−}(g; ε) = γ_{F}(x, ψ)εΦ_{+}(g) and Γ_{+}(g; ε) = γ_{F}(x, ψ)εΦ_{−}(g). Then (5.2) becomes

Θ_{L(π,ν_{0})}(g̃) = \frac{b}{m} [Γ_{−}(g̃)ν_{0}(x)^{−1}Θ_{π}(h) + Γ_{+}(g̃)ν_{0}(−1)ν_{0}(−x)^{−1}Θ_{π}(h′)] .

Recall that 2b/m = 1, so that b/m = 1/2. Recall that det h = x and det h′ = −x. We have proved the following proposition. Proposition 5.3. Let ν_{0} be a character of F^{×} which satisfies χ_{π}(xI) = ν_{0}(x^{2}). If g = diag(x, 1/x), h = diag(x, 1) and h′ = diag(−x, 1) then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{πν^{−1}}(h) + ν_{0}(−1)Γ_{+}(g̃)Θ_{πν^{−1}}(h′)] .

5.2 Elliptic Set

Let E = F(δ) be a quadratic extension of F, δ^{2} = Δ ∈ F. Fix an embedding ι of E^{×} in GL_{2}(F) and write S_{Δ} = ι(E^{1}) (see Section 2.2). Set δ′ = ι(δ).

Suppose g is an elliptic element of SL_{2}(F). Without loss of generality, g ∈ S_{Δ} for some Δ. By the inversion formula, we have that

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{m} ∑_{α∈F^{×}/F^{×2}} χ_{ν_{0}}(z_{α})^{−1}Θπ̃ (z_{α}g̃) .

The analysis is divided into two cases, the case of g a square and g not a square.

5.2.1 Elliptic Set, Case I: g = h^{2}

Suppose that g is the square of an element h in GL_{2}(F). By Proposition 2.4, ξg is a square if and only if ξ = 1 or Δ. In light of Proposition 2.72, the inversion formula becomes

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{m} [Θπ̃(g̃) + χ_{ν_{0}}(ΔI; 1)^{−1}Θπ̃((ΔI; 1)g̃)]  (5.4)

By Flicker’s formula on the elliptic set, we have that

\begin{aligned}Θπ̃(g̃) & = εs(g)Θπ̃(g; s(g)^{−1}) \\ & = bεs(g)\frac{D(h)}{D(g)}Θ_{π}(h). \end{aligned}

By (4.13) this becomes

Θπ̃(g̃) = bεs(g)Φ_{det h}(g)Θ_{π}(h).  (5.5)

We now consider the term χ_{ν_{0}}(ΔI; 1)^{−1}Θπ̃((ΔI; 1)g̃). By definition, we have

χ_{ν_{0}}(ΔI; 1)^{−1} = ν_{0}(Δ)^{−1}γ_{F}(Δ, ψ)(−1, Δ)_{F}.  (5.6)

Flicker’s formula for the character on the elliptic set yields

Θπ̃((ΔI; 1)g̃) = εs(Δg)c(ΔI, g)Θπ̃(Δg; s(Δg)^{−1})

= bεs(Δg)c(ΔI, g)\frac{D(δ′h)}{D(Δg)}Θ_{π}(δ′h).

Equation (2.60) implies that the right–hand side equals

bεs(Δg)c(ΔI, g)\frac{D(δ′h)}{D(g)}Θ_{π}(δ′h).

Hence, by (4.15),

Θπ̃((ΔI; 1)g̃) = bεs(Δg)c(ΔI, g)Φ_{− det h}(g)Θ_{π}(δ′h).  (5.7)

Putting (5.6) and (5.7) together we obtain

χ_{ν_{0}}(ΔI; 1)^{−1}Θπ̃((ΔI; 1)g̃) =
bε(−1, Δ)_{F}s(Δg)c(ΔI, g)ν_{0}(Δ)^{−1}γ_{F}(Δ, ψ)Φ_{− det h}(g)Θ_{π}(δ′h).

We condense the result using (2.38) and obtain

χ_{ν_{0}}(ΔI; 1)^{−1}Θπ̃((ΔI; 1)g̃) =
bεs(g)c_{β}(ΔI, −g)ν_{0}(Δ)^{−1}γ_{F}(Δ, ψ)Φ_{− det h}(g)Θ_{π}(δ′h).

(5.8)

Plugging (5.5) and (5.8) into (5.4), we see

Θ_{L(π,ν_{0})}(g̃) = \frac{b}{m}[εs(g)Φ_{det h}(g)Θ_{π}(h)
+ εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)Φ_{− det h}(g)ν_{0}(Δ)^{−1}Θ_{π}(δ′h)].

Recall that 2b/m = 1, so that b/m = 1/2. Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2}[εs(g)Φ_{det h}(g)ν_{0}(det h)Θ_{πν^{−1}}(h)

+ εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)Φ_{− det h}(g)ν_{0}(− det h)Θ_{πν^{−1}}(δ′h)].

By Theorem 4.41, we have that Γ_{− det h}(g̃) = εs(g)Φ_{det h}(g) and Γ_{det h}(g̃) = εs(g)γ_{F}(Δ, ψ)c_{β}(ΔI, −g)Φ_{− det h}(g). This implies the next proposition. Proposition 5.9. Suppose that g = h^{2} ∈ S_{Δ} is elliptic. Let ν_{0} be a character of F^{×} which satisfies χ_{π}(xI) = ν_{0}(x^{2}). Put λ = det h. Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−λ}(g̃)ν_{0}(λ)Θ_{πν^{−1}}(h) + Γ_{λ}(g̃)ν_{0}(−λ)Θ_{πν^{−1}}(δ′h)] .

5.2.2 Elliptic Set, Case II: g ≠ h^{2}

Assume now that g is not the square of an element in GL_{2}(F). Put ι(z) = g. By Proposition 2.4, we may write ξ_{0}z = w^{2} with ξ_{0} = Nw. Let h = ι(w) and g̃ = (g; ε). Note that ξ_{0}Δz = (δw)^{2} and if ξz is a square, then ξ = ξ_{0} or ξ_{0}Δ. By Proposition 2.72, the inversion formula becomes

\begin{aligned}Θ_{L(π,ν_{0})}(g̃) = \frac{1}{m} & [χ_{ν_{0}}(ξ_{0}I; 1)^{−1}Θπ̃((ξ_{0}I; 1)g̃) \\ & + χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θπ̃((ξ_{0}ΔI; 1)g̃)] \end{aligned}  (5.10)

First we calculate

χ_{ν_{0}}(ξ_{0}I; 1)^{−1}Θπ̃((ξ_{0}I; 1)g̃).

By definition, we have that

χ_{ν_{0}}(ξ_{0}I; 1)^{−1} = ν_{0}(ξ_{0})^{−1}γ_{F}(ξ_{0}, ψ)(−1, ξ_{0})_{F}.  (5.11)

By Flicker’s formula on the elliptic set, we have that

\begin{aligned}Θπ̃((ξ_{0}I; 1)g̃) & = εs(ξ_{0}g)c(ξ_{0}I, g)Θπ̃(ξ_{0}g; s(ξ_{0}g)^{−1}) \\ & = bεs(ξ_{0}g)c(ξ_{0}I, g)\frac{D(h)}{D(ξ_{0}g)}Θ_{π}(h). \end{aligned}

By (4.26) this becomes

Θπ̃((ξ_{0}I; 1)g̃) = bεs(ξ_{0}g)c(ξ_{0}I, g)Φ_{+}(g)Θ_{π}(h).  (5.12)

Patching (5.11) and (5.12) together, we get

\begin{aligned}χ_{ν_{0}}(ξ_{0}I; 1)^{−1} & Θπ̃((ξ_{0}I; 1)g̃) = \\ & bεγ_{F}(ξ_{0}, ψ)(−1, ξ_{0})_{F}s(ξ_{0}g)c(ξ_{0}I, g)ν_{0}(ξ_{0})^{−1}Φ_{+}(g)Θ_{π}(h). \end{aligned}

We condense this using (2.38) and obtain

\begin{aligned}χ_{ν_{0}}(ξ_{0}I; 1)^{−1} & Θπ̃((ξ_{0}I; 1)g̃) = \\ & bεγ_{F}(ξ_{0}, ψ)s(g)c_{β}(ξ_{0}I, −g)ν_{0}(ξ)^{−1}Φ_{+}(g)Θ_{π}(h). \end{aligned}  (5.13)

Next we calculate

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θπ̃((ξ_{0}ΔI; 1)g̃).

By definition, we have that

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1} = ν_{0}(ξ_{0}Δ)^{−1}γ_{F}(ξ_{0}Δ, ψ)(−1, ξ_{0}Δ)_{F}.  (5.14)

By Flicker’s formula on the elliptic set, we have that

Θπ̃((ξ_{0}ΔI; 1)g̃) = εs(ξ_{0}Δg)c(ξ_{0}ΔI, g)Θπ̃(ξ_{0}Δg; s(ξ_{0}Δg)^{−1})

= bεs(ξ_{0}Δg)c(ξ_{0}ΔI, g)\frac{D(δ′h)}{D(ξ_{0}Δg)}Θ_{π}(δ′h).

By (4.28) this becomes

Θπ̃((ξ_{0}ΔI; 1)g̃) = bεs(ξ_{0}Δg)c(ξ_{0}ΔI, g)Φ_{−}(g)Θ_{π}(δ′h).  (5.15)

Patching (5.14) and (5.15) together, we get

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θπ̃((ξ_{0}ΔI; 1)g̃) =

bεγ_{F}(ξ_{0}Δ, ψ)(−1, ξ_{0}Δ)_{F}s(ξ_{0}Δg)c(ξ_{0}ΔI, g)ν_{0}(ξ_{0}Δ)^{−1}Φ_{−}(g)Θ_{π}(δ′h).

We condense this using (2.38) and obtain

χ_{ν_{0}}(ξ_{0}ΔI; 1)^{−1}Θπ̃((ξ_{0}ΔI; 1)g̃) =
bεγ_{F}(ξ_{0}Δ, ψ)s(g)c_{β}(ξ_{0}ΔI, −g)ν_{0}(ξ_{0}Δ)^{−1}Φ_{−}(g)Θ_{π}(δ′h).

(5.16)

By (5.13) and (5.16), (5.10) becomes

Θ_{L(π,ν_{0})}(g̃) =\frac{b}{m}[εs(g)c_{β}(ξ_{0}I, −g)γ_{F}(ξ_{0}, ψ)Φ_{+}(g)ν_{0}(ξ_{0})^{−1}Θ_{π}(h)

+ εs(g)c_{β}(ξ_{0}ΔI, −g)γ_{F}(ξ_{0}Δ, ψ)Φ_{−}(g)ν_{0}(ξ_{0}Δ)^{−1}Θ_{π}(δ′h)].

Recall that 2b/m = 1, so that b/m = 1/2. Then

Θ_{L(π,ν_{0})}(g̃) =\frac{1}{2}[εs(g)c_{β}(ξ_{0}I, −g)γ_{F}(ξ_{0}, ψ)Φ_{+}(g)Θ_{πν^{−1}}(h)

+ εs(g)c_{β}(ξ_{0}ΔI, −g)γ_{F}(ξ_{0}Δ, ψ)Φ_{−}(g)ν_{0}(−1)Θ_{πν^{−1}}(δ′h)].

By Theorem 4.41,

Γ_{−}(g̃) = εs(g)c_{β}(ξ_{0}I, −g)γ_{F}(ξ_{0}, ψ)Φ_{+}(g)

and

Γ_{+}(g̃) = εs(g)c_{β}(ξ_{0}ΔI, −g)γ_{F}(ξ_{0}Δ, ψ)Φ_{−}(g).

This implies the following proposition. Proposition 5.17. Suppose that g ∈ S_{Δ} is regular elliptic and is not a square. Write ξ_{0}g = h^{2} with ξ_{0} = det h. Let ν_{0} be a character of F^{×} which satisfies χ_{π}(xI) = ν_{0}(x^{2}). Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2}[Γ_{−}(g̃)Θ_{πν^{−1}}(h) + ν_{0}(−1)Γ_{+}(g̃)Θ_{πν^{−1}}(δ′h)].

5.3 Recapitulation

The following theorem summarizes our work thus far. Theorem 5.18. Suppose that π is an irreducible representation of GL_{2}(F) such that χ_{π}(−I) = 1. Let ν_{0} be a character of F^{×} which satisfies χ_{π}(xI) = ν_{0}(x^{2}). Assume that g ∈ SL_{2}(F) and let g̃ = (g; ε).

1. Assume that g = diag(x, 1/x) is hyperbolic. Let h = diag(x, 1) and h′ = diag(−x, 1). Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{πν^{−1}}(h) + ν_{0}(−1)Γ_{+}(g̃)Θ_{πν^{−1}}(h′)] .

2. Assume that g = h^{2} ∈ S_{Δ} is elliptic and that det h = 1. Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{πν^{−1}}(h) + ν_{0}(−1)Γ_{+}(g̃)Θ_{πν^{−1}}(δ′h)] .

3. Assume that g = h^{2} ∈ S_{Δ} is elliptic and that det h = −1. Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [ν_{0}(−1)Γ_{+}(g̃)Θ_{πν^{−1}}(h) + Γ_{−}(g̃)Θ_{πν^{−1}}(δ′h)] .

4. Assume that g ∈ S_{Δ} is elliptic and is not a square. Write ξ_{0}g = h^{2} with ξ_{0} = det h. Then

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2}[Γ_{−}(g̃)Θ_{πν^{−1}}(h) + ν_{0}(−1)Γ_{+}(g̃)Θ_{πν^{−1}}(δ′h)].

Notice that the character of L(π, ν_{0}) always contains two terms. On the hyperbolic set, this is because the inversion formula contributes a single term while Flicker’s character identity contributes two. On the elliptic set, the inversion formula contributes two terms. But on the elliptic elements Flicker’s identity consists of only one term, so the total number of terms for the elliptic set is also two.

In Chapter 4, we derived character identities for the oscillator representation restricted to G̃L_{2}(F)_{+}. These are related to the above identities through the following corollary. Corollary 5.19. Let ω(μ) denote the oscillator representation of G̃L_{2}(F). Then

Θ_{L(μ,μ_{0})}(g̃) = μ_{0}(−1)Θ_{ω(μ)(μ_{0})}(g̃).

Proof. This follows from (4.10), (4.23), (4.38) and Theorem 5.18.

5.4 Pushing Down to SO_{1,2}(F): The Character

Formula

We would like to write our formulas in Theorem 5.18 in terms of characters of representations on SO_{1,2}(F). We will do this by “pushing” the representation πν^{−1} to a representation of SO_{1,2}(F) via the map p from Section 2.3. Definition 5.20. Suppose π is a representation of GL_{2}(F) with trivial central character. Define a representation π′ of SO_{1,2}(F) by pushing π down to SO_{1,2}(F) via p, i.e.,

Θ_{π}(g) = Θ_{π′}(p(g)).

From (2.8), we get

Θ_{π}(diag(a, b)) = Θ_{π′}(diag(a/b, b/a, 1)).  (5.21)

Suppose h ∈ S_{Δ}. Then h = ι(w) for some w ∈ F(\sqrt{Δ}). By Proposition 2.13 it follows that

Θ_{π}(h) = Θ_{π}(ιw) = Θ_{π′}(pιw) = Θ_{π′}(ι′φw).  (5.22)

Proposition 5.23. Suppose that π is a representation of GL_{2}(F) with trivial central character.

1. Suppose that g = diag(x, 1/x), h = diag(x, 1), and h′ = diag(−x, 1). Then

Θ_{π}(h) = Θ_{π′}(τ(g))  (5.24)

and

Θ_{π}(h′) = Θ_{π′}(τ(−g)).  (5.25)

2. Suppose that g = h^{2} and det h = ±1. Then

Θ_{π}(h) = Θ_{π′}(τ(±g))  (5.26)

and

Θ_{π}(δ′h) = Θ_{π′}(τ(∓g))  (5.27)

3. Suppose that g is not a square of an element in GL_{2}(F). Write ξ_{0}g = h^{2},

ξ_{0} = det h. Then

Θ_{π}(h) = Θ_{π′}(τ(g))  (5.28)

and

Θ_{π}(δ′h) = Θ_{π′}(τ(−g)).  (5.29)

Proof. Formulas (5.24) and (5.25) follow immediately from (5.21).

Suppose that g = h^{2} with det h = ±1. If h = ι(w) then det h = Nw. Consequently, by (5.22),

Θ_{π}(h) = Θ_{π′}(ι′φw)
= Θ_{π′}(ι′(w^{2}/Nw))
= Θ_{π′}(τ(g/ det h))
= Θ_{π′}(τ(±g))

and (5.26) follows. Also,

Θ_{π}(δ′h) = Θ_{π′}(ι′φδw)
= Θ_{π′}(ι′((δw)^{2}/N(δw)))
= Θ_{π′}(τ(Δg/ − Δ det h))
= Θ_{π′}(τ(∓g)).

So (5.27) follows.

Finally, suppose that ξ_{0}g = h^{2} with ξ_{0} = det h. Write h = ι(w). Note that ι(w^{2}/Nw) = g. By (5.22), we have

Θ_{π}(h) = Θ_{π′}(ι′φw)
= Θ_{π′}(ι′(w^{2}/Nw))
= Θ_{π′}(τ(g))

and (5.28) follows. The proof of (5.29) is similar to the proof of (5.27) and is omitted.

In the character formulas for L(π, ν_{0}), we required that χ_{π}(xI) = ν_{0}(x^{2}). Then πν^{−1} has trivial central character and the representation (πν^{−1})′ of SO_{1,2}(F) is defined. The main character identity which we have been seeking is contained in the following theorem. Theorem 5.30. Suppose that π is an irreducible representation of GL_{2}(F). Suppose that ν_{0} is a character of F^{×} satisfying χ_{π}(xI) = ν_{0}(x^{2}). Then for every regular semisimple g̃ ∈ S̃L_{2}(F),

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)) + ν_{0}(−1)Γ_{+}(g̃)Θ_{(πν^{−1})′}(τ(−g))] . (5.31)

Proof. This follows immediately by substituting the equalities from Proposition 5.23 (with π replaced by πν^{−1}) into the character identities we obtained in Theorem 5.18.

We could have stated the next corollary after Theorem 5.18, but it is more convenient to do so here. Corollary 5.32. Let β_{0} be a character of F^{×} satisfying β_{0}(−1) = 1. Then

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

In particular, for any character β_{0},

Θ_{L(πβ^{2},ν_{0}β_{0}^{2})} = Θ_{L(π,ν_{0})}.

Definition 5.33. Let R be the set of all pairs (π, ν_{0}) where π is an irreducible representation of GL_{2}(F) satisfying χ_{π}(−I) = 1 and ν_{0} is a character of F^{×} satisfying χ_{π}(xI) = ν_{0}(x^{2}) for all x ∈ F^{×}. We define an equivalence relation ∼ on R by (π, ν_{0}) ∼ (σ, λ_{0}) if there exists a character β_{0} with β_{0}(−1) = 1 such that σ = πβ and λ_{0} = ν_{0}β_{0}. We write

X = R/ ∼ .

Lemma 5.34. L(π, ν_{0}) = L(σ, λ_{0}) if and only if (π, ν_{0}) ∼ (σ, λ_{0}).

Proof. Suppose (π, ν_{0}) ∼ (σ, λ_{0}). Then L(σ, λ_{0}) = L(π, ν_{0}) by the character formula.

Conversely, suppose that L(π, ν_{0}) = L(σ, λ_{0}). By the condition on central characters, this implies that ν_{0}(−1) = λ_{0}(−1).

Recall that

Lift_{F}(π)|_{G̃L_{2}(F)_{+}} = ∑_{i=1}^{|F^{×}/F^{×2}|} π_{i}, Lift_{F}(π)|_{S̃L_{2}(F)} = ∑_{i=1}^{|F^{×}/F^{×2}|} τ_{i}

and that

Lift_{F}(σ)|_{G̃L_{2}(F)_{+}} = ∑_{i=1}^{|F^{×}/F^{×2}|} σ_{i}, Lift_{F}(σ)|_{S̃L_{2}(F)} = ∑_{j=1}^{|F^{×}/F^{×2}|} μ_{j}.

Our assumption shows that there exist i and j such that τ_{i} = μ_{j} and hence that ∑ τ_{i} = ∑ μ_{j}. Since G̃L_{2}(F)_{+} = Z(G̃L_{2}(F)_{+})S̃L_{2}(F), there exists a character β such that ∑ σ_{i} = (∑ π_{i}) ⊗ β, i.e., that Lift_{F}(σ) = Lift_{F}(π)β. But Lift_{F}(π)β = Lift_{F}(πβ^{2}). Since Flicker’s lifting is an injection, we have that σ = πβ^{2}. Hence L(π, ν_{0}) = L(πβ^{2}, ν_{0}) = L(π, ν_{0}β_{0}^{−2}). By the central character condition, we may find a quadratic character α_{0} such that β_{0}^{2} = α_{0}λ_{0}/ν_{0}. That is, setting γ = β^{2}, we can take σ = πγ, λ_{0} = α_{0}ν_{0}γ_{0}, and γ_{0}(−1) = 1. This implies that α_{0}(−1) = 1, since ν_{0}(−1) = λ_{0}(−1).

If α_{0} is trivial, then we are done. Otherwise, since

L(π, ν_{0}) = L(σ, λ_{0})
= L(πγ, λ_{0})
= L(π, λ_{0}γ_{0}^{−1})

and λ_{0}γ_{0}^{−1} = α_{0}ν_{0} we have reduced to the case L(π, ν_{0}) = L(π, ν_{0}α_{0}). As α is quadratic with α_{0}(−1) = 1, we get L(π, ν_{0}) = L(πα, ν_{0}). This in turn implies that L(π, μ_{0}) = L(πα, μ_{0}) for all μ_{0}, so that Lift_{F}(π) = Lift_{F}(πα). Since Flicker’s lifting is an injection, we get π = πα and (π, ν_{0}) ∼ (π, ν_{0}α_{0}) ∼ (σ, λ_{0}). This completes the proof.

The character formula obtained in Theorem 5.30 suggests that we examine the map

\begin{matrix}Irr(S̃L_{2}(F)) & → Irr(SO_{1,2}(F)) \\ L(π, ν_{0}) & ↦ (πν^{−1})′. \end{matrix}

This map is clearly surjective, since if π′ ∈ Irr(SO_{1,2}(F)), then π has trivial central character, and L(π, 1) ↦ π′. Fix a character β_{0} satisfying β_{0}(−1) = −1. Suppose that L(π, ν_{0}) and L(πβ, ν_{0}β_{0}) are both nonzero. In general, L(π, ν_{0}) and L(πβ, ν_{0}β_{0}) are inequivalent because they have different central characters. However, they both map to (πν^{−1})′. Therefore, this map is not injective. We remedy this situation in the next section when we “stabilize” (5.31).

5.5 Stable Representations of S̃L_{2}(F)

Theorem 5.35. Let β_{0} be a character of F^{×} satisfying β_{0}(−1) = −1. Then

Θ_{L(π,ν_{0})+L(πβ,ν_{0}β_{0})}(g̃) = Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)).  (5.36)

Proof. We apply Theorem 5.30 to (π, ν_{0}) and (πβ, ν_{0}β_{0}). In the second case, we replace ν with β^{−1}ν^{−1}, so that (πβ)(β^{−1}ν^{−1}) equals πν^{−1}. It follows that

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)) + ν_{0}(−1)Γ_{+}(g̃)Θ_{(πν^{−1})′}(τ(−g)).]
Θ_{L(πβ,ν_{0}β_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)) + β_{0}ν_{0}(−1)Γ_{+}(g̃)Θ_{(πν^{−1})′}(τ(−g)).]

= \frac{1}{2} [Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)) − ν_{0}(−1)Γ_{+}(g̃)Θ_{(πν^{−1})′}(τ(−g)).]

Adding the first and third equations gives the result. Definition 5.37. Suppose that π is an irreducible representation of GL_{2}(F) satisfying χ_{π}(−I) = 1. Let ν_{0} be a character of F^{×} such that χ_{π}(xI) = ν_{0}(x^{2}) for all x ∈ F^{×}. Fix a character β_{0} of F^{×} satisfying β_{0}(−1) = −1. Define

L_{st}(π, ν_{0}) = L(π, ν_{0}) + L(πβ, ν_{0}β_{0}).

As before, let R be the collection of pairs (π, ν_{0}) where π is an irreducible representation of GL_{2}(F) and ν_{0} is a character of F^{×} with the property that χ_{π}(xI) = ν_{0}(x^{2}). Let

Gr_{st}(S̃L_{2}(F)) = span{L_{st}(π, ν_{0}) : (π, ν_{0}) ∈ R}.

Any ρ ∈ Gr_{st}(S̃L_{2}(F)) will be called a stable virtual representation of S̃L_{2}(F). We repeat Theorem 5.35 in this notation:

Θ_{L_{st}(π,ν_{0})}(g̃) = Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)).

Definition 5.38. We define an equivalence relation ∼_{st} on R by (π, ν_{0}) ∼_{st} (σ, λ_{0}) if there exists a character β_{0} such that σ = πβ and λ_{0} = ν_{0}β_{0}. Set

X_{st} = R/∼_{st}.

Lemma 5.39. L_{st}(π, ν_{0}) = L_{st}(σ, λ_{0}) if and only if (π, ν_{0}) ∼_{st} (σ, λ_{0}).

Proof. Assume first that (π, ν_{0}) ∼_{st} (σ, λ_{0}). It follows immediately from Theorem 5.35 that L_{st}(π, ν_{0}) = L_{st}(σ, λ_{0}). Conversely, if L_{st}(π, ν_{0}) = L_{st}(σ, λ_{0}) then (5.36) implies that (πν^{−1})′ = (σλ^{−1})′. The pullback is unique, so πν^{−1} = σλ^{−1}. Therefore π = σλ^{−1}ν. Take β_{0} = λ_{0}^{−1}ν_{0} in the definition of ∼_{st} to conclude that (π, ν_{0}) ∼_{st} (σ, λ_{0}). Corollary 5.40. We have bijections

1. X_{st} ↔ {L_{st}(π, ν_{0})},

2. X_{st} ↔ Irr(SO_{1,2}(F)) given by (π, ν_{0}) ↦ (πν^{−1})′.

Proof. The obvious map X_{st} → {L_{st}(π, ν_{0})} gives the first bijection. The second follows from the first and Theorem 5.35. Corollary 5.41. The map

L_{st} : Gr(SO_{1,2}(F)) → Gr_{st}(S̃L_{2}(F))

defined by

L_{st}(π) = L_{st}(π ∘ p, 1)

is a bijection.

Properties of the Correspondence

In this chapter, we discuss the nature of the bijection

L_{st} : Gr(SO_{1,2}(F)) → Gr_{st}(S̃L_{2}(F)).

We begin with the full principal series representations of S̃L_{2}(F) and SO_{1,2}(F). Then we get the correspondence for one–dimensional representations.We conclude with the special and supercuspidal representation and after this we summarize the results.

6.1 Principal Series Representations

We explain how the principal series fit into our correspondence. The parameterization of the principal series of S̃L_{2}(F) and SO_{1,2}(F) is given in the following definition. (The parameterization of principal series of GL_{2}(F) and G̃L_{2}(F) has already been given in Section 2.6.)

Recall that

\begin{aligned}A & = \{diag(x, 1/x) : x ∈ F^{×}\}, \\ N & = \{(\begin{matrix}1 & y \\ 0 & 1 \end{matrix}) : y ∈ F\} , \\ B & = AN, \end{aligned}

and that B̃ = ÃN. We also write

\begin{aligned}A′ & = \{diag(a, 1) : a ∈ A\}, \\ N′ & = \{diag(n, 1) : n ∈ N\}, \\ B′ & = \{diag(b, 1) : b ∈ B\}. \end{aligned}

Definition 6.1. Let

λ̃(x; ε) = γ_{F}(x, ψ)ε;

this is a genuine character of F̃^{×} (see Proposition 3.14). Any genuine character of Ã is of the form ν̃_{0}(diag(x, 1/x); ε) = ν_{0}(x)λ̃(x; ε). Extend this trivially to N and define

P SS̃L(ν̃_{0}) = IndS̃L_{2}(F) ; B̃(ν̃_{0})

for the full principal series representation of S̃L_{2}(F) with parameter ν̃_{0}. Definition 6.2. Let ν_{0} be any character of F^{×}. We identify this with a character of A′ in the obvious way. We extend ν_{0} to N′ and write

P S_{SO}(ν_{0}) = Ind_{B′}^{SO_{1,2}(F)}(ν_{0})

for the full principal series representation of SO_{1,2}(F) with parameter ν_{0}.

Let D_{SO}, D_{SL}, and D_{GL} be the Weyl denominators for SO_{1,2}(F), SL_{2}(F), and GL_{2}(F) respectively.

Lemma 6.3. Let x ∈ F^{×} and put g = diag(x, 1/x), g̃ = (g; ε). Then

Γ_{±}(g̃) = \frac{D_{SO}(τ(∓g))}{D_{SL}(g)}γ(x, ψ)ε.

Proof. By Lemma 4.1, D_{SO}(τ(∓g))/D_{SL}(g) = | det(1∓g)|^{−1/2}. This, in conjunction with Theorem 4.41 completes the proof. Lemma 6.4. Suppose x ∈ F^{×} and put g = diag(x, 1/x), h = diag(x, 1), h′ = diag(−x, 1). Then

\frac{D_{SO}(τ(g))}{D_{GL}(h)} = \frac{D_{SO}(τ(−g))}{D_{GL}(h′)} = 1.

Proof. This follows immediately from the definitions of D_{SO} and D_{GL}.

We state the induced character formulas for GL_{2}(F), S̃L_{2}(F) and SO_{1,2}(F) in terms of our parameterization of their principal series. Let

w = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Proposition 6.5 (Induced Character Formula). We have the following formulas.

1. Let π(η_{1}, η_{2}) be a principal series representation of GL_{2}(F) (see Definition 2.51). Suppose g is conjugate to diag(x, y). Then

Θ_{π(η_{1},η_{2})}(diag(x, y)) = \frac{η_{1}(x)η_{2}(y) + η_{1}(y)η_{2}(x)}{D_{GL}(diag(x, y))}.

If g is not conjugate to diag(x, y) for some x and y then Θ_{π(η_{1},η_{2})}(g) = 0.

2. Suppose that h is conjugate to g = diag(x, 1/x) and let h̃ = (h; ε). Then

\begin{aligned}Θ_{P SS̃L(ν̃_{0})}(h̃) & = \frac{ν̃_{0}(g) + ν̃_{0}(wgw^{−1})}{D_{SL}(g)} \\ & = γ_{F}(x, ψ)ε\frac{ν_{0}(x) + ν_{0}^{−1}(x)}{D_{SL}(g)}, \end{aligned}

If h is not conjugate to g = diag(x, 1/x) for some x then Θ_{P SS̃L(ν̃_{0})}(h̃) = 0.

3. Suppose that h ∈ SO_{1,2}(F) is conjugate to τ(g). Then

\begin{aligned}Θ_{P S_{SO}(ν_{0})}(h) & = \frac{ν_{0}(τ(g)) + ν_{0}(τ(g)^{τ(w)})}{D_{SO}(g)} \\ & = \frac{ν_{0}(x) + ν_{0}^{−1}(x)}{D_{SO}(g)}. \end{aligned}

If h ∈ SO_{1,2}(F) is not conjugate to τ(g) for some g, then Θ_{P S_{SO}(ν_{0})}(h) = 0.

Corollary 6.6. For all g regular semisimple, g̃ = (g; ε),

Θ_{P SS̃L(ν̃_{0})}(g̃) = Γ_{−}(g̃)Θ_{P S_{SO}(ν_{0})}(τ(g)).  (6.7)

Proof. If g is not conjugate to an element of A, then both sides are 0. Otherwise, by the induced character formulas,

Θ_{P SS̃L(ν̃_{0})}(g̃) = \frac{D_{SO}(τ(g))}{D_{SL}(g)}γ_{F}(x, ψ)εΘ_{P S_{SO}(ν_{0})}(τ(g)).

Then

Θ_{P SS̃L(ν̃_{0})}(g̃) = Γ_{−}(g̃)Θ_{P S_{SO}(ν_{0})}(τ(g))

by Lemma 6.3.

Suppose that π(η_{1}, η_{2}) is a strongly even principal series representation of GL_{2}(F), so that Lift_{F}(π(η_{1}, η_{2})) is non–zero (see Proposition 2.68). This implies that there exist characters μ_{1} and μ_{2} of F^{×2} satisfying μ_{i}(x^{2}) = η_{i}(x) for all x ∈ F^{×}. Extend each μ_{i} arbitrarily to F^{×}. In order that L(π(η_{1}, η_{2}), ν_{0}) be defined, i.e., χ_{π(η_{1},η_{2})}(xI) = ν_{0}(x^{2}), we must have that

ν_{0}^{2} = η_{1}η_{2}.

This implies that there is a quadratic character α_{0} such that

ν_{0} = α_{0}μ_{1}μ_{2}.  (6.8)

Let g = diag(x, 1/x), g̃ = (g; ε), h = diag(x, 1), h′ = diag(−x, 1). Put π = π(η_{1}, η_{2}). By Theorem 5.18, we have

Θ_{L(π,ν_{0})}(g̃) = \frac{1}{2} [Γ_{−}(g̃)Θ_{π⊗ν^{−1}}(h) + ν_{0}(−1)Γ_{+}(g̃)Θ_{π⊗ν^{−1}}(h′)] .  (6.9)

By Theorem 5.18 and the induced character formula for GL_{2}(F), Θ_{L(π,ν_{0})}(g̃) = 0 for all other g̃ with g not conjugate to diag(x, 1/x) for some x. When we write our character formulas below, we assume that g̃ = (diag(x, 1/x); ε) and that the character formulas are zero for all other g̃ with g not conjugate to an element of this form. Proposition 6.10. Suppose that π = π(η_{1}, η_{2}) is a principal series such that η_{1} and η_{2} are even. Then

Θ_{L(π,ν_{0})}(g̃) = Γ_{−}(g̃)Θ_{(π⊗ν^{−1})′}(τ(g)).

That is, if π is a strongly even principal series representation of GL_{2}(F), we have

L(π, ν_{0}) = L_{st}(π, ν_{0})

Proof. By Lemma 6.3, we have that

ν_{0}(−1)Γ_{+}(g̃)Θ_{π⊗ν^{−1}}(h′) = ν_{0}(−1)\frac{D_{SO}(τ(−g))}{D_{SL}(g)}Θ_{π⊗ν^{−1}}(h′).

By the induced character formula, this becomes

\begin{aligned}ν_{0}(−1)Γ_{+}(g̃)Θ_{π⊗ν^{−1}}(h′) & = ν_{0}(−1)\frac{η_{1}(−x)ν_{0}^{−1}(−x) + η_{2}(−x)ν_{0}^{−1}(−x)}{D_{GL}(h′)} \\ & = γ_{F}(x, ψ)ε\frac{D_{SO}(τ(−g))}{D_{SL}(g)} \frac{η_{1}(−x)ν_{0}^{−1}(x) + η_{2}(−x)ν_{0}^{−1}(x)}{D_{GL}(h′)}. \end{aligned}

By assumption, η_{1}(−1) = η_{2}(−1) = 1. Furthermore, by Lemma 6.4 we have that D_{SO}(τ(−g))/D_{GL}(h′) = 1. Incorporating this information, we see

ν_{0}(−1)Γ_{+}(g̃)Θ_{π⊗ν^{−1}}(h′) = γ_{F}(x, ψ)ε\frac{η_{1}(x)ν_{0}^{−1}(x) + η_{2}(x)ν_{0}^{−1}(x)}{D_{SL}(g)}.

By Lemma 6.4, we have that D_{SO}(τ(g))/D_{SL}(g) = 1. Then the above equation becomes

ν_{0}(−1)Γ_{+}(g̃)Θ_{π⊗ν^{−1}}(h′) = γ_{F}(x, ψ)ε\frac{D_{SO}(τ(g))}{D_{SL}(g)} \frac{η_{1}(x)ν_{0}^{−1}(x) + η_{2}(x)ν_{0}^{−1}(x)}{D_{GL}(g)}.

By Lemma 6.3 and the induced character formula, we obtain

Γ_{−}(g̃)Θ_{π⊗ν^{−1}}(h) = ν_{0}(−1)Γ_{+}(g̃)Θ_{π⊗ν^{−1}}(h′).

Then (6.9) implies that

Θ_{L(π,ν_{0})}(g̃) = Γ_{−}(g̃)Θ_{πν^{−1}}(h) = Γ_{−}(g̃)Θ_{(πν^{−1})′}(τ(g)).

The last expression equals Θ_{L_{st}(π,ν_{0})}. This completes the proof. Lemma 6.11. Let μ_{0} any character of F^{×} for which L(π(η_{1}, η_{2}), μ_{0}) is defined. Then

L(π(η_{1}, η_{2}), μ_{0}) = L(π(η_{1}η_{2}^{−1}, 1), μ_{0}η_{2}^{−1}).

Proof. Write η for η_{2} ∘ det. The computation

(η_{1}, η_{2})η^{−1}(diag(x, y)) = η_{1}(x)η_{2}(y)η_{2}^{−1}(xy) = η_{1}η_{2}^{−1}(x)

shows that π(η_{1}, η_{2})η^{−1} = π(η_{1}η_{2}^{−1}, 1). By the remarks after Definition 5.33, the lemma follows. Lemma 6.12. Let ν_{0} = α_{0}μ_{1}μ_{2} be as in (6.8). Then

(π(η_{1}, η_{2}) ⊗ ν^{−1})′ = P S_{SO}(ν_{0}/η_{2}) = P S_{SO}(η_{1}/ν_{0}).

Remark. Θ_{P S_{SO}(λ_{1})} = Θ_{P S_{SO}(λ_{2})} if and only if λ_{2} = λ_{1} or λ_{1}^{−1}.

Proof. We compute

\begin{aligned}(η_{1}, η_{2})ν^{−1}(diag(x, y)) & = η_{1}(x)η_{2}(y)ν_{0}^{−1}(xy) \\ & = η_{1}(x)η_{2}(y)α_{0}^{−1}(xy)μ_{1}^{−1}(xy)μ_{2}^{−1}(xy). \end{aligned}

Recall that μ_{1}(x^{2}) = η_{1}(x) and μ_{2}(x^{2}) = η_{2}(x) and that α_{0} is quadratic. We get

(η_{1}, η_{2})ν^{−1}(diag(x, y)) = α_{0}(x/y)μ_{1}(x/y)μ_{2}(y/x)
= α_{0}μ_{1}/μ_{2}(x/y)
= ν_{0}/η_{2}(x/y) = η_{1}/ν_{0}(x/y).

This completes the proof. Corollary 6.13.

Θ_{L_{st}(π(η_{1},η_{2}),ν_{0})}(g̃) = Γ_{−}(g̃)Θ_{P S_{SO}(ν_{0}/η_{2})}(τ(g)).

Hence

P SS̃L({ \atop ν_{0}/η_{2}}) = L(π(η_{1}, η_{2}), ν_{0}).

Proof. Since μ_{0} is a character of the form given in (6.8), Lemma 6.12 in conjunction with Proposition 6.10 implies that

Θ_{L(π(η_{1},η_{2}),ν_{0})}(g̃) = Θ_{L(π(η_{1}η_{2}^{−1},1),ν_{0}η_{2}^{−1})}(g̃) = Γ_{−}(g̃)Θ_{P S_{SO}(ν_{0}η_{2}^{−1})}(τ(g)).

Corollary 6.14.

L_{st}(P S_{SO}(ν_{0})) = L(π(ν_{0}^{2}, 1), ν_{0}).

Proof. By Lemma 6.12, π(ν_{0}^{2}, 1)ν^{−1} is P S_{SO}(ν_{0}).

From these two corollaries, we easily deduce the following statement.

Corollary 6.15. Let π be a strongly even principal series of GL_{2}(F). Then the correspondence

L(π, ν_{0}) ↔ (πν^{−1})′

is

P SS̃L(ν̃_{0}) ↔ P S_{SO}(ν_{0}).

Remark. If π = π(η_{1}, η_{2}) is a principal series of GL_{2}(F) with the η_{i} both even, then

L_{st}(π(η_{1}, η_{2}), ν_{0}) = L(π(η_{1}, η_{2}), ν_{0}).

Suppose that η_{1} and η_{2} are odd. If β_{0} is a character of F^{×} with β_{0}(−1) = −1,

L_{st}(π(η_{1}, η_{2}), ν_{0}) = L(π(η_{1}β_{0}, η_{2}β_{0}), ν_{0}β_{0}).

If one of the η_{i} is even and the other is odd, then L_{st}(π(η_{1}, η_{2}), ν_{0}) is zero. However, in order to get all principal series of S̃L_{2}(F) it suffices to consider L(π(η_{1}, η_{2}), ν_{0}) with η_{1} and η_{2} both even.

6.2 One Dimensional Representations and the

Dependence on ψ

Suppose that μ′ is a one dimensional representation of SO_{1,2}(F). Then μ is a one dimensional representation of GL_{2}(F) with trivial central character. That is, there is a character μ_{0} of F^{×} such that μ = μ_{0} ∘ det and μ(xI) = 1 for all x ∈ F^{×}. This in turn implies that μ_{0}(x^{2}) = 1 for all x ∈ F^{×}, i.e., that μ_{0} is a quadratic character. Then there exists y such that μ_{0}(x) = q_{y}(x) for all x ∈ F^{×}. Here q_{y}(x) = (x, y)_{F} as in Proposition 3.19. Hence the one dimensional representations of SO_{1,2}(F) are parameterized by y ∈ F^{×}/F^{×2}. We put Q_{y} = q_{y} ∘ det and write Q′_{y} for the corresponding one–dimensional representation of SO_{1,2}(F).

In Chapter 3, we fixed an additive character ψ of F. The maps L_{st}, L_{st}, Γ_{−}, Γ_{+} and χ_{ν_{0}} all depended on ψ. We will now vary the choice of ψ, and we write L_{st}^{ψ}, L_{st}^{ψ}, Γ_{+}^{ψ}, Γ_{−}^{ψ} and χ_{ν_{0}}^{ψ} to indicate the dependence on ψ.

The following proposition lists some formal properties showing the role of the choice of the additive character ψ. At this point we refer the reader to Sections 2.5 and 3.3 where we have established the relevant notation. Proposition 6.16. We have that χ_{ν_{0}}^{ψ_{y}} = χ_{ν_{0}q_{y}}^{ψ} and

L_{st}^{ψ}(π, ν_{0})^{y} = L_{st}^{ψ}(π, ν_{0}q_{y}) = L_{st}^{ψ}(πQ_{y}, ν_{0}) = L_{st}^{ψ_{y}}(π, ν_{0})

The relationships between Γ_{±}^{ψ} and Γ_{±}^{ψ_{y}} are given in the next two lemmas. Lemma 6.17. Suppose g ∈ SL_{2}(F).

1. Assume that g = diag(x, 1/x) is hyperbolic. Then for each y ∈ F^{×}, q_{y}(x)Γ_{+}^{ψ}(g̃) = Γ_{+}^{ψ_{y}}(g̃).

2. Assume that g = h^{2} ∈ S_{Δ} is elliptic and that det h = 1. Then for each

y ∈ F^{×},

q_{y}(Δ)Γ_{+}^{ψ}(g̃) = Γ_{+}^{ψ_{y}}(g̃).

3. Assume that g = h^{2} ∈ S_{Δ} is elliptic and that det h = −1. Then for each

y ∈ F^{×},

Γ_{+}^{ψ}(g̃) = Γ_{+}^{ψ_{y}}(g̃).

4. Assume that g ∈ S_{Δ} is elliptic and is not a square. Write ξ_{0}g = h^{2} with

ξ_{0} = det h. Then for each y ∈ F^{×},

q_{y}(ξ_{0}Δ)Γ_{+}^{ψ}(g̃) = Γ_{+}^{ψ_{y}}(g̃).

Proof. This follows from the fact that

q_{y}(x)γ_{F}(x, ψ) = γ_{F}(x, ψ_{y})

(see Section 2.5). Suppose that we are in case 1. By Theorem 4.41, we have that

q_{y}(x)Γ_{+}^{ψ}(g̃) = \frac{q_{y}(x)γ_{F}(x, ψ)ε}{| det(1 − g)|^{1/2}}

= \frac{γ_{F}(x, ψ_{y})ε}{| det(1 − g)|^{1/2}}

= Γ_{+}^{ψ_{y}}(g̃).

For case 2, we have that

q_{y}(Δ)Γ_{+}^{ψ}(g̃) = \frac{εs(g)(y, Δ)_{F}γ_{F}(Δ, ψ)c_{β}(ΔI, −g)}{| det(1 − g)|^{1/2}}

= \frac{εs(g)γ_{F}(Δ, ψ_{y})c_{β}(ΔI, −g)}{| det(1 − g)|^{1/2}}

= Γ_{+}^{ψ_{y}}(g̃).

By Theorem 4.41, case 3 does not depend on ψ so the assertion is clear. The proof of case 4 is similar to the proofs of cases 1 and 2 and is omitted. Lemma 6.18. Suppose g ∈ SL_{2}(F).

1. Assume that g = diag(x, 1/x) is hyperbolic. Then for each y ∈ F^{×},

q_{y}(x)Γ_{−}^{ψ}(g̃) = Γ_{−}^{ψ_{y}}(g̃).

2. Assume that g = h^{2} ∈ S_{Δ} is elliptic and that det h = 1. Then for each

y ∈ F^{×},

Γ_{−}^{ψ}(g̃) = Γ_{−}^{ψ_{y}}(g̃).

3. Assume that g = h^{2} ∈ S_{Δ} is elliptic and that det h = −1. Then for each

y ∈ F^{×},

q_{y}(Δ)Γ_{−}^{ψ}(g̃) = Γ_{−}^{ψ_{y}}(g̃).

4. Assume that g ∈ S_{Δ} is elliptic and is not a square. Write ξ_{0}g = h^{2} with ξ_{0} = det h. Then for each y ∈ F^{×},

q_{y}(ξ_{0})Γ_{−}^{ψ}(g̃) = Γ_{−}^{ψ_{y}}(g̃).

Proof. The proof is the same as the proof of the previous lemma. Suppose that we are in case 1. By Theorem 4.41, we have that

q_{y}(x)Γ_{−}^{ψ}(g̃) = \frac{q_{y}(x)γ_{F}(x, ψ)ε}{| det(1 + g)|^{1/2}}

= \frac{γ_{F}(x, ψ_{y})ε}{| det(1 + g)|^{1/2}}

= Γ_{−}^{ψ_{y}}(g̃).

By Theorem 4.41, case 2 does not depend on ψ so the assertion is clear. Cases 3 and 4 follow similarly by the argument for case 1 and their proofs are omitted. Proposition 6.19. Let ν_{0} be a character of F^{×}. Then

L^{ψ}(ν, ν_{0}) = \begin{cases}ω_{e}(ψ) & if ν_{0}(−1) = 1, \\ −ω_{o}(ψ) & if ν_{0}(−1) = −1. \end{cases}

Proof. Assume β_{0} is a character of F^{×} satisfying β_{0}(−1) = −1. We have

Θ_{L^{ψ}(ν,ν_{0})}(g; ε) + Θ_{L^{ψ}(νβ,ν_{0}β_{0})}(g; ε) = Θ_{ω_{e}(ψ)}(g; ε) − Θ_{ω_{o}(ψ)}(g; ε).  (6.20)

We also have

Θ_{L^{ψ}(ν,ν_{0})}(−g; ε) + Θ_{L^{ψ}(νβ,ν_{0}β_{0})}(−g; ε) = Θ_{ω_{e}(ψ)}(−g; ε) − Θ_{ω_{o}(ψ)}(−g; ε).

Recall that (−g; ε) = (−I; 1)(g; εc(−I, g)). So

\begin{matrix}γ_{F}(−1, ψ)c(−I, g)ν_{0}(−1) & (Θ_{L^{ψ}(ν,ν_{0})}(g; ε) − Θ_{L^{ψ}(νβ,ν_{0}β_{0})}(g; ε)) = \\ γ_{F}(−1, ψ)c(−I, g)(Θ_{ω_{e}(ψ)}(g; ε) + Θ_{ω_{o}(ψ)}(g; ε)), \end{matrix}

i.e.,

ν_{0}(−1)(Θ_{L^{ψ}(ν,ν_{0})}(g; ε) − Θ_{L^{ψ}(νβ,ν_{0}β_{0})}(g; ε)) = Θ_{ω_{e}(ψ)}(g; ε) + Θ_{ω_{o}(ψ)}(g; ε). (6.21)

If ν_{0}(−1) = 1, then adding (6.20) and (6.21) yields L^{ψ}(ν, ν_{0}) = ω_{e}(ψ). On the other hand if ν_{0}(−1) = −1, then subtracting (6.21) from (6.20) yields L^{ψ}(ν, ν_{0}) = −ω_{o}(ψ). Remark. This is consistent with (4.9), (4.22), and (4.37). Note that this could have been proved using the computations given in Lemmas 6.17 and 6.18.

The next corollary gives the correspondence for one–dimensional representations. Corollary 6.22. For each y ∈ F^{×}, we have that

L_{st}^{ψ}(Q′_{y}) = ω_{e}(ψ_{y}) − ω_{o}(ψ_{y}).

Proof. This is obvious from Proposition 6.19 because

L_{st}^{ψ}(1, q_{x}) = L^{ψ}(1, q_{x}) + L^{ψ}(ν, ν_{0}q_{x})

with ν_{0}(−1) = −1.

We also have the following decompositions which are already well known (see Gelbart and Piatetski–Shapiro [10]). Corollary 6.23. Let μ_{0} be a character of F^{×} such that μ_{0}(−1) = 1. Then we have

ω(μ)|_{S̃L_{2}(F)} = ∑_{x∈F^{×}/F^{×2}} ω_{e}(ψ_{x}).

If μ′_{0}(−1) = −1 then

ω(μ′)|_{S̃L_{2}(F)} = ∑_{x∈F^{×}/F^{×2}} ω_{o}(ψ_{x}).

Proof. The first assertion follows from Proposition 3.19. The second assertion follows easily from the previous corollary.

6.3 Special and Supercuspidal Representations

6.3.1 Special Representations

A reducible principal series on GL_{2}(F) takes the form π(λ| · |^{1/2}, λ| · |^{−1/2}) (see Definition 2.51). Thus the reducible principal series with trivial central character are in one to one correspondence with π(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}).

Let σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}) and μ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}) be the associated special and one–dimensional representations respectively, and σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2})′ be the push–down of σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}) to SO_{1,2}(F). Proposition 6.24.

\begin{aligned}L_{st}(σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2})′) & = P SS̃L({ \atop q_{x}| · |^{1/2}}) − ω_{e}(ψ_{x}) + ω_{o}(ψ_{x}) \\ & = L_{st}(π(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}), q_{x}) − L_{st}(1, q_{x}) \end{aligned}

Proof. By Jacquet–Langlands [12], p. 275, we have that

Θ_{π(q_{x}|·|^{1/2},q_{x}|·|^{−1/2})} = Θ_{μ(q_{x}|·|^{1/2},q_{x}|·|^{−1/2})} + Θ_{σ(q_{x}|·|^{1/2},q_{x}|·|^{−1/2})}.

We have shown that π(q_{x}|·|^{1/2}, q_{x}|·|^{−1/2}) pushes down to P S_{SO}(q_{x}|·|^{1/2}) and that this corresponds with P SS̃L({ \atop q_{x}| · |^{1/2}}). On the other hand, Jacquet–Langlands have shown (see [12], p. 102) that μ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}) is equivalent to the one dimensional representation Q_{x} = q_{x} ∘det. Thus μ(q_{x}|·|^{1/2}, q_{x}|·|^{−1/2}) pushes down to Q′_{x}. We have shown that L_{st}(Q′_{x}) = ω_{e}(ψ_{x}) − ω_{o}(ψ_{x}). Hence

L_{st}(σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2})′) = P SS̃L({ \atop q_{x}| · |^{1/2}}) − ω_{e}(ψ_{x}) + ω_{o}(ψ_{x}),

i.e.,

L_{st}(σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2})′) = L_{st}(π(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2}), q_{x}) − L_{st}(1, q_{x}).

The reducible principal series P SS̃L({ \atop q_{x}| · |^{1/2}}) of S̃L_{2}(F) is an even oscillator representation plus a special representation. If σ̃(q_{x}| · |^{1/2}) denotes this special

representation, then P SS̃L({ \atop q_{x}| · |^{1/2}}) − ω_{e}(ψ_{x}) + ω_{o}(ψ_{x}) = ω_{o}(ψ_{x}) + σ̃(q_{x}| · |^{1/2}).

Corollary 6.25.

L_{st}(σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2})′) = ω_{0}(ψ_{x}) + σ̃(q_{x}| · |^{1/2}).

In particular, a special representation of SO_{1,2}(F) corresponds with a a sum of a supercuspidal and a special representation of S̃L_{2}(F).

6.3.2 Supercuspidal Representations

Suppose π′ is an irreducible supercuspidal representation of SO_{1,2}(F). Then π is an irreducible supercuspidal representation of GL_{2}(F) and Flicker’s lifting Lift_{F}(π) is supercuspidal by Theorem 2.67. Its restriction to S̃L_{2}(F) breaks up into a sum of supercuspidal representations. Proposition 6.26. Suppose π′ is an irreducible supercuspidal representation of SO_{1,2}(F). Then L_{st}(π′) is the sum of two supercuspidal representations.

6.4 Summary

Theorem 6.27. Consider the correspondence

\begin{matrix}L_{st}^{ψ} : Gr(SO_{1,2}(F)) & ↔ Gr_{st}(S̃L_{2}(F)) \\ (π ⊗ ν^{−1})′ & ↔ L_{st}^{ψ}(π, ν_{0}). \end{matrix}

This satisfies:

• Principal series on S̃L_{2}(F) correspond to principal series on SO_{1,2}(F). That is, L_{st}^{ψ}(P S_{SO}(ν_{0})) = P SS̃L(ν̃_{0}).

• The one–dimensional representation Q′_{x} of SO_{1,2}(F) corresponds with the difference of the two halves of the oscillator representation of S̃L_{2}(F) attached to ψ_{x}. That is, L_{st}^{ψ}(Q′_{x}) = ω_{e}(ψ_{x}) − ω_{o}(ψ_{x}).

• A special representation on SO_{1,2}(F) corresponds with the sum of an odd oscillator representation and an odd special representation of S̃L_{2}(F). More precisely, L_{st}^{ψ}(σ(q_{x}| · |^{1/2}, q_{x}| · |^{−1/2})′) = ω_{o}(ψ_{x}) + σ̃(q_{x}| · |^{1/2}).

• If π′ is an irreducible supercuspidal representation of SO_{1,2}(F), then L_{st}^{ψ}(π′) is the sum of two supercuspidal representations of S̃L_{2}(F).

BIBLIOGRAPHY

[1] J. Adams, Characters of non–linear groups, Preprint.

[2] , The character of the oscillator representation, Israel. J. Math. 98 (1997), 229–252.

[3] , Lifting of characters on orthogonal and metaplectic groups, Duke Math. J. 92 (1998), no. 1, 129–178.

[4] C. Blondel, Les représentations supercuspidales de groupes métaplectiques sur GL(2) et leurs caractères, Mémoire de la Société Mathématique de France 113 (1985), 1–119.

[5] T. Bröcker and T. tom Dieck, Representation theory of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer–Verlag, 1985.

[6] Y. Flicker, Automorphic forms on covering groups of GL(2), Inventiones math. 79 (1980), 119–182.

[7] Y. Flicker and D.A. Kazhdan, Metaplectic correspondences, Publ. Math. I.H.E.S. 64 (1986), 53–110.

[8] S. Gelbart, Automorphic forms on adele groups, Annals of Mathematics Studies, vol. 83, Princeton University Press, 1975.

[9] , Weil’s representation and the spectrum of the metaplectic group, Lecture Notes in Mathematics, vol. 530, Springer–Verlag, 1976.

[10] S. Gelbart and I. Piatetski-Shapiro, Distinguished representations and modular forms of half-integral weight, Inventiones math. 59 (1980), 145–188.

[11] , Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger, Israel. Journ. of Math. 44 (1983), no. 2, 97–126.

[12] H. Jacquet and R. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114, Springer–Verlag, 1970.

[13] Harish–Chandra, Harmonic analysis on reductive p–adic groups, Lecture Notes in Mathematics, vol. 162, Springer–Verlag, 1970.

[14] , Admissible invariant distributions on reductive p–adic groups, Queen’s Papers in Pure and Applied Math. 48 (1978), 281–347.

[15] R. Howe, The character of the oscillator representation, Preprint.

[16] , θ–series and invariant theory, Proceedings of Symposia in Pure Mathematics, vol. 33, AMS, 1979, pp. 275–285.

[17] N. Jacobson, Basic Algebra I, W.H. Freeman and Co., 1985.

[18] J.P. Labesse and R.P. Langlands, L–indistinguishability for SL(2), Can. J. Math. 31 (1979), 726–785.

[19] D.A. Kazhdan and S.J. Patterson, Metaplectic forms, Publ. Math. I.H.E.S. 59 (1984), 35–142.

[20] , Towards a generalized Shimura correspondence, Adv. in Math. 60 (1986), 161–234.

[21] A. Knapp and S. Gelbart, L-indistinguishability and R groups for the special linear group, Adv. in Math. 43 (1982), no. 2, 101–121.

[22] T. Kubota, Topological covering of SL(2) over a local field, J. Math. Soc. Japan 19 (1967), no. 1, 114–121.

[23] , On automorphic functions and the reciprocity law in a number field, Kinokuniya Book–Store Co., Ltd., 1969.

[24] D. Manderscheid, Supercuspidal duality for the two–fold cover of SL(2) and the split O(3), Ph.D. Thesis, Yale University, 1981.

[25] C. Moore, Extensions and low dimensional cohomology theory of locally compact groups, Trans. Amer. Math. Soc. 113 (1964), 40–86.

[26] , Group extensions of p–adic and adelic linear groups, Publ. Math. I.H.E.S. 35 (1968), 157–222.

[27] S. Niwa, Modular forms of half integral weight and the integral of certain theta–functions, Nagoya Math. J. 56 (1974), 147–161.

[28] I. Piatetski-Shapiro, Work of Waldspurger, Lie Group representations II, vol. 1041, Springer Lecture Notes, 1984, pp. 280–302.

[29] D. Prasad, On the theta correspondence, Preprint.

[30] S. Rallis and G. Schiffman, Représentations supercuspidales du group métaplectique, J. Math. Kyoto Univ. 17 (1977), no. 3, 567–603.

[31] R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pac. J. Math. 157 (1993), no. 2, 335–371.

[32] J.P. Serre, A course in arithmetic, Graduate Texts in Mathematics, vol. 7, Springer–Verlag, 1973.

[33] , Local fields, Graduate Texts in Mathematics, vol. 67, Springer– Verlag, 1979.

[34] G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440–481.

[35] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126.

[36] A. Silberger, Introduction to harmonic analysis on reductive p–adic groups, Mathematical Notes, vol. 23, Princeton University Press, 1979.

[37] M. Tádic, Notes on representations of non–archimedean SL(n), Pacific J. Math. 152 (1992), no. 2, 375–396.

[38] P. Torasso, Sur le caractère de la représentation de Shale–Weil de Mp(nR) et Sp(n, C), Math. Ann. 252 (1980), 53–86.

[39] J. L. Waldspurger, Correspondance de Shimura, J. Math. pures et appl. 59 (1980), 1–133.

[40] , Sur les coefficients de Fourier des formes modulaires de poids demi– entier, J. Math. pures et appl. 60 (1981), 375–484.

[41] , Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219–307.

[42] A. Weil, Sur certaines groupes d’opérateurs unitaires, Acta Math. 11 (1964), 143–211.