A note on the Global Langlands Conjecture

Erez M. Lapid

Received: April 4, 1998

Communicated by Don Blasius

Abstract. The theory of base change is used to give some new examples of the Global Langlands Conjecture. The Galois representations involved have solvable image and are not monomial, although some multiple of them in the Grothendieck group is monomial. Thus, it gives nothing new about Artin’s Conjecture itself. An application is given to a question which arises in studying multiplicities of cuspidal representations of SL_{n}. We explain how the (conjectural) adjoint lifting can prove GLC for a family of representations containing the tetrahedral 2-dimensional ones.

1991 Mathematics Subject Classification: Primary 11R39; Secondary 11F70, 22E55.

1 Introduction

The Global Langlands Conjecture asserts that for any n-dimensional irreducible representation σ of the absolute Galois group (or more generally, the Weil group) of a number field F, there corresponds a cuspidal representation π of GL_{n}(𝔸_{F}) with matching Langlands parameters almost everywhere. Specifically, if π_{v} = Ind_{B}^{G}(|·|^{s_{1}}, …, |·|^{s_{n}}) then the condition is that

σ(F r_{v}) ∼ (\begin{matrix}q_{v}^{−s_{1}} \\ & … \\ & & q_{v}^{−s_{n}} \end{matrix} ) .

Such a cuspidal representation π is unique by virtue of strong multiplicity one. This conjecture implies that the (partial) L-functions of σ and π are the same, and hence Artin’s Conjecture for σ is true. The case of n = 1 is essentially global class field theory. In ([L]) Langlands observed that the theory of base change, initiated by Saito and Shintani, can lead to the GLC for some irreducible 2-dimensional Galois representations with solvable image. This was extended to all such representations in [Tu]. Later, Arthur and Clozel ([AC]) proved cyclic base change for GL_{n}. Thus, the GLC is preserved under induction from a cyclic extension. In particular, it holds for representations of the type

Ind_{W_{E}}^{W_{F}} θ, F ⊂_{sc} E, θ a Hecke character of E,  (∗)

where F ⊂_{sc} E means that the extension is obtained by a series of cyclic extensions. It is then a consequence of results of Jacquet and Shalika ([JS]) to deduce GLC for irreducible representations which are linear combinations over ℤ in the Grothendieck group of representations of the above type. Unfortunately, by a result of Dade ([Da]), such representations are themselves of type (∗). The purpose of this note is to indicate other cases for which the GLC can be proved, using base change. In all our cases the representation is a linear combination over ℚ of representations of type (∗). Again, by modifying the result of Dade, some multiple of such a representation is monomial, so that Artin’s Conjecture is automatically satisfied for it. However, it is not necessarily monomial itself. The prototype of the new examples is an irreducible n-dimensional representation σ, factoring through a group G, whose image in P GL_{n}(ℂ) has order n^{2} (which is the least possible). Equivalently, nσ = Ind_{Z}^{G} ζ in the Grothendieck group where Z is the center of G and ζ is the central character of σ. We call such representations minimal. If G is solvable then we know that there exists an automorphic representation Π of GL_{n^{2}}(F) induced from cuspidals corresponding to Ind_{Z}^{G} ζ. If we can show that Π is “isotypic”, i.e. it is induced from π ⊗ ⋯ ⊗ π for π on GL_{n}(F), then π ↔ σ and GLC is valid for σ. This is not immediate; if Π = ⊞m_{i}π_{i} is the “decomposition” of Π then a standard argument of L-functions gives us that ∑ m_{i}^{2} = n^{2}. However, this is not enough to conclude that there is only one summand. On the other hand, there are lucky situations where it is not difficult to prove the “purity” of Π, even though σ may not be itself monomial. The main point is that base change and automorphic induction can sometime be used to provide analogues in the automorphic side of classical results from representation theory of finite groups, such as Frobenius reciprocity and Clifford theory. Even if we limit ourselves to the solvable case this does not always work, because we only have the constructions for normal subgroups. The following is a typical case of our main result.

Theorem 1. Let σ be a minimal representation factoring through G. Suppose that either

1. G/Z has a composition series whose quotients have pairwise coprime orders (Theorem 4), or,

2. mσ = Ind_{N}^{G} τ in R(G) where τ is minimal, N is normal in G, m^{2} = |G/N|, and both N and G/N are nilpotent (Theorem 5).

Then σ is automorphic.

As an application, we address a question which was posed in [La]. The problem is pertaining to calculating multiplicities of representations of SL(n) in the cuspidal spectrum. There is a heuristic analogue of the multiplicity formula of Labesse-Langlands for L-packets coming from representations of the Weil group. One would like to compare the two formulas. Our result here is that the comparison is valid for L-packets coming from irreducible representations of W_{F} induced from a Hecke character on E where F ⊂ E is nilpotent.

In the last section, we focus on the simplest representations for which Artin’s Conjecture is not known. These are higher dimensional analogues of the tetrahedral 2-dimensional representations. In more detail, for any prime power q we consider q-dimensional irreducible representations σ whose image in P GL_{q}(ℂ) is isomorphic to 𝔽_{q} ⊕ 𝔽_{q} ⋊ B where B ≤ SL_{2}(𝔽_{q}) is solvable and q |̸|B|. The argument in [L] that proves GLC in the case q = 2 carries over to the general case, provided that we know that the adjoint lifting from automorphic representations on GL_{q} to GL_{q^{2}−1} exists. While this is far from being proved, even for q = 3, the example is given as an illustration of a “simple” case to stare at, while working on Artin’s Conjecture.

1.1 Notations and preliminaries

Throughout this paper 𝔸 = 𝔸_{F} (resp. 𝕀_{F}) will denote the ring of adeles (resp. group of ideles) over a number field F, W_{F} is the Weil group of F. For a cyclic extension F ⊂ E we let ω_{E/F} be a character of 𝕀_{F} with kernel F^{∗} Nm_{F}^{E} 𝕀_{E}. An extension F ⊂ E is called subcyclic (written F ⊂_{sc} E) if there exists a sequence of cyclic extensions F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = E. For any extension F ⊂ E we let Γ = Γ_{E/F} be the homogeneous space (or the group in the normal case) Gal(F̄/F)/ Gal(F̄/E).

For any group G, let Irr_{n}(G) be the set of equivalence classes of n-dimensional irreducible representations of G. Let also Irr(G) = ⋃_{n} Irr_{n}(G). We write R(G) for the Grothendieck group of the category of finite dimensional representations of G. Let c : R(G) × R(G) → ℤ be the canonical pairing. The determinant character of σ ∈ Irr(G) will be denoted by χ_{σ}. We will often encounter the follow situation. Suppose that H is a normal subgroup of G of finite index and σ ∈ Irr(H). We let (G/H)(σ) = {g ∈ G/H : σ^{g} ≃ σ}. Suppose that G/H(σ) = G/H. Choose a transversal {g_{x}}_{x∈G/H} and let A_{x} : V_{σ} → V_{σ} be intertwining operators between (σ, V_{σ}) and (σ^{g_{x}}, V_{σ}). The cocycle given by

α_{σ}(x, y) = A_{x}A_{y}A_{xy}^{−1}σ(g_{x}g_{y}g_{xy}^{−1})^{−1} ∈ ℂ^{∗}  (1)

defines an element in the Schur multipliers of G/H, which depends only on σ. It is the obstruction to extending σ to G. We have End_{G}[Ind_{H}^{G} σ] ≃ ℂ[G/H, α_{σ}] where the latter is the twisted group algebra of G/H. In particular, Ind_{H}^{G} σ is isotypic if and only if ℂ[G/H, α_{σ}] is simple. For the central character we have

α_{χ_{σ}} = α_{σ}^{m}  (2)

where m = deg σ.

We will deal with automorphic representations π of GL_{n} which are induced from cuspidals, i.e. there exists a parabolic P of type (n_{1}, …, n_{r}) and a cuspidal representation ⊗π_{i} of M_{P} = GL_{n_{1}}(F) × ⋯ × GL_{n_{r}}(F) such that π = ⊞π_{i} = Ind_{P (𝔸)}^{G(𝔸)} ⊗ π_{i}. By the results of Jacquet-Shalika ([JS]) the π_{i}’s are uniquely determined and we call them the components of π. We will denote by Cusp_{n}(F) the set of cuspidal representations of GL_{n}(F). Let also Cusp(F) = ⋃ Cusp_{n}(F). Let R_{cusp}(F) be the semigroup of automorphic representations induced from cuspidal representations of GL_{n}(F). If π = ⊞π_{i}, τ = ⊞τ_{j} are the decompositions of π, τ ∈ R_{cusp}(F), we let

c(π, τ) = #{(i, j) : π_{i} ≃ τ_{j}}.

Then c(π, τ) is the order of the pole at s = 1 of the partial Jacquet-Shalika Lfunction L^{S}(π ⊗τ^{∨}, s) where τ^{∨} is the contragredient of τ ([JS]). We call π, τ disjoint if c(π, τ) = 0. Similarly to the notations above, if π ∈ R_{cusp}(E) and F ⊂ E is normal we let Γ(π) = {γ ∈ Γ_{E/F} : π^{γ} ≃ π}. Also, the central character of an automorphic representation π will be denoted by χ_{π}. If π ∈ Cusp_{n}(F) and σ ∈ Irr_{n}(W_{F}) have the same Langlands parameters almost everywhere we write π ↔ σ. Then of course χ_{π} = χ_{σ}, where we identify characters of W_{F} and 𝕀_{F}. Let us call a Weil group representation automorphic if there exists an automorphic representation, necessarily unique, with matching Langlands parameters almost everywhere. We say that the corresponding automorphic representation is of Galois type. Finally we call an extension F ⊂ E p-subnormal if it can be embedded in a normal extension of F of p-power order.

2 Base Change

In this section we recollect some facts about base change and automorphic induction. Let F ⊂ E be an extension of degree m and GL_{n}(E) = RES_{E/F} GL_{n}. Recall that the L-group of GL_{n}(E) is isomorphic to GL_{n}(ℂ)^{Γ_{E/F}} ⋊ Gal(F̄/F) where Gal(F̄/F) acts through its action on Γ. There are two “dual” homomorphisms

bc : ^{L}GL_{n}(F) ⟶ ^{L}GL_{n}(E)
ai : ^{L}GL_{n}(E) ⟶ ^{L}GL_{nm}(F)

of L-groups in the theory of base change, corresponding to restriction and induction of representations of the Weil groups. They are defined by

bc(g, σ) = ((g, …, g), σ)
ai((g_{1}, …, g_{m}), σ) = (diag(g_{1}, …, g_{m})R_{σ}, σ)

where R_{σ} is the permutation matrix on the n × n blocks corresponding to σ. Langlands functoriality predicts the existence of liftings BC_{F}^{E} and AI_{E}^{F} of automorphic representations compatible with these homomorphisms.

Theorem 2 ([AC]). Let F ⊂_{sc} E. Then BC_{F}^{E} and AI_{E}^{F} exist and define additive morphisms

BC_{F}^{E} : R_{cusp}(F) ⟶ R_{cusp}(E)
AI_{E}^{F} : R_{cusp}(E) ⟶ R_{cusp}(F).

Furthermore, if F ⊂ E is cyclic and ρ_{1}, ρ_{2} ∈ Cusp(F), then BC(ρ_{1}) ≃ BC(ρ_{2}) if and only if ρ_{2} ≃ ρ_{1} ⊗ ω_{E/F}^{i} for some i. Similarly, if π_{1}, π_{2} ∈ Cusp(E), then AI(π_{1}) ≃ AI(π_{2}) if and only if π_{2} ≃ π_{1}^{γ} for some γ ∈ Γ_{E/F}.

These maps enjoy the following properties which are analogous to those of restriction and induction. Let F ⊂_{sc} E, K, π ∈ R_{cusp}(E), and ρ ∈ R_{cusp}(F). Then:

AI_{E}^{F} π = AI_{L}^{F} AI_{E}^{L} π for F ⊂_{sc} L ⊂_{sc} E and similarly for BC .  (3)
(BC_{F}^{E} ρ)^{g} = BC_{F^{g}}^{E^{g}} ρ^{g} for g ∈ Gal(ℚ̄/ℚ) and similarly for AI .  (4)
BC_{F}^{E}(AI_{K}^{F}(π)) = ⊞_{γ∈W_{K}\W_{F} /W_{E}} AI_{E·K^{γ}}^{E} (BC_{K}^{E^{γ^{−1}} ·K} π)^{γ}.  (5)

These properties are mentioned, at least implicitly, in [AC], and follow easily from an unramified computation. Furthermore, the L-function identity

L^{S}(AI_{E}^{F} π ⊗ ρ^{∨}, s) = L^{S}(π ⊗ BC_{F}^{E} ρ^{∨}, s) gives the following form of Frobenius reciprocity:

c(AI_{E}^{F} π, ρ) = c(π, BC_{F}^{E} ρ).

Finally, the same argument as in the group case proves that if π ∈ Cusp(E) then AI_{E}^{F}(π) is cuspidal if and only if BC_{E^{γ}}^{E·E^{γ}} (π^{γ}) and BC_{E}^{E·E^{γ}} (π) are disjoint for any γ ∈ Γ_{Ē/F} − Γ_{Ē/E}, where Ē is the normal closure of F ⊂ E.

Let F ⊂ E be Galois, σ ∈ Irr(W_{E}) and F ⊂ K ⊂ E be defined by Γ(σ). Recall that by Clifford’s Theory, the induction gives a bijection between the subsets of Irr(W_{K}) and Irr(W_{F}) of those representations whose restriction to W_{E} contains σ in their decomposition. The same will be true in the automorphic setup, if we assume that F ⊂ E is solvable and F ⊂ K is sub-normal.

We also need the following fact from group theory. Let H < G be a normal subgroup with G/H nilpotent. Let λ be a character of H. Then any irreducible constituent of Ind_{H}^{G} λ is induced from a character on some subgroup H < K < G.

3 The main results

The heart of the matter is the following simple Lemma.

Lemma 1. Let H be a normal subgroup of G with [G : H] = p^{k}, p prime, and let σ ∈ Irr_{m}(H) where p |̸m. Suppose that σ^{g} ≃ σ for any g ∈ G. Then

1. There exists a subgroup H < K < G with [K : H]^{2} ≥ [G : H] such that σ extends to K.

2. The following conditions are equivalent

(a) Ind_{H}^{G} σ is isotypic.

(b) Ind_{H}^{G} χ_{σ} is isotypic.

(c) If H < K < G and σ extends to K then [K : H]^{2} ≤ [G : H].

(d) If H < K < G and χ_{σ} extends to a character of K then [K : H]^{2} ≤ [G : H].

3. Under these conditions, if Σ is an extension of σ to K with [K : H]^{2} = [G : H] then Ind_{K}^{G} Σ and Ind_{K}^{G} χ_{Σ} are irreducible and

Ind_{H}^{G} σ = [K : H] Ind_{K}^{G} Σ, Ind_{H}^{G} χ_{σ} = [K : H] Ind_{K}^{G} χ_{Σ}.

Proof. Let α_{σ} be as in (1). The relation (2) together with the fact that p |̸m implies that 2a ⟺ 2b and 2c ⟺ 2d. Decompose Ind_{H}^{G} χ_{σ} as ⊕_{i=1}^{r}m_{i}λ_{i} with λ_{i} ∈ Irr(G). Then ∑ m_{i}^{2} = [G : H] and ∑ m_{i} dim(λ_{i}) = [G : H]. We also know (see end of §2) that for all i, λ_{i} = Ind_{K_{i}}^{G} θ_{i} for some 1-dim character θ_{i} on K_{i} > H extending χ_{σ}. For some i, dim(λ_{i}) ≤ m_{i} ≤ [G : H]^{1/2}, and hence [K_{i} : H]^{2} ≥ [G : H]. Moreover, if r > 1 then we get a strict inequality. This proves the first part and that 2d implies 2b. Suppose that r = 1. Then

m_{1} = dim(λ_{1}) = [K_{1} : H] = [G : H]^{1/2}.

If λ were an extension of χ_{σ} to K with [K : H] > [G : H]^{1/2} then Ind_{K}^{G} λ would be a subrepresentation of Ind_{H}^{G} χ_{σ} of dimension < [G : H]^{1/2} which is absurd. Finally, to prove the last statement, note that by the considerations above Ind_{H}^{G} σ = [K : H]ρ with ρ irreducible. It remains to observe that Ind_{K}^{G} Σ is a subrepresentation of Ind_{H}^{G} σ of dimension m[K : H].

The analogue of the following Lemma to the group case (valid for any Galois extension) is proved easily using Schur’s lemma.

Lemma 2. Let F ⊂ E be a solvable extension, and let ρ_{1}, ρ_{2} ∈ Cusp(F). Assume that BC_{F}^{E} ρ_{1} is cuspidal and BC_{F}^{E} ρ_{2} ≃ BC_{F}^{E} ρ_{1}. Then ρ_{2} ≃ ρ_{1} ⊗ ω for some character ω of W_{F} /W_{E}.

Proof. We use induction on [E : F], the cyclic case being covered by Theorem 2. Let F ⊂ K be a cyclic extension in E. Using

BC_{K}^{E} BC_{F}^{K} ρ_{1} ≃ BC_{K}^{E} BC_{F}^{K} ρ_{2}

and the induction hypothesis we get BC_{F}^{K} ρ_{1} ≃ BC_{F}^{K} ρ_{2} ⊗ ω for some character ω of W_{K}/W_{E}. Conjugating by γ ∈ Γ_{K/F} we also have BC_{F}^{K} ρ_{1} ≃ BC_{F}^{K} ρ_{2} ⊗ω^{γ}, from which

BC_{F}^{K} ρ_{1} ⊗ ω^{γ}ω^{−1} ≃ BC_{F}^{K} ρ_{1}.

If ω^{γ} ≠ ω then BC_{F}^{E} ρ_{1} = BC_{K}^{E} BC_{F}^{K} ρ_{1} is not cuspidal, contradicting our assumption. Thus ω is Γ_{K/F} -invariant. Hence, we can extend ω to a character μ of W_{F} /W_{E}. We have BC_{F}^{K} ρ_{1} ≃ BC_{F}^{K}(ρ_{2} ⊗ μ) and we can appeal to the cyclic case.

Let us now give a simple descent criterion for base change, whose analogue in the group case is well-known.

Proposition 1. Let F ⊂ E be a solvable extension and let π ∈ Cusp_{n}(E) with ([E : F], n) = 1. Suppose that π^{γ} ≃ π for any γ ∈ Γ_{E/F} and that there exists a character χ of W_{F} which extends χ_{π}. Then there exists a unique ρ ∈ Cusp_{n}(F) such that BC_{F}^{E}(ρ) ≃ π and χ_{ρ} = χ.

Proof. The uniqueness part follows immediately from the Lemma above. To prove the existence we proceed by induction. Let F ⊂ K be a cyclic extension contained in E. Using the induction hypothesis for K ⊂ E we extend π to K as ρ′ with χ_{ρ′} = χ∘Nm_{F}^{K}. Now, for any γ ∈ Γ_{K/F}

BC_{K}^{F} ρ′^{γ} ≃ π^{γ} ≃ π ≃ BC_{K}^{F} ρ′.

By uniqueness, ρ′^{γ} ≃ ρ′ and we can use the descent criterion for cyclic extensions (Theorem 2). After a possible twist by a character, we get the required central character.

Theorem 3. Let E/F be a p-extension of number fields and suppose that GLC holds for σ ∈ Irr_{m}(W_{E}) with p |̸m. Assume that Ind_{W_{E}}^{W_{F}} σ = n · τ with τ ∈ Irr(W_{F}). Then τ is automorphic.

Proof. By Clifford, Ind_{W_{E}}^{W_{L}} σ = n · τ′ and Ind_{W_{L}}^{W_{F}} τ′ = τ, where L is the subfield corresponding to Γ_{E/F} (σ). Let π ↔ σ. Suppose that we know that

AI_{E}^{L}(π) = nρ′  (6)

with ρ′ ∈ Cusp(L). We can then conclude, by comparing the parameters, that ρ′ ↔ τ′ and AI_{L}^{F} ρ′ ↔ τ. Let us prove (6). We can assume that L = F. By Lemma 1 we know that τ = Ind_{W_{K}}^{W_{F}} Σ where F ⊂ K ⊂ E and Σ ∈ Irr_{m}(W_{K}) extends σ. By the Proposition above, there exists Π ∈ Cusp_{m}(K) such that BC_{K}^{E} Π = π and χ_{Π} = χ_{Σ}. We claim that ρ = AI_{K}^{F} Π is cuspidal. If not, then the condition for cuspidality (§2), and the fact that BC_{K}^{E} Π is cuspidal imply that we have BC_{K^{γ}}^{K·K^{γ}} Π^{γ} ≃ BC_{K}^{K·K^{γ}} Π for some γ ∈ Γ_{E/F} − Γ_{E/K}. In particular,

χ_{Π}^{γ}|_{W_{K}∩W_{K}^{γ}} = χ_{Π}|_{W_{K}∩W_{K}^{γ}} .  (7)

However, Ind_{W_{K}}^{W_{F}} χ_{Σ} is irreducible. This contradicts (7), because χ_{Σ} = χ_{Π}. Finally,

c(ρ, AI_{E}^{F} π) = c(BC_{F}^{E}(AI_{K}^{F} Π), π) = ∑_{γ∈Γ_{E/F} /Γ_{E/K}} c(BC_{K}^{E}(Π)^{γ}, π) = [K : F] = n

so that AI_{E}^{F} π = nρ as required.

Theorem 4. Let F ⊂ E be a normal extension of number fields with the property that there exists a sequence of distinct primes p_{i}, i = 1, 2, … r and a sequence of extensions F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = E where F_{i−1} ⊂ F_{i} is a p_{i}-extension. Let σ ∈ Irr_{m}(W_{E}) with p_{i} |̸m for any i, be such that Ind_{W_{E}}^{W_{F}} σ = nτ for some n and τ ∈ Irr(W_{F}). Then τ is automorphic.

Proof. Using induction and the previous Theorem we have to show that the conditions of the Theorem hold for F = F_{1}. Suppose on the contrary that Σ_{1} = Ind_{W_{E}}^{W_{F_{1}}} σ is not isotypic. First, we claim that the irreducible constituents of Σ_{1} lie in the same orbit under Γ_{F_{1}/F}. Indeed, if τ_{1} is an irreducible component of Σ_{1}, then by our condition Ind_{W_{F_{1}}}^{W_{F}} τ_{1} is isotypic and the type does not depend on τ_{1}. In particular, for any

irreducible component τ′ of Σ_{1} we have c(Ind_{W_{F_{1}}}^{W_{F}} τ′, Ind_{W_{F_{1}}}^{W_{F}} τ_{1}) > 0. This implies that τ′ = τ_{1}^{γ} for some γ ∈ Γ_{F_{1}/F}. Next, note that for γ ∈ Γ_{F_{1}/F}, c(Σ_{1}^{γ}, Σ_{1}) = 0 if γ does not lie in the image Γ̄ of Γ_{E/F} (σ) under Γ_{E/F} → Γ_{F_{1}/F}, and Σ_{1}^{γ} ≃ Σ_{1} otherwise. Thus Σ_{1} = k(⊕τ_{1}^{γ}) for some k where the sum is over the orbit of τ_{1} under Γ̄. Since p_{1} |̸ dim Σ_{1}, this orbit is a singleton and we get Σ_{1} = kτ_{1} as required.

For the application we have in mind, we would also like to have the dual statement, which is proved in a similar way.

Theorem 4’. Let F ⊂ E be as before. Suppose that π ↔ σ and AI_{E}^{F} π = nρ with ρ cuspidal. Then Ind_{W_{E}}^{W_{F}} σ = nτ with τ irreducible and ρ ↔ τ.

Proof. The reduction to the case where F ⊂ E is a p-extension is as in Theorem 4, using only the formal properties of base change described in §2. Then again, by using ‘Clifford Theory’ in the automorphic side, we are reduced to the case where Γ(σ) = Γ(π) = Γ. Assume on the contrary that Ind_{W_{E}}^{W_{F}} σ is not isotypic. According to Lemma 1, we can extend σ to a representation Σ ∈ Irr_{m}(W_{K}) where [K : F]^{2} < [E : F]. By Proposition 1 we have a cuspidal representation Π ∈ Cusp_{m}(K) with BC_{K}^{E} Π ≃ π. But then,

c(ρ, AI_{K}^{F} Π) = \frac{1}{n}c(AI_{E}^{F} π, AI_{K}^{F} Π) = \frac{1}{n}c(π, BC_{F}^{E} AI_{K}^{F} Π) ≥ \frac{1}{n}c(π, BC_{K}^{E} Π) > 0.

This contradicts the fact dim AI_{K}^{F} Π = [K : F]m < dim ρ.

Lemma 3. Let F ⊂ E be a nilpotent extension and let σ ∈ Irr(W_{E}). Then Ind_{W_{E}}^{W_{F}} σ

is isotypic if and only if for any p, Ind_{W_{E}}^{W_{Ep}} σ is isotypic, where E_{p} is the field defined by the p-Sylow subgroup Γ_{p} of Γ = Γ_{E/F}. The analogous statement for cuspidal representations also holds.

Proof. If Ind_{W_{E}}^{W_{Ep}} σ = n_{p}τ_{p} with τ_{p} irreducible then n_{p}^{2} = |Γ_{p}(σ)|. Thus Ind_{W_{E}}^{W_{F}} σ =

Ind_{W_{Ep}}^{W_{F}} Ind_{W_{E}}^{W_{Ep}} σ is divisible by n_{p} in R(W_{F}). Hence, it is divisible by n where

n^{2} = |Γ(σ)|. Thus, Ind_{W_{E}}^{W_{F}} σ is isotypic. The converse was proved in the proof of Theorem 4. The cuspidal side is similar, with R_{cusp}(F) playing the role of R(W_{F}).

Theorem 5. Let again σ ∈ Irr_{m}(W_{E}) satisfy GLC and suppose that there exists F ⊂ K ⊂ E such that Ind_{W_{E}}^{W_{K}} σ = kρ and Ind_{W_{K}}^{W_{F}} ρ = lτ for ρ ∈ Irr(W_{K}), τ ∈ Irr(W_{F}). Assume that F ⊂ E, F ⊂ K are normal and both Γ_{E/K} and Γ_{K/F} are nilpotent. Also, assume that (m, [E : F]) = 1. Then τ is automorphic.

Proof. By Theorem 4, we know that ρ is automorphic. Let ξ be the corresponding cuspidal representation, and Ξ = AI_{K}^{F} ξ. We have to prove that Ξ is isotypic. By the previous Lemma it is enough to consider the case where F ⊂ K is a p-extension. Let K ⊂ Ep̄ ⊂ E be the subfield corresponding to the p̄-Hall subgroup of Γ_{E/K}. Let p_{1}, …, p_{r} be the other prime divisors of [E : K]. The sequence F ⊂ Ep̄ ⊂ E\overline{\{p,p_{1}\}} ⊂ ⋯ ⊂ E satisfies the conditions of Theorem 4.

Again, we also have the dual statement.

Theorem 5’. Let F ⊂ K ⊂ E and m be as before, and let σ ∈ Irr_{m}(W_{E}). Suppose that π ↔ σ and that both AI_{E}^{F} π and AI_{E}^{K} π are isotypic. Then Ind_{W_{E}}^{W_{F}} σ is isotypic. If μ is the cuspidal type of AI_{E}^{F} π and τ is the irreducible constituent of Ind_{W_{E}}^{W_{F}} σ, then μ ↔ τ.

Example. Recall that for any Abelian group A, the Schur multipliers H^{2}(A, ℚ/ℤ) can be canonically identified with the alternating bilinear forms on A with values in Z = ℚ/ℤ. As in [La], let (·, ·) be a non-degenerate form, and H be the corresponding Heisenberg group. That is, H sits in an exact sequence

0 ⟶ Z ⟶ H ⟶ A ⟶ 0

and the commutator pairing induces (·, ·) on A. Let σ be the Stone–von-Neumann representation with central character ψ(z) = e^{2πiz}. Let α : B → Aut(A, (·, ·)) be an action of the Abelian group B on A by symplectic automorphisms. Assume that (|A|, |B|) = 1. Then, we can lift α to an action on H, and σ extends to an irreducible representation of H ⋊ B. Suppose now that [·, ·] is a non-degenerate alternating form on B. Let γ ∈ H^{2}(B, Z) be the corresponding cocycle, and G be the extension of H by B defined by γ. Then Ind_{H}^{G} σ = mτ with τ irreducible and m^{2} = |B|. Let τ′ be the restriction of τ to a finite subgroup G′ of G with G = ZG′. According to Theorem 4 any Galois representation which factors through τ′ satisfies GLC. However, τ′ is not monomial, unless there exist maximal isotropic subgroups A_{1},B_{1} of A and B respectively so that A_{1} is invariant under B_{1}.

4 An Application

As an application, we refer to a problem considered in [La]. Recall that the global multiplicity M(L) of an L-packet L of SL_{n} was defined to be the sum of multiplicities of L-packets which coincide with L almost everywhere. For its computation we considered two equivalence relations on cuspidal representations of GL_{n}:

1. π̃∼_{s}ρ̃ if there exists a Hecke character ω of 𝕀_{F} /F^{∗} such that ρ̃ ≃ π̃ ⊗ ω,

2. π̃∼_{w}ρ̃ if for almost every place v there exists a character ω_{v} of F_{v}^{∗} such that ρ̃_{v} ≃ π̃_{v} ⊗ ω_{v}.

Let L(π̃) be the L-packet defined by a cuspidal representation π̃ of GL_{n}. By the multiplicity formula of Labesse and Langlands ([LL])

M(L(π̃)) = |{π̃′ : π̃′∼_{w}π̃}/∼_{s}|.

There are also analogous equivalence relations for projective representations of a group G. Let ϕ_{i} : G → P GL_{n}(ℂ), i = 1, 2. Define

1. ϕ_{1}∼_{s}ϕ_{2} if there exists x ∈ P GL_{n}(ℂ) such that ϕ_{1}(g) = x^{−1}ϕ_{2}(g)x for all g ∈ G,

2. ϕ_{1}∼_{w}ϕ_{2} if for any g ∈ G there exists x ∈ P GL_{n}(ℂ) such that ϕ_{1}(g) = x^{−1}ϕ_{2}(g)x.

We can also define

M(ϕ) = |{ϕ′ : ϕ′∼_{w}ϕ}/∼_{s}|

for any ϕ : G → P GL_{n}(ℂ) (the latter is always finite, and in fact bounded in terms of n only). We denote by σ̄ the projective representation obtained from an ordinary representation σ by the projection GL_{n}(ℂ) → P GL_{n}(ℂ). The problem is to show that for automorphic representations σ of W_{F} with π̃ ↔ σ we have M(L(π̃)) = M(σ̄). This was proved in the case where σ is induced from a character of an extension which is either Abelian or a sub-p-extension for some p. We can now show

Theorem 6. Let σ = Ind_{W_{K}}^{W_{F}} θ where θ is a Hecke character of K and F ⊂ K is nilpotent. Let π̃ ↔ σ. Then M(L(π̃)) = M(σ̄).

Proof. The statement in the Theorem is equivalent to the following two statements:

1. If π̃′∼_{w}π̃ then π̃′ is of Galois type.

2. If \overline{σ′}∼_{w}σ̄ then σ′ is automorphic.

Let us prove the second statement (the dual statement is proved similarly). As in the proof of Theorem 2 in [La] we have the following properties for σ′:

1. σ′|_{W_{K}} = d(⊕_{g∈Γ_{K/F} (ρ)\Γ_{K/F}} ρ^{g}) with ρ ∈ Irr_{d}(W_{K}).

2. Ind_{W_{K}}^{W_{F}} ρ ≃ dσ′.

3. The kernel of ρ̄ is W_{E} where E is an Abelian extension of order d^{2} over K, normal over F.

4. ρ|_{W_{E}} = dζ for a Hecke character ζ of E and Ind_{W_{E}}^{W_{K}} ζ = dρ.

We can now use Theorem 5 to conclude the proof.

Unfortunately, the general case where F ⊂ E is solvable lies beyond the limitations of the method described in this paper.

5 A generalized tetrahedral representation

We conclude by analyzing the simplest case of a Galois representation no multiple of which is monomial. Let 𝔽_{q} be the finite field with q elements and V be a 2-dimensional vector space over 𝔽_{q} with a non-degenerate 𝔽_{q}-bilinear alternating form. Let H be the corresponding Heisenberg group. Let Aut_{c}(H) be the automorphisms of H which act trivially on the center. The exact sequence

0 ⟶ Inn(H) ⟶ Aut_{c}(H) ⟶ Sp(V) ⟶ 0  (8)

splits if 2 |̸q. In any case, it splits over any subgroup B of Sp(V) ≃ SL_{2}(𝔽_{q}) with q |̸|B|. The classification of these subgroups B is well known (e.g. [Di]) and runs parallel to the case of SL_{2}(ℂ). Let B ≠ 1 be a solvable group of this kind. Then, the image of B in P SL_{2}(𝔽_{q}) is either cyclic, dihedral, A_{4} or S_{4}. The Stone–von-Neumann representation of H extends to a q-dimensional irreducible representation of G = H ⋊ B. In fact, if q is odd, it extends to H ⋊ SL_{2}(𝔽_{q}) and the restriction to SL_{2}(𝔽_{q}) is the Weil representation. Suppose that σ ∈ Irr_{q}(W_{F}) factors through G. (By abuse of notation we will also regard σ ∈ Irr_{q}(G).) The image of σ̄ is isomorphic to V ⋊ B. Let F ⊂ E ⊂ K be the extensions corresponding to the inverse image of V and the kernel of σ̄ respectively. We know that GLC holds for σ|_{W_{E}}; let π_{E} ↔ σ|_{E}. Clearly π_{E}^{γ} ≃ π_{E} for any γ ∈ Γ_{E/F} ≃ B. By Proposition 1 there exists a unique π ∈ Cusp_{q}(F) so that BC_{F}^{E} π ≃ π_{E} and χ_{π} = χ_{σ}. What are the obstacles to proving that π ↔ σ? If v is a place in F which splits completely in E, then clearly g(π_{v}) ∼ σ(F r_{v}). However, if v has relative degree d then we only know that

g(π_{v})^{d} ∼ σ(F r_{v})^{d}.  (9)

At this point we must assume some functoriality hypothesis, which looks inaccessible by today’s methods.

Assumption 1. There exists a lifting of automorphic representations corresponding to the adjoint representation Ad : GL_{q} ⟶ GL_{q^{2}−1}.

Granting the assumption, GLC for σ would follow from the following

Proposition 2. Let Π be the adjoint lift of π. Then

1. Π ↔ Ad(σ)

2. Ad(σ(F r_{v})) ∼ Ad(g(π_{v}))

3. σ̄(F r_{v}) ∼ \overline{g(π_{v})}

4. σ(F r_{v}) ∼ g(π_{v}) and thus π ↔ σ.

Proof. 1. Note that

Ad(σ)|_{W_{E}} ≃ ⊕_{1≠θ∈{ \atop W_{E}/W_{K}}} θ.

Since B acts freely on V − 0, there is a unique representation of W_{F}, namely Ad(σ), whose restriction to W_{E} is Ad(σ)|_{W_{E}}. Moreover, Ad(σ) = ⊕_{O} Ind_{W_{E}}^{W_{F}} θ where O is a set of representatives of the B-orbits of non-trivial characters of W_{E}/W_{K}. Similarly, Π′ = ⊞_{O} AI_{E}^{F} θ is the unique element of R_{cusp}(F) satisfying BC_{F}^{E} Π′ = ⊞_{1≠θ∈{ \atop W_{E}/W_{K}}} θ.

Clearly Π′ ↔ Ad(σ). On the other hand, by functoriality, BC_{F}^{E} Π ↔ Ad(σ)|_{W_{E}} ≃ ∑_{1≠θ∈{ \atop W_{E}/W_{K}}} θ and thus Π ≃ Π′.

2. This follows immediately from 1.

3. We can assume that v does not split completely in E. Then, since B acts freely on V − 0, σ̄(F r_{v}) ∼ σ̄(g) for some g ∈ B. If q is odd, the Weil representation Θ decomposes as the sum of the two irreducible representations (uniquely determined up to conjugation by GL_{2}(𝔽_{q})) of SL_{2}(𝔽_{q}) of dimensions \frac{q±1}{2}. From the character table one sees that

Θ|_{T} ≃ R_{reg}(T) + (−1)^{ϵ(T)}η

where T is a torus of SL_{2}(𝔽_{q}) (split or non-split), R_{reg} is the regular representation, ϵ(T) is 0 if T is split, and 1 otherwise, and finally η is the unique character of T of order 2. Since σ|_{B} = Θ|_{B} we conclude that

σ̄(F r_{v}) ∼ \overline{(\begin{matrix}1 \\ & ζ \\ & & … \\ & & & ζ^{q−1} \end{matrix})}  (10)

where ζ is a root of unity of order |g|. If q is even, (10) still holds (for a more general setup, see [Is]). It is now easy to see that σ̄(F r_{v}) is the unique element in P GL_{q}(ℂ), up to conjugacy, which maps under the adjoint representation to the conjugacy class of the diagonal element consisting of all roots of unity of order |g|, each appearing (q^{2} − 1)/|g| times. Thus, 2 implies 3.

4. This follows from 3, (9), and the fact that χ_{σ} = χ_{π}.

Remark. 1. The case q = 2 is the classical dihedral case, proved by Langlands in [L], using the adjoint lifting for GL(2) ([GJ]). The only difference in the argument above is that we use base change for GL_{n} with n > 2 in step 1. Langlands avoids this (which was not known then) by using an L-function argument. This argument uses the equality

L^{S}(Π ⊗ Π′^{∨}, s) = L^{S}(Π′ ⊗ Π′^{∨}, s)  (11)

to conclude that Π ≃ Π′ since Π, Π′ are cuspidal in that case. However, if q > 2, Π, Π′ are not cuspidal, and the relation (11), which can be proved in the same way, is not sufficient to conclude that Π ≃ Π′.

2. In the case q = 3 and |B| = 2, we get a three-dimensional monomial representation. Thus, GLC follows from [JPS].

3. No other cases seem to be known.

Acknowledgment

I would like to thank the Department of Mathematics of the University of California, Los Angeles, and especially Jonathan Rogawski for their kind hospitality. I would like to thank Don Blasius, Stephen Gelbart and Jonathan Rogawski for helpful discussions and suggestions.

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Department of Mathematics University of California Los Angeles CA 90095-1555 USA Current address: School of Mathematics Institute for Advanced Study Olden Lane Princeton, NJ 08540 USA erezl@math.ias.edu