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Let F be a number field. For each place v of F, let Fv be the completion of F at
- v. For each finite v, let Ov be the ring of
integers of Fv and denote by Pv its maximal ideal. Let ̟v be a generator of Pv. Let qv = [Ov: Pv] and fix an absolute value | |v such that |̟v|v = q−1
v .
Let G be a split connected reductive al- gebraic group over F. This simply means a Zariski closed subgroup of GLN(F) for some N, where F is the algebraic closure
- f F, whose radical (maximal normal con-
nected solvable subgroup) consists of only 1
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semisimple elements. The radical is then equal to the connected component of the center of G. Being split simply means that G has a maximal abelian subgroup consist- ing entirely of diagonalizable elements, a maximal torus, which is isomorphic over F to some power of F
∗.
Let B be a Borel subgroup of G (over F), i.e., a maximal connected solvable subgroup
- f G. Let T be a maximally split torus of G
contained in B. Then B = TU, where U is the unipotent radical of B. The unipotent subgroup U determines a set of simple roots ∆ and positive roots R+ for T, upon acting
- n the Lie algebra g of G.
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Let P be a parabolic subgroup of G, i.e., a conjugate of a closed subgroup of G con- taining B. We will assume P is standard by P ⊃ B. We let N be the unipotent radi- cal (maximal connected normal unipotent subgroup) of P. Then P = MN, where M is a reductive subgroup, called a Levi sub-
- group. We will fix M by assuming T ⊂ M.
Let A be the split component of M, i.e., the connected component of the center of M (the maximally split subtorus of the cen- ter of M, if the group is not necessarily split over F). The parabolic subgroup P is maximal if the dimension of A/A ∩ ZG is one, where ZG is the center of G. Then
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the adjoint action of A on the Lie algebra
- f N has a unique reduced eigenfunction α,
the simple root of A in N. There exists a unique simple root of T whose restric- tion to A is α. We will denote this root
- f T also by α and always identify them
with each other. Other roots of A in N are simply multiples (considered additively)
Throughout these lectures P is al- ways assumed to be maximal. We refer to [B2,Sat,Sp] as our main references for alge- braic groups and their structure theory. Let H to be any connected reductive al- gebraic group defined over F. Considering H as a group over Fv, for each v, we let
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Hv = H(Fv). If AF is the ring of ade- les of F, we set H = H(AF ). It may be considered as a restricted product of groups Hv with respect to H(Ov) for all v, where H splits over an unramified extension [B1]. There will be no restriction if H splits over
- F. Moreover, in this case each Kv = H(Ov)
is a maximal compact subgroup of Gv. We let K =
v Kv, where each Kv is a good
maximal compact subgroup of Hv and Kv = H(Ov) for almost all v(cf. [B1,Sh1]). Then G = PK. For every algebraic group H over F, let X(H)F denote the group of F rational char- acters of H. We let X(H) = X(H)F . Note
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that if T is a split torus over F, then X(T)F = X(T). We set a = Hom(X(M)F , R) = Hom(X(A)F , R). Then a∗ = X(M)F ⊗ZR = X(A)F ⊗ZR and a∗
C = a∗ ⊗R C is the complex dual of a, via
λ, χ ⊗ z = λ(χ)z, λ ∈ a, χ ∈ X(M)F and z ∈ C. For each v, the embedding X(M)F ֒ → X(M)Fv induces a map from av = Hom(X(M)Fv, R) to a. There exists a homomorphism HM: M − → a defined by expχ, HM(m) =
|χ(mv)|v for every χ ∈ X(M)F and m = (mv). Ex- tend HM to HP on G by making it trivial
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If ρP is half the sum of roots in N, we set ˜ α = ρP, α−1ρP ∈ a∗. It is a fundamental weight for T (cf. [Sh1,Sh2]). Finally having fixed M with M ⊃ T, let θ ⊂ ∆ denote the subset of simple roots, generating M. We sometimes write Mθ for
- M. Let W be the Weyl group of T in G.
We use WM to denote its Weyl group in
- M. There exists a unique element ˜
w0 ∈ W such that ˜ w0(θ) ⊂ ∆, while ˜ w0(α) < 0. We will always choose a representative w0 for ˜ w0 in G(F) and use w0 to denote each of its components. We will be more specific about the choice of w0 later. Finally, let M′ be the Levi subgroup of G generated
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by ˜ w0(θ). There exists a parabolic subgroup P′ ⊃ B which has M′ as a Levi factor, in fact the unique one containing T. Let N′ be the unipotent radical of P′ (cf. [La1,Sh3]).
and Generic Representations Denote by X∗(T) = X(T) the character group of T which is the same as X(T)F . Let X∗(T) be the group of cocharacters of T, i.e., homomorphisms from Gm = F
∗ into
T = T(F). Let ∆∨ = ∆∨(T) be the set
- f simple coroots of T, i.e., α∨: Gm −
→ T satisfying α(α∨(t)) = t2. Let (2.1) ψ0(G) = (X∗(T), ∆(T), X∗(T), ∆∨(T))
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9
denote the based root datum. By Chevalley–Grothendieck theorem [Sp], there exist a complex connected reductive group G∨ with a maximal torus T ∨ such that ψ0(G)∨ = (X∗(T), ∆∨(T), X∗(T), ∆(T)) = ψ0(G∨) = (X∗(T ∨), ∆(T ∨), X∗(T ∨), ∆∨(T ∨)). The L–group LG of G is LG = G∨ × ΓF , where ΓF = Gal(F/F). In general, one carries the action of ΓF on roots and co- roots dually to G∨ and let LG = G∨ ⋊ ΓF . (Observe that G∨ = LG0.) In fact, we have an exact sequence
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(2.2) 0 − → Int(G) − → Aut(G) − → Autψ0(G) − → 0, where Int(G) is the subgroup of inner auto-
- morphisms. One can show that if {Xβ}β∈∆′,
∆′ being the set of simple roots of T, is a set
- f simple root vectors invariant under ΓF ,
then Aut ψ0(G) = Aut(G, B, T, {Xβ}β∈∆′). The set {Xβ}β∈∆′ is called a splitting, as it splits (2.2). The map ΓF − → Aut(G, B, T, {Xβ}β) defines a map ΓF − − − − → Autψ0(G)∨ = Aut ψ0(G∨)
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which is a subset of Aut(G∨). This de- fines the action of ΓF on G∨ and defines
LG (cf. [B1, Sat]).
Given a connected reductive algebraic group H over F, let LH be its L–group. Considering H as a group over Fv, we then denote by LHv its L–group over Fv. If G is split over F and if we decide to only consider
LG0 = G∨, then we may assume that the L–
groups are all the same, no matter the place
- v. Finally, the natural map ΓFv → ΓF leads
to a map ηv: LMv → LM for all v. In our setting LM is a Levi subgroup of
LG and one can define a unipotent group LN (cf. [B1]) so that LM LN is a parabolic
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subgroup of LG with unipotent radical LN. The L–group LM acts on the (complex) Lie algebra Ln of LN. Let r be this represen- tation. Decompose r =
m
ri to its irre- ducible subrepresentations, indexed accord- ing to the values ˜ α, β = i as β ranges among the positive roots of T. More pre- cisely, Xβ∨ ∈ Ln lies in the space Vi of ri if and only if ˜ α, β = i. Here Xβ∨ is a root vector attached to the coroots β∨, consid- ered as a root of the L–group. Clearly the integer m is equal to the nilpotence class of
- Ln. We let ri,v = ri · ηv for each i. Again if
G is split over F, we may assume ri,v = ri for all v and i (cf. [La,Sh1,Sh2]).
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Let π = ⊗vπv be a cuspidal representa- tion of M = M(AF ). Then for almost all v, πv is an unramified representation of Mv = M(Fv). This means that πv has a vector which remains invariant under M(Ov). In this case, the class of πv is determined by a semisimple LM–conjugacy class Av ⊂ LMv =
- LM. Given a complex analytic (finite dimen-
sional) representation ρ of LM, we define the local Langlands L–function attached to πv, ρ and a complex number s by (2.3) L(s, πv, ρ) = det(I − ρ(Av)q−s
v )−1.
When M is quasisplit over Fv, to split over an unramified extension Lw/Fv, and τv is
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the unique Frobenius conjugacy class in Gal (Lw/Fv), then Av must be replaced by tv ⋊ τv, tv ∈ LT 0. We may moreover assume that tv is fixed by τv ([B1, La1, Sh2]). In these lectures, we will mainly be con- cerned with the case where ρ = ri,v, i = 1, . . . , m. We shall now discuss the notion of generic
- representations. We will first assume that F
is a local field. Fix a F–splitting {Xβ}β∈∆′ as before. This then determines a map (2.4) φ: U →
→ Ga, where the product runs over all β ∈ ∆′, sending exp(xβXβ) to Σxβ.Let ψF be a non–
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trivial character of F. We shall then define a generic character χ of U = U(F) by (2.5) χ(u) = ψF (Σβxβ), where φ(u) = (xβ)β and the β–component
- f u is exp(xβXβ). Conversely, given a
generic character χ of U, i.e., one which is non–trivial on every simple root group, there exists an F–splitting {Xβ}β such that χ is defined by (2.4) and (2.5). If π is an irreducible (admissible) uni- tary representation of M = M(F), then π is called generic, or more precisely χ– generic for a generic character χ of U 0 = (U ∩ M)(F), if there exists a functional λ
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- n the space of π, called a Whittaker func-
tional, which is continuous with respect to the seminorm topology defined by the Hilbert space norm on the space H(π) of π if F is archimedean (cf. [S,Sh4,Sh5]) (continu-
- us with respect to the trivial locally convex
topology on H(π) for which every seminorm is continuous if F is non–archimedean) and satisfies (2.6) π(u)x, λ = χ(u)x, λ, u ∈ U 0, x ∈ H(π)∞, the subspace of C∞– vectors. By a theorem of Shalika [S], the space of all the Whittaker functionals on H(π) is at most one–dimensional. Changing
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17
the splitting, we may assume χ is defined by (2.4) and (2.5). Now, assume F is global. Let ψ = ⊗vψv be a non–trivial character of F\AF . Then each ψv is non–trivial. Moreover, for al- most all v, ψv is unramified, i.e., Ov is the largest ideal on which ψv is trivial. The map (2.4) is F–rational and therefore extends to a map from U = U(AF ) into Ga(AF ), send- ing U(F) into Ga(F). We then define a character χ of U(F)\U by (2.5). Conse- quently, if χv(uv) is defined by (2.5) and ψv for each v, then χ(u) =
v χv(uv), where
u = (uv)v ∈ U. Now, let π = ⊗vπv is a cuspidal represen-
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tation of M = M(AF ). Choose a function ϕ in the space of π and set (2.7) Wϕ(m) =
We shall say π is (globally) χ–generic if Wϕ = 0 for some ϕ. Then each πv is χv–generic. If ϕ = ⊗vϕv, ϕv ∈ H(πv), then Wϕ(m) =
πv(mv)ϕv, λv =
Wϕv(mv) (2.8) for some χv–Whittaker functional λv on H(πv). We finally point out that every generic character χ of U(F)(U(F)\U(AF ) if F is
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19
global) points to a F–splitting and conversely, each F–splitting defines a generic character, both by means of (2.4) and (2.5). Now assume F is local and πv is an ir- reducible admissible representation of Mv. Let s ∈ C and denote by I(s, πv) = I(s˜ α, πv), the induced representation (2.9) I(s, πv) = IndMvNv↑Gvπv⊗qs˜
α,HMv ( ) v
⊗1. This is the right regular action of Gv on the space of smooth functions f from Gv to H(πv), satisfying: (2.10) f(mng) = πv(m)qs˜
α+ρP,HMv (m) v
f(g). If v = ∞, smooth would mean that f is
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infinitely many times differentiable. On the
- ther hand for p < ∞, it simply means that
f is locally constant, i.e., f(gk) = f(g) for k in some open compact subgroup depending
One can now define a Whittaker func- tional λχv(s, πv) for I(s, πv) as follows: (2.11) λχv(s, πv)(f) =
v
f(w−1
0 n′), λχ(n′)dn′,
where λ is a fixed Whittaker functional on H(πv). The integral converges as a princi- pal value integral and stabilizes as N ′
v is ap-
proached by an increasing sequence of open compact subgroups (cf. [CS,Sh3]), i.e., we
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may replace N ′
v with a sufficiently large open
compact subgroup depending on f.
- 3. Eisenstein Series and Intertwining
Operators; The Constant Term Let π = ⊗vπv be a cusp form on M. Given a KM–finite function ϕ in the space
- f π, we extend ϕ to a function ˜
ϕ on G as
- follows. The representation π is a subrepre-
sentation of L2
0(Z0 MM(F)\M, ρ), where Z0 M
is the AF –points of the connected compo- nent Z0
M of the center of M and ρ is a
character of Z0
M(F)\Z0
is then in this L2–space and being KM– finite, its right translations by elements in
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22
KM = ΠvKM,v, KM,v = Kv ∩ Mv, generate a finite dimensional representation τ of KM (cf. [Car]). We may assume ϕ is so chosen that τ is irreducible and write τ = ⊗vτv, where for almost all v, τv is trivial. Next we will choose irreducible (finite dimensional) representations ˜ τv of each Kv, containing τv. Moreover, we assume ˜ τv is the triv- ial representation for almost all Kv. Set ˜ τ = ⊗v˜ τv (cf. [Sh6]). Let Pτ be the projection on the space of τ and fix measures dkv on each KM,v whose total mass is 1. Let dk be the product mea- sure on KM. Set
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(3.1) ˜ ϕ(m) =
ϕ(mk)˜ τ(k)dk · Pτ. Observe that ˜ ϕ(mk−1) = ˜ τ(k) ˜ ϕ(m). This is a ˜ τ–function on M in Harish–Chandra’s ter- minology [HC]. We extend ˜ τ to all of G by ˜ ϕ(nmk) = ˜ τ(k−1) ˜ ϕ(m). It is easily checked to be a well–defined operator valued func- tion on G ([Sh6]). Next, set (3.2) ˜ Φs(g) = ˜ ϕ(g) exps˜ α + ρP, HP (g) and let Φs be a matrix coefficient of this
- perator valued function. (See (3.7) below.)
The corresponding Eisenstein series is then
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defined by (3.3) E(s, Φs, g, P) =
Φs(γg). We will also define the operator valued Eisenstein series by (3.4) ˜ E(s, ˜ Φs, g, P) =
˜ Φs(γg). They both converge for Re(s) >> 0 and have a finite number of simple poles for Re(s) ≥ 0, none with Re(s) = 0 (cf. [HC,La2,MW1]). Let I(s, π) = ⊗vI(s, πv) be the represen- tation of G = G(AF ) induced from (3.5) π ⊗ exps˜ α + ρP, HM( ) ⊗ 1.
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Let f ∈ V (s, π) = ⊗vV (s, πv) be defined by (3.6) f(n0m0k0) = exps˜ α + ρP, HP (m0)·
˜ τ(k−1
0 )τ(k)x, ˆ
xπ(m0k)ϕdk, with m0 ∈ M, n0 ∈ N and k0 ∈ K. Here x ∈ H(τ) and ˆ x ∈ H(ˆ τ), where ˆ τ is the contragredient of τ. Moreover (3.7) f(n0m0k0)(e) = ˜ Φs(g)x, ˆ x = Φs(g), where the left hand side is the value of the cusp form f(n0m0k0) at identity and g = n0m0k0 (cf. [Sh6]).
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Observe that (3.8) E(s, Φs, g, P) = ˜ E(s, ˜ Φs, g, P)x, ˆ x. Given f ∈ V (s, π) and Re(s) >> 0, de- fine the global intertwining operator A(s, π) by (3.9) A(s, π)f(g) =
0 n′g)dn′.
Finally, if at each v we define a local inter- twining operator by (3.10) A(s, πv, w0)fv(g) =
v
fv(w−1
0 n′g)dn′,
then (3.11) A(s, π) = ⊗vA(s, πv, w0).
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27
Observe that (3.12) A(s, π): I(s, π) → I(−s, w0(π)) and (3.13) A(s, πv, w0): I(s, πv) → I(−s, w0(πv)). Using (3.7), we now define (3.14) (M(s, π)Φs)(g) = A(s, π)f(g)(e), where by the left hand side we understand the value of the cusp form A(s, π)f(g) at e. This is basically the Langlands’ M(s, π) in- troduced in [La2], or as denoted by Harish– Chandra in [HC], his function c(s, π).
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Constant Term Theorem. The constant term (3.15) EP′(s, Φs, g, P) =
- N′(F )\N ′ E(s, Φs, n′g, P)dn′
is equal to (3.16) EP′(s, Φs, g, P) = δM,M′Φs(g)+(M(s, π)Φs)(g). Here δM,M′ is the Kronecker δ–function. Its analytic properties and therefore those
- f M(s, π) are exactly the same as E(s, Φs, −, P).
In the split case the proof of the first part is in [La1], as in elsewhere. But the rest is among the main properties of Eisen-
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stein series and included in several places [La2,HC,MW1]. To prove the first part, one just substi- tutes (3.3) in (3.15) and expands and uses Bruhat decomposition and cuspidality of ϕ. We finally express the functional equa- tion of Eisenstein series by (cf. [HC,La2]), (3.17) E(−s, M(π, s)Φs, g, P ′) = E(s, Φs, g, P).
and Automorphic L–Functions It follows immediately from the Constant Term Theorem that ⊗vA(s, πv, w0) is a mero- morphic function of s with a finite num-
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ber of simple poles for Re(s) > 0, none on Re(s) = 0. Assume v is an unramified place for π, i.e., that πv is spherical. Take f 0
v ∈ V (s, πv)
such that f 0
v (k) is a fixed vector invariant
under M(Ov) for all k ∈ G(Ov). With no- tation as in Section 2, let r =
m
ri be the adjoint action of LM on Ln. We have Lemma 4.1 [La1]. Assume πv is unrami-
A(s, πv, w0)f 0
v (ev)
=
m
L(is, πv, ˜ ri)/L(1 + is, πv, ˜ ri)f 0
v (ev).
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With a clever induction [La1,La2,Sh3] the problem reduces to that of SL2, which we will now explain. Here G = SL2 and M = T = Gm = F
∗ v.
We need to calculate (4.1.1)
f 0
v
−1 1 x 1
=
f 0
v
x 1
since Rwf 0
v = f 0 v , where w =
1
We then have (4.1.2)
f 0
v
1 x 1 dx =
dx +
f 0
v
1 x 1 dx.
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We now write (4.1.3)
x 1
1 x −1 1 x−1
and therefore for |x|v > 1, which implies x−1 ∈ Ov, f 0
v
1 x 1 = f 0
v
x−1 1 x = η−1
v (x)|x|−1−s v
, where ηv is the character of F ∗ (with cusp form η = ⊗vηv on A∗
F ) defining the induced
- representation. Then (4.1.2) equals
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1 +
∞
ηv(̟n
v )q−n−ns v
v
)−(̟−n+1
v
)
|x|vd∗x = 1 +
∞
ηv(̟n
v )q−n−ns v
|̟−n
v |v
v
d∗x = 1 +
∞
ηv(̟n
v )(1 − q−1 v )q−ns v
. Write ηv(x) = |x|µv
v
to get (4.1.4) 1 +
∞
q−nµv−ns
v
(1 − q−1
v )
= 1 + (1 − q−1
v ) · q−µv−s v
· 1 1 −
1 qµv+s
v
= 1 −
1 qµv+1+s
v
1 −
1 qµv+s
v
.
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34
Now if α∨ is the standard coroot of SL2, then qµv
v
= α∨(Av), following our HT identification. (See the remark below). Here Av ∈ PGL2(C) repre- sents the semisimple conjugacy class parametriz- ing πv and α∨ is the root of PGL2(C), or the coroot of SL2. Clearly α∨(Av) is the eigenvalue for the adjoint action of LT on
Ln, evaluated at Av. Therefore
(4.1.5) q−µv
v
= α∨(Av)−1 = ˜ r(Av) We thus get that (4.1.1) equals (4.1.6) (1−α∨(Av)−1q−s
v )−1/(1−α∨(Av)−1q−s−1 v
)−1
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which equals (4.1.7) L(s, ηv, ˜ r1)/L(1 + s, ηv, ˜ r1) since m = 1, i.e., r = r1. SL2(R). It is instructive to also compute the case of SL2(R)(GL2(R), respectively). We again need to calculate (4.1.12)
f 0
v
1 x 1 dx. Here K = SO2(R)(O2(R), respectively) and we can write
x 1
y b
where k(θ) =
sin θ − sin θ cos θ
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We can then take tan θ = −x, b = a−1 = √ x2 + 1 and y = x/ √ x2 + 1. We need to calculate ∞
−∞
√ x2 + 1 s1 (
where η a b = |a|s1|b|s2, or 2 ∞ (x2 + 1)−(s1−s2+1)/2dx. Let s = s1 − s2 and set x = tan θ, we need to calculate 2 π/2 (cos θ)s−1dθ.
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Using the standard formula π/2 cosp θ sinq θdθ = Γ p + 1 2
q + 1 2
p + q 2 + 1
where Γ(s) = ∞ tse−td∗t, (4.1.12) then equals (4.1.13) Γ(1/2)Γ(s/2)/Γ((s + 1)/2). Using Γ(1/2) = √π, (4.1.13) equals π−s/2Γ(s/2)/π−(s+1/2)Γ(s + 1/2)
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which is again L(s)/L(s + 1), where L(s) is the archimedean Hecke–Tate L–function at- tached to the character |x|s, the R–component
The main result of Langlands in [La1] can be stated as follows. Let S be a finite set of places with the property that if v ∈ S, πv is an unramified representation. Every f ∈ V (s, π) is of the form f ∈ ⊗v∈SV (s, πv)
v },
where f 0
v is Kv–spherical for some S. To
be precise, the decomposition π = ⊗vπv de- pends on a choice of M(Ov)–invariant vec- tors {xv} for all the unramified places which
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- ne fixes once for all (cf. [S]). The functions
f 0
v must then satisfy f 0 v (kv) = xv for all
kv ∈ Kv and all v ∈ S. Assume further that f = ⊗vfv, with fv = f 0
v for all v ∈ S.
For each i, let (4.1) LS(s, π, ri) =
L(s, πv, ri). Then by Lemma 4.1 A(s, π)f(e)=(
m
LS(is, π, ˜ ri)/LS(1 + is, π, ˜ ri)) ⊗v∈Sf 0
v (ev)
(4.2) It now follows from the properties of the constant term A(s, π) that
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Theorem 4.2 (Langlands) [La1]. The product quotient
m
LS(is, π, ˜ ri)/LS(1 + is, π, ˜ ri) is meromorphic on all of C. Clearly one needs an induction to get this to lead to meromorphic continuation of each L–function in the product. We will soon discuss this induction.
We shall now give a number of important examples of L–functions which appear in constant terms for appropriate pairs (G, M). We refer to [La1,Sh2] for the complete list.
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5.1. Let G = GLn+t, M = GLn × GLt, π = ⊗vπv cusp form on GLn(AF ) and π′ = ⊗vπ′
v one on GLt(AF ). Then m = 1
and we get L(s, π × ˜ π′), the Rankin–Selberg product L–function for the pair (π, ˜ π′) (cf. [JPSS1] and [Sh10]). It will be discussed by Cogdell in more length. 5.2. Let G be a classical group, split
- ver F and let M = GLn × G′, where G′ is
a classical group of the same type, but lower
- rank. Let π and π′ be cuspidal representa-
tions of GLn(AF ) and G′. Then m = 2. One gets L(s, π × ˜ π′) as its first L–function. For i = 2, we get L(s, π, ρ), where ρ = Λ2 if LG is orthogonal and ρ = Sym2 if LG is
SLIDE 42
42
symplectic (cf. [CKPSS1,2]. 5.3. Let G = GSpinn+t, M = GLn × GSpint, π and π′ cusp forms on GLn(AF ) and GSpint(AF ), respectively. Then m = 2. Again we get L(s, π × ˜ π′) as our first L–function. The second L–function is then an appropriate twist of either L(s, π, Λ2) or L(s, π, Sym2). 5.3.a. Let G = GSpin5+2n, M = GLn× Gpin5 = GLn × GSp4, and let (π, π′) be a cusp form on GLn(AF )×GSp4(AF ). Again we get L(s, π × ˜ π′) as our first L–function. This is very important. 5.3.b. G = GSpin6+2n, M = GLn ×
SLIDE 43 43
- GSpin6. We have 0 → {±1} → GL4(C) →
GSO6(C) → 0. Suppose π is on GLn(AF ) and π′ on GL4(AF ). We then get L(s, π ⊗ ˜ π′, ρn⊗Λ2ρ4) as our first L–function (cf. [K4]). 5.4. Let G be a simply connected group
- f either type E6 or E7. Choose M such
that MD, the derived group of M, is ei- ther SL3 ×SL2 ×SL3 or SL3 ×SL2 ×SL4,
- respectively. There exist F–rational injec-
tions from M into GL3 × GL2 × GLt, t = 3
- r 4, which are identity on SL3×SL2×SLt.
Let π1 ⊗π2 ⊗σ be a cuspidal representation
- f GL3(AF ) × GL2(AF ) × GLt(AF ). Then
m = 3 or 4 according as if G = E6 or E7. The first L–function is then L(s, π1×π2ט σ)
SLIDE 44 44
(cf. [KS2]). All these L–functions will be revisited later in connection with functoriality.
- 6. Local Coefficients, Non–constant
Term and the Crude Functional Equation Changing the splitting in U, we may as- sume each χv is defined by means of ψv through equation (2.5). At each v, let λχv (s, πv) be the Whittaker functional defined by equation (2.11). Next, let A(s, πv, w0) be the local intertwining operator defined by (3.10). Finally, let λχv(−s, w0(πv)) be the corresponding functional defined for I(−s, w0(πv)) by means of (2.11). Using our
SLIDE 45 45
assumption on χv, Rodier’s theorem points to the existence of a complex function Cχv(s, πv)
- f s, depending on πv, χv and w0 such that
(6.1) λχv(s, πv) = Cχv(s, πv) λχv(−s, w0(πv)) · A(s, πv, w0). This is what we call the “Local Coeffi- cient” attached to s, πv, χv and w0 (cf. [Sh2,Sh3]). The choice of w0 is now specified by our fixed splitting as in [Sh4]. Now, let (6.2) Eχ(s, Φs, g, P) =
E(s, Φs, ug, P)χ(u)du, the χ–nonconstant term of E(s, Φs, g, P) (cf. [Sh2,Sh3,Sh6]), where χ = ⊗vχv.
SLIDE 46 46
If we substitute for E(s, Φs, ug, P) its def- inition in (3.3) and do some telescoping, we get, using orthogonality of χ, that (6.3) Eχ(s, Φs, e, f) =
λχv(s, πv)(fv), where fv is the local component of f defined by (3.6) in which ϕ = ⊗vϕv is identified with Wϕ = ⊗vWv, where Wϕ is defined by (2.7). As explained in [S], Wv(ev) = 1 for almost all v. We now appeal to the following formula of Casselman–Shalika [CS]: Theorem 6.1(Casselman-Shalika [CS]). Assume πv and ψv are both unramified and if fv(ev) defines a Whittaker function Wv in
SLIDE 47 47
the Whittaker model of πv, assume Wv(ev) =
- 1. Observe that this is the case for almost
all v. Then (6.4) λχv(s, πv)(fv) =
m
L(1 + is, πv, ˜ ri)−1. In fact, if Wfv(gv) = λχv(s, πv)(Iv(gv)fv) is the Whittaker function attached to fv, then (6.3) can be written as Theorem 6.2. One has (6.5) Eχ(s, Φs, e, f) =
Wfv(ev)
m
LS(1 + is, π, ˜ ri)−1.
SLIDE 48 48
- Remark. As opposed to intertwining oper-
ators, Whittaker functions are by no means multiplicative and therefore proof of Theo- rem 6.1 cannot be reduced to rank one cal- culations by means multiplicativity (cocycle relations). It should be pointed out that in the case of SL2, Theorem 6.1 is an easy ex- ercise. Corollary 6.3 [Sh3]. The product
m
LS(1 + is, π, ri) = 0 for Re(s) = 0. In particular, if π and π′ are cusp forms on GLn(AF ) and GLt(AF ), then L(1, π × π′) = 0,
SLIDE 49 49
where local L–functions at every place are corresponding Artin ones ([HT], [He1]).
- Proof. Modulo non–vanishing of Wfv(e) for
Re(s) = 1, which is highly non–trivial if v = ∞ (cf. Casselman–Wallach [Ca2,W]), this follows from unitarity (and therefore holomorphy) of M(s, π) for Re(s) = 0, and Theorem 6.2. Observe that the integration in (6.2) is over a compact set. Now, computing the non–constant terms from the two sides of the functional equa- tion (3.17), Lemma 4.1 and Theorems 6.1 and 6.2, together with Definition (6.1) im- plies:
SLIDE 50 50
Theorem 6.4 (Crude Functional Equa- tion) [Sh3,Sh6]. We have (6.6)
m
LS(is, π, ri) =
Cχv(s, ˜ πv)
m
LS(1 − is, π, ˜ ri). We just point out that by Lemma 4.1, Theorem 6.1 and Definition (6.1) (6.7) Cχv(s, ˜ πv) =
m
L(1−is, πv, ˜ ri)/L(is, πv, ri), whenever πv is unramified.
SLIDE 51 51
- 7. The Main Induction, Functional
Equations and Multiplicativity To prove the individual functional equa- tions, i.e., for each L(s, π, ri) with precise root numbers and L–functions , we appeal to the following induction statement (cf. [Sh1,Sh2]). It is crucial in all the results that we prove from now on. Proposition 7.1 [Sh1]. Given 1 < i ≤ m, there exists a split group Gi over F, a maximal F–parabolic subgroup Pi = MiNi and a cuspidal automorphic representation π′ of Mi = Mi(AF ), unramified for every v ∈ S, such that if the adjoint action of
SLIDE 52 52
LMi on Lni decomposes as r′ = m′
r′
j, then
LS(s, π, ri) = LS(s, π′, r′
1).
Moreover m′ < m. It was observed by Arthur [A], that each Mi can be taken equal to M and π′ = π. More precisely: Proposition 7.2 [A]. Given i, 1 < i ≤ m, there exists a split connected reductive F– group Gi with M as a Levi subgroup and m′ < m for which r′
1 = ri. Each Gi can be
taken to be an endoscopic group for G. (Its L–group is the centralizer of a semisimple element in LG.)
SLIDE 53 53
Next, we need the following variant of a result of Henniart and Vigneras. In it, we assume that the defining additive character ψ for χ is local component of a global one. Here we shift the ramification to archimedean
- places. Consequently, we need to use a re-
sult of Dixmier–Malliavin on convolution al- gebras for real semisimple groups. Proposition 7.3 [Sh1]. Let σ be an ir- reducible χ–generic supercuspidal represen- tation of G = G(F), where F is a non– archimedean local field and G is defined over
- F. Let B = TU be the Borel subgroup of
G defining χ. Then there exists a num- ber field K with a ring integers O, a split
SLIDE 54 54
group H over K, a non–degenerate charac- ter ˜ χ = ⊗v ˜ χv of UH(K)\ UH(AK), and a globally ˜ χ–generic cusp form π = ⊗vπv on H = H(AK) such that: a) Kv0 = F for some place v0 of K, b) ˜ χv0 = χ, c) as a group over F, H = G, d) πv0 = σ, and finally e) for every other finite place v of K, v = v0, πv is of class one with respect to a special maximal compact subgroup Qv
- f H(Kv). Here UH is the unipotent
radical of a Borel subgroup of H for which UH as a group over F equals U.
SLIDE 55 55
Proposition 7.3 [Sh1,Sh4]. Assume ei- ther Fv is archimedean or πv has a vector fixed by an Iwahori subgroup. Let ϕv: W ′
Fv → LMv be the homorphism of the Deligne–Weil
group parametrizing πv. For each i, let L(s, ri· ϕv) and ε(s, ri·ϕv, ψv) be the Artin L–function and root number attached to ri · ϕv. Then (7.1) Cχv(s, πv)=
m
ε(is, ˜ ri·ϕv, ψv)L(1 − is, ri · ϕv) L(is, ˜ ri · ϕv) .
- Remark. If G is quasisplit, but not split,
then a product of Langlands λ–functions (Hilbert symbols) will also appear on the right hand side of (7.1).
SLIDE 56 56
Applying these propositions inductively and using the Crude Functional Equa- tions (6.6) then implies: Theorem 7.4 [Sh1]. Assume G is a split reductive algebraic group over a local field F
- f characteristic zero. Let P = MN, P ⊃
B, be a maximal parabolic subgroup as be-
- fore. Let χ be a generic character defined
by the splitting and ψF ∈ ˆ
ducible admissible χ–generic representation σ of M = M(F), these exist m complex functions γ(s, σ, ri, ψF ), 1 ≤ i ≤ m, such that:
SLIDE 57 57
1) If F and σ satisfy the conditions of Propo- sition 7.3, then (7.2) γ(s, σ, ri, ψF ) = ε(s, ri · ϕ, ψF ) L(1 − s, ˜ ri · ϕ)/L(s, ri · ϕ). 2) Equation (7.1) holds (in the form
m
γ(is, σ, ˜ ri, ψF )). 3) γ(s, σ, ri, ψF ) is multiplicative under in- duction (to be discussed below). 4) Whenever σ becomes a local component
- f a globally generic cusp from, then γ’s
become the local factors needed in their functional equations. Moreover 1), 3) and 4) determine the γ–functions uniquely.
SLIDE 58 58
What is multiplicativity? We will discuss this only in examples since the gen- eral formulation is complicated. It simply says that the γ–functions are multiplicative under parabolic induction and is a conse- quence of multiplicativity of intertwining op- erators (3.10) under that (cf. [Sh3]). This is very deep from the point of view of Rankin– Selberg method and usually quite hard to
- prove. Here are some examples:
Example 1 (cf. [Sh7]). Suppose G = Sp(2n + 2t) and M = GLn × Sp(2t), where n and t are positive integers. Write σ = σ1 ⊗ τ. Suppose M′ = GLn1 × . . . × GLnk × GLt1 × . . . × GLtℓ × Sp(2a), where
SLIDE 59 59
n1+. . .+nk = n and t1+. . .+tℓ+a = t. By case Cn of [Sh2], r1 is equal to the tensor product of the standard representation of GLn(C) and SO2t+1(C). If σ′ =
k
σ′
j ⊗ ℓ
σ
′′
b ⊗ τ ′, then multi-
plicativity simply means that, if σ ⊂ Ind
M ′N ′↑Gσ′ ⊗ 1,
then (7.3) γ(s, σ1 × τ, ψF ) =
k
ℓ
γ(s, σ′
j × σ
′′
b , ψF )
γ(s, σ′
j × ˜
σ
′′
b , ψF ) k
γ(s, σ′
j × τ ′, ψF ).
SLIDE 60 60
If ρn is the standard representation of GLn(C), then r2 = Λ2ρn. With σ1 and σ′
j
as before, multiplicativity for r2 means γ(s, σ1, Λ2ρn, ψF ) =
k
γ(s, σ′
j, Λ2ρnj, ψF )
γ(s, σ′
i × σ′ j, ψF ).
No more ri beyond r2 shows up and this is the case for all the classical groups. Equal- ity of the dimension on both sides of (7.4) simply means the following trivial identity: k
ni 2 − k
ni
k
(n2
i −ni)+2
ninj.
SLIDE 61
61
L–function and root number L–functions are now defined using γ–functions. When σ is tempered, we define L(s, σ, ri) as the inverse of the normalized polynomial P(q−s) in q−s satisfying P(0)= 1 and (7.5) γ(s, σ, r1, ψF ) = ε(s, σ, ri, ψF ) L(1 − s, σ, ˜ ri)/L(s, σ, ri). The L–function L(s, σ, ˜ ri) and the root number ε(s, σ, ri, ψF ) are also uniquely de- fined by (7.5). To proceed we need the fol- lowing theorem.
SLIDE 62
62
Theorem 7.5. Suppose σ is tempered. Then L(s, σ, ri)are all holomorphic for Re(s) > 0. With this theorem in hand, L(s, σ, ri) are now also multiplicative if σ is tempered. (See below.) To define L(s, σ, ri) for any irreducible χ– generic representation, we appeal to Lang- lands classification [La3,Si]. We embed σ ⊂ IndM ′(N ′∩M)↑Mσ′
ν ⊗ 1, where σ′ ν is quasi–
tempered with a negative Langlands param- eter ν. Then σ′
0 is tempered.
By multi- plicativity, we then write γ(s, σ, ri, ψF ) as a product of appropriate γ–functions γ(s, σ′
ν,j, rij, ψF ), where j runs over a finite
SLIDE 63 63
index set determined by M′ and M, i.e., (7.6) γ(s, σ, ri, ψF ) =
γ(s, σ′
ν,j, rij, ψF ).
More precisely, for each j, there exist Levi subgroups (not necessarily maximal) M′
j and ˜
Mj of G with T ⊂ M′
j ⊂ ˜
Mj as a maximal Levi subgroup. The representa- tion σ′
ν,j is a quasi–tempered representation
j for which σ′ 0,j is tempered. The repre-
sentation rij of LM ′
j is an irreducible consti-
tutent of the action of LM ′
j on the Lie alge-
bra of the L–group of ˜ Mj ∩N′
j, where N′ j ⊂
U is the unipotent radical of P′
j = M′ jN′ j.
Thus the γ–function γ(s, σ′
ν,j, rij, ψF ) is a
SLIDE 64 64
γ–function attached to the pair ( ˜ Mj, M′
j).
When ν = 0, by Conjecture 7.5, the prod- uct of the numerators of γ(s, σ′
0,j, rij, ψF )
equals to the numerator of the product and if L(s, σ′
0,j, rij)−1 denotes the normalized
numerator of γ(s, σ′
0,j, rij, ψF ), i.e., the re-
ciprocal of the tempered L–function attached to σ′
0,j and rij by means of the pair ( ˜
Mj, M′
j),
we then use L(s, σ′
ν,j, rij) to denote its an-
alytic continuation to ν. We now set (7.7) L(s, σ, ri) =
L(s, σ′
ν,j, rij).
This agrees with the way Artin L–functions are defined [La3,KSh,Sh1,T]. Details are given in [Sh3]; also see the discussion in pages
SLIDE 65 65
862 and 863 of [KS2]. The root number is then defined uniquely to satisfy (7.5). We should point out that in Definition (7.7) we do not need to assume the validity of Con- jecture 7.5. But if valid, it implies that L– functions are also multiplicative, if the rep- resentations are tempered. Having defined our L–functions and root numbers everywhere, we set L(s, π, ri) =
L(s, πv, ri) and ε(s, π, ri) =
ε(s, πv, ri, ψv). We then have:
SLIDE 66
66
Theorem 7.6(Functional Equation [Sh1]). For each i, 1 ≤ i ≤ m, (7.8) L(s, π, ri) = ε(s, π, ri)L(1 − s, π, ˜ ri). Exercise 1. Use the pair (G, M), G = GL3 and M = GL2 × GL1, to get the stan- dard L–function for GL2. Determine L– functions using our method and show that they are equal to those of Jacquet–Langlands. Exercise 2. Let G = Esc
6
and M be such that MD = SL3 × SL2 × SL3. Fact 1. There exists a F–rational map (in- jection) f: M → GL3 × GL2 × GL3 whose restriction to MD is identity.
SLIDE 67 67
Fact 2. m = 3 and if π2⊗π1⊗σ is an unram- ified representation of GL3(F) × GL2(F) × GL3(F), where F is a local field, then L(s, π2×π1ט σ) = L(s, (π2⊗π1⊗σ)·f, r1). Define γ(s, π2 × π1 × ˜ σ) to be γ(s, (π2 ⊗ π1 ⊗ σ) · f, r1), using our method for arbi- trary local representations π2 ⊗ π1 ⊗ σ. As- sume ˜ σ = Ind
(F ∗)3×U↑GL3(F )(µ1 ⊗ µ2 ⊗ µ3) ⊗ 1.
Show that multiplicativity implies: γ(s, π2 × π1 × ˜ σ) =
3
γ(s, π2 × (π1 ⊗ µj)), where the γ–function on the right are those
- f Rankin–Selberg for GL3(F) × GL2(F).
SLIDE 68 68
(This is crucial to Kim–Shahidi’s proof of functoriality [KS2] of the inclusion GL2(C)⊗ GL3(C) ֒ → GL6(C), to be discussed later.) Exercise 3. Let G = SO(2m + 2n + 1) and M = GLm × SO(2n + 1). Let σ ⊗ π be an irreducible admissible χ–generic representa- tion of GLm(F)×SO2n+1(F), where F is a local field. Assume σ ֒ → Ind
(F ∗)m×U↑GLm(F ) m
µj. Show that multiplicativity implies: γ(s, σ × π) =
m
γ(s, π ⊗ µj). (This is crucial to Cogdell–Kim–Piatetski– Shapiro–Shahidi’s proof [CKPSS1] of func-
SLIDE 69 69
torial transfer from generic cusp forms on classical groups to GL2n(AF ).)
- 8. Twists by Highly Ramified Characters
Holomorphy and Boundedness Since our aim is to establish those an- alytic properties of L–functions from our method which are crucial in proving the strik- ing new cases of functoriality, we will limit
- ur discussion on the issue of holomorphy of
L–functions only to twists by highly rami- fied characters. In fact, as we explained in earlier sections, the functional equations for L–functions within our method are proved quite generally and multiplicativity and the
SLIDE 70 70
related machinary necessary for applying con- verse theorems to our L–functions are in perfect shape. Nothing that general can be said about the holomorphy and possible poles of these L–functions. On the other hand, there has recently been some remarkable new progress
- n the question of holomorphy of these L–
functions, mainly due to Kim [K2,K3,KS1]. They rely on reducing the existence of the poles to that of existence of certain unitary automorphic forms, which in turn point to the existence of certain local unitary rep- resentations. One then disposes of these representations, and therefore the pole, by
SLIDE 71 71
checking the corresponding unitary dual of the local group. In view of the functional equation, this needs to be checked only for Re(s) ≥ 1/2, if a certain local assumption
- n normalized local intertwining operators
is valid. To explain, let A(s, πv, w0) be the local intertwining operator attached by equa- tion (3.10) to our inducing representation πv. We recall that we are dealing with a pair (G, M) and a χ–generic cuspidal rep- resentation π = ⊗vπv of M = M(AF ). Let, for each i, 1 ≤ i ≤ m, L(s, πv, ri) and ε(s, πv, ri, ψv) be the corresponding L–function and root number specified earlier. We de-
SLIDE 72 72
fine a normalized operator N(s, πv, w0) by (8.1) N(s, πv, w0) = r(s, πv, ψv)A(s, πv, w0), where the normalizing factor is defined as [Sh1] (8.2) r(s, πv, ψv) =
m
ε(is, πv, ˜ ri) L(1 + is, πv, ˜ ri)/L(is, πv, ˜ ri). Theorem 8.1. The operator N(s, πv, w0) is holomorphic and non–zero for Re(s) ≥ 1/2. It should be mentioned that it is a result
- f Yuanli Zhang [Z] that, if Theorem 7.5
SLIDE 73 73
is valid for the pair (G, M), then the non– vanishing of N(s, πv, w0) for Re(s) ≥ 1/2 follows from its holomorphy over the same range. Arguments given in [CKPSS1,K4,KS2], then proceed under the validity of Theorem 8.1 for (G, M) as well as for all other related lower rank pairs (that come into the multi- plicativity), which consequently are verified in each of the cases in [CKPSS1,K4,KS2]. The main issue with this argument is that
- ne cannot always get such a contradiction
and rule out the pole. In fact, there are many unitary duals whose complementary series extend all the way to Re(s) = 1, mak-
SLIDE 74 74
ing the results far from general. On the other hand, if one considers a highly ramified twist πη (see below) of π, then it can be shown quite generally that every L(s, πη, ri) is entire (cf. [Sh8] for its local analogue). In fact, if η is highly ram- ified, then by checking central characters, w0(πη) ≃ πη, whose negation is a necessary condition for M(s, πη) to have a pole, a ba- sic fact from Langlands spectral theory of Eisenstein series (Lemma 7.5 of [La2]). This lemma was used by Kim in [K2] and in view
- f the present powerful converse theorems of
Cogdell and Piatetski–Shapiro [CPS1,2,3], that is all one needs to prove our cases of
SLIDE 75 75
functoriality [CKPSS1,K4,KS2]. We formal- ize this by quoting the following (Proposi- tion 2.1) from [KS2]. Theorem 8.2. There exists a rational char- acter ξ ∈ X(M)F = X(M), with the follow- ing property. Let S be a non–empty finite set of finite places of F. For every globally generic cuspidal representation π of M = M(AF ), there exist non–negative integers fv, v ∈ S, such that for every gr¨
η = ⊗vηv of F for which the conductor of ηv, v ∈ S, is larger than or equal to fv, every L–function L(s, πη, ri), i = 1, . . . , m, is entire, where πη = π ⊗ (η · ξ). The ra- tional character ξ can be simply taken to be
SLIDE 76 76
ξ(m) = det(Ad(m)|n), m ∈ M, where n is the Lie algebra of N. The last ingredient in applying converse theorems is that of boundedness of each L(s, π, ri) in every vertical strip of finite width, away from its finite number of poles. The finiteness of poles is again a consequence
- f the finiteness of the poles of M(s, π) for
Re(s) ≥ 0 and the functional equation satis- fied by each L(s, π, ri), but under the valid- ity of Assumption 8.1 (cf. [GS1]). In this full generality, the boundedness in finite vertical strips, away from their poles, were proved by Gelbart–Shahidi in [GS1], again using
- ur method. The main difficulty in proving
SLIDE 77 77
this result is having to deal with reciprocals
- f each L(s, π, ri), 2 ≤ i ≤ m, near and on
the line Re(s) = 1, the edge of critical strip, whenever m ≥ 2, which is unfortunately the case for each of the L–functions appearing in our cases of functoriality. We handle this by appealing to equation (6.5) (Theorem 6.2) and estimating the non–constant term (6.2) by means of Langlands [HC,La2] and M¨ uller [Mu], and a non–trivial result from complex function theory (Matsaev’s theo- rem). Here is the statement of the main re- sult of [GS1] as formulated for πη to avoid the issue of poles.
SLIDE 78 78
Theorem 8.3. Let ξ and η be as in The-
Assume η is sufficiently rami- fied so that each L(s, πη, ri) is entire. Then, given a finite real interval I, each L(s, πη, ri) remains bounded for all s with Re(s) ∈ I.
- Remark. Another proof of Theorem 8.3 is
due to Gelbart and Lapid, following some ideas of Sarnak.
- 9. Examples of Functoriality with Applications
Consider the embedding i: GL2(C) ⊗ GL3(C) ֒ → GL6(C). This is a homomorphism from LGL2×LGL3 into LGL6. Accordingly functoriality pre-
SLIDE 79 79
dicts a map Aut (GL2(AF )) × Aut (GL3(AF )) → Aut (GL6(AF )). More precisely, let π1 = ⊗vπ1v and π2 = ⊗vπ2v be cusp forms on GL2(AF ) and GL3(AF ), respectively, with π1v given by t1v ∈ GL2(C) and π2v by t2v ∈ GL3(C) for almost all v. Let
v be the irreducible admissible spher-
ical representation of GL6(Fv) defined by {t1v ⊗ t2v} ⊂ GL6(C). Then we can state the functoriality in this case as:
- Functoriality. There exists an automor-
phic representation ′ = ⊗v ′
v of GL6(AF )
such that ′
v = v for almost all v.
SLIDE 80 80
More precisely, let ρiv: W ′
Fv → GLi+1(C),
i = 1, 2, parametrize πiv (Harris–Taylor [HT], Henniart [He1]) for all v. Let ρ1v ⊗ ρ2v be the six dimensional representation of W ′
Fv,
i.e., the homomorphism ρ1v ⊗ ρ2v: W ′
Fv → GL6(C).
Denote by π1v ⊠ π2v the irreducible admis- sible representation of GL6(Fv) attached to ρ1v ⊗ ρ2v. Let π1 ⊠ π2 = ⊗v(π1v ⊠ π2v). Theorem 9.1 (Kim–Shahidi [KS2]). The irreducible admissible representation π1⊠π2
- f GL6(AF ) is automorphic. Thus GL2(C)⊗
GL3(C) ֒ → GL6(C) is functorial.
SLIDE 81
81
For the proof, one applies an appropriate version of the converse theorem [CPS2] to the following cases of our method. In each case G and MD, the derived group of M, are given as follows. 1) G = SL(5)(orGL(5)), MD = SL2× SL3 2) G = Spin(10), MD = SL3 × SL2 × SL2 3) G = Esc
6 ,
MD = SL3 × SL2 × SL3 4) G = Esc
7 ,
MD = SL3 × SL2 × SL4 We then get the necessary analytic prop- erties of the highly ramified twisted L–functions L(s, π1 × π2 × (σ ⊗ η)), σ ⊗ η = σ ⊗ η · det,
SLIDE 82 82
η a highly ramified gr¨
σ’s are appropriate cusp forms on GLj(AF ), j = 1, 2, 3, 4, respectively. Observe that L(s, π1v ×π2v ×σv) = L(s, (π1v ⊠π2v)×σv) for almost all v. Similarly for root numbers. We in fact prove these equalities for all v. This is immediate from the fact that the local components of the weak transfer es- tablished through the converse theorem (The-
- rem 3.8 of [KS2]) is in fact π1v ⊠ π2v for
each v (Theorem 5.1 of [KS2]). Proof of this is quite delicate and beside the tech- niques already discussed (functional equa- tions, multiplicativity,...), relies on several
SLIDE 83 83
- ther techniques such as base change, both
normal [AC] and non–normal [JPSS2], as well as certain results from the theory of types [BH]. Next, let π = ⊗vπv be a cusp form on GL2(AF ). Let Ad(π) be its Gelbart–Jacquet transfer [GJ]. Then π ⊠ Ad(π) = Sym3(π) ⊗ ω−1
π
⊞ π implies that Sym3(π) is an automorphic rep- resentation of GL4(AF ). More precisely, for every v let ρv: W ′
Fv →
GL2(C) be the two dimensional represen- tation of the Deligne–Weil group W ′
Fv at-
tached to πv (cf. [Ku]).
SLIDE 84 84
Consider: Sym3 · ρv = Sym3ρv: W ′
Fv → GL4(C).
Let Sym3πv be the irreducible admissible representation of GL4(Fv) attached to Sym3ρv. Set Sym3π = ⊗vSym3πv. Theorem 9.2 (Kim–Shahidi [KS2]). Sym3π is an automorphic representation of GL4(AF ). It is cuspidal unless π is of di- hedral or tetrahedral type. Next, let = ⊗v
sentation of GL4(AF ). Let Λ2: GL4(C) → GL6(C) be the exterior square map. Let
SLIDE 85 85
ϕv: W ′
Fv → GL4(C) parametrize v for all
v [HT,He1]. Then Λ2ϕv: W ′
Fv → GL6(C)
parametrizes Λ2
v, an irreducible admis-
sible representation of GL6(Fv). Theorem 9.3 (Kim [K4]). There exists an automorphic representation ′ = ⊗v ′
v
v = Λ2 v.
Corollary 9.4 [K4]. The representation Sym4(π) is automorphic, where Sym4(π) = ⊗vSym4(πv) and π = ⊗vπv is a cusp form on GL2(AF ).
- Proof. Apply Λ2 to Sym3π;
Λ2(Sym3π) = Sym4π ⊗ ωπ ⊞ ω3
π
SLIDE 86
86
Proposition 9.5 (Kim-Shahidi [KS3]). Sym4(π) is cuspidal unless π is of dihedral, tetrahedral or octahedral type. Theorem 9.3 is proved by using G=Spin2k+8,MD =SLk+1×SL4, k=0, 1, 2, 3. We get L(s, π ⊗ σ, Λ2ρ4 ⊗ ρk+1) for each cusp form σ on GLk+1(AF ), k = 0, 1, 2, 3. If we use the isogeny SL4 → SO6 → 0, we note that functoriality established in Theorem 9.3 is a special case of functoriality for the embedding GSO2n(C) ֒ → GL2n(C).
SLIDE 87
87
The result is the transfer Aut(GSpin2n(AF )) Aut(GL2n(AF )). This is now established by Asgari–Shahidi in Duke J. in the weak form. The full trans- fer is now being written. It needs general- ization of the descent of Ginzburg–Ralis– Soundry to GSpin groups. This is half– done by Hundley–Sayag. We have now com- pleted the rest of it and expect to prove results as strong as those of [CKPSS] for GSpinm, m = odd or even. We need some results on LS(s, π, Λ2⊗χ) and LS(s, π, Sym2⊗ χ) when π is a cusp form on GLn(AF ). (We need holomorphy for both for Re(s) >
SLIDE 88 88
- 1. This implies non–vanishing for both for
Re(s) > 1. Much stronger results now seem to be available through the work of D. Belt for Λ2 ⊗ χ and S. Takeda for Sym2 ⊗ χ, us- ing Jacquet–Shalika and Bump–Ginzburg, respectively.) In the cases of Asgari–Shahidi we have G = GSpin2(m+n), M = GLm × GSpin2n for 1 ≤ m ≤ 2n − 1. The L–functions are product L–functions. Apply the converse theorem. Applications: Theorem 9.6 (Kim–Shahidi [KS3]). Let F be an arbitrary number field. Let π =
SLIDE 89 89
⊗vπv be a cusp form on GL2(AF ). Assume πv is parametrized by tv =
βv
Then q−1/9
v
< |αv| and |βv| < q1/9
v
. Similar inequality holds at archimedean places (λ > 0.23765432...). This is proved using the techniques of [Sh2] (which led to 1/5 using Sym2π and groups of either type F4 or E6, by apply- ing a general theorem from [Sh2] which im- plies L(s, πv, Sym5(ρ2)) is holomorphic for
SLIDE 90
90
Re(s) ≥ 1 for all such v). When this general theorem is applied to a simply connected group of type E8 with a Levi M for which MD = SL5×SL4, together with a represen- tation related to Sym4π⊗ Sym3π leading to L(s, πv, Sym9(ρ2)), one gets 1/9. We refer to [CPSS] for an important application of Theorem 9.6. Next, we have Theorem 9.7 (Kim–Sarnak [KSa]). Suppose F = Q, then p−7/64 ≤ |αp| and |βp| ≤ p7/64. At the archimedean places we get the esti- mate λ ≥
975 4096 = 0.2380371 for the first
positive eigenvalue of ∆.
SLIDE 91 91
- Proof. This is proved by means of analytic
methods of Duke–Iwaniec [DI] applied to L(s, Sym4π, Sym2), (cf. [BG]) along the lines
- f Bump–Duke–Hoffstein–Iwaniec [BDHI]
which led to 5/28 + ε over Q, when applied to L(s, Sym2π, Sym2).
SLIDE 92 92
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Department of Mathematics Purdue University West Lafayette, IN 47907 USA shahidi@math.purdue.edu
