Table Of ContentMATCHING OF HECKE OPERATORS FOR EXCEPTIONAL DUAL
PAIR CORRESPONDENCES
GORDAN SAVIN AND MICHAEL WOODBURY
to Steve Rallis, in memoriam
3
1 Abstract. Let G be a split algebrac group of type En defined over a p-adic field. This
20 groupcontains a dual pair G×G′ where one of the groups is of type G2. The minimal rep-
resentationofG,whenrestrictedtothedualpair,givesacorrespondenceofrepresentations
n
of the two groups in the dual pair. We prove a matching of spherical Hecke algebras of G
Ja and G′, when acting on the minimal representation. This implies that the correspondence
is functorial, in the sense of Arthur and Langlands, for spherical representations.
3
2
]
T
N 1. Introduction
.
h Let F be a p-adic field with ring of integers O and fixed inverse of uniformizer ̟. Let q be
t
a the order of the residual field. We fix an absolute value on F so that |̟| = q. We consider
m exceptional dual pairs G×G′ inside of an adjoint group G where each group consists of the
[ F points of a split reductive algebraic group. Denote by K and K′ hyperspecial maximal
2 compact subgroups of G and G′ respectively. Let (Π,V) be the minimal representation of
v
G.
2
4 If σ′ is an irreducible representation of G′, we call an irreducible representation σ of G
5 a Θ-lift of σ′ if σ ⊗ σ′ is a quotient of Π. If σ,σ′ are spherical, we will prove that the
4
correspondence σ ↔ σ′ is functorial with respect to a natural injection on the dual groups
.
1
0 r : G′(C) → G(C).
3
1 To be more precise, let C be the centrablizer of rb(G′) in G. Then C is a reductive, possibly
:
v finite, group. Let f : SL (C) → C be a map corresponding to the regular unipotent orbit in
i 2 b b
X C by Jacobson-Morozov. Let
ar s = f q1/2 0 .
0 q−1/2
(cid:16) (cid:17)
Let H and H′ denote the spherical Hecke algebras of G and G′ respectively. Let T ∈ H
correspond, via the Satake isomorphism, to a finite dimensional representation V of G(C).
(We describe this in more detail in Section 2 below.) Write V = V′ ⊗V′′, the restriction
b
of V to G′ ⊗C. We define a map r˜: H → H′ by P
(1) b r˜(V) = TrV′′(s)V′.
XV′
Note that if C is finite, then s = 1, the identity in G, and r˜(V) is just the restriction of V
to G′. We consider the following dual pairs:
b
b
Partially supported by NSF grant DMS 0852429.
1
G D D E E E
4 5 6 7 8
G′ S PGL PGL G G
3 2 3 2 2
G G G G PGSp F
2 2 2 6 4
Table 1. Dual pairs G×G′ ⊂ G
Theorem 1.1. For the dual pairs G × G′ ⊂ G in Table 1, T ∈ H and r given by (1),
G
Π(T) = Π(r(T)) as operators on VK×K′.
e
As a matter of terminology, if the actions of T and r(T) agree on a space V, or, more
e
precisely, on a subset of fixed vectors, for all T ∈ H we will say there is a matching of Hecke
G
operators of HG and HG′ on V or, more concisely, matcheing on V. We trust that the precise
space of fixed vectors will be clear from context.
An analogue of Theorem 1.1 is well known in the case of classical theta correspondences
[5]. For exceptional groups, the first example of matching was obtained by Rallis and Soudry
in [6].
The proof of Theorem 1.1 is by induction on the rank of G. The main tool is Jacquet
functors of V with respect to maximal parabolic subgroups of dual pairs. Most of the needed
functors were computed in [4]. One remaining, but rather remarkable case (for G = E ), is
8
computed in the last section.
2. The Satake isomorphism
Before giving the proof of Theorem 1.1 we review the facts about the Hecke algebra and
the Satake isomorphism (most of which can be found in [2]) which will be relevant, and we
give some general lemmas which will be key in the proof of Theorem 1.1.
2.1. The Hecke algebra. Let G be a split reductive group, K a hyperspecial maximal
compact subgroup, B = TU aBorel subgroup. Thereis anIwasawa decompositionG = BK,
and the choice of Borel gives a set Φ+ of positive roots.
We identify the cocharacters of T, X (T), with the coweight lattice Λ so that for every
• c
cocharacter λ : F× → T, the adjoint action of λ(t) on the root subgroup of U, corresponding
to a root α, is given by the multiplication by the scalar thλ,αi. We have a Cartan decompo-
sition:
Proposition 2.1. The group G is the disjoint union of double cosets Kλ(̟)K for λ ∈ Λ+
c
where
Λ+ = {λ ∈ Λ | hλ,αi ≥ 0 for all α ∈ Φ+}.
c c
Example. G = GL (F), G = GL (C). If α = (1,−1,0) and α = (0,1,−1) are the simple
3 3 1 2
roots, then P+ consists of λ = (l,m,n) such that l ≥ m ≥ n. Then
b
tl
λ(t) = tm
(cid:16) tn(cid:17)
and it is easy to verify directly, from the theory of elementary divisors, that
̟l
GL3(F) = GL3(O) ̟m GL3(O).
G (cid:16) ̟n(cid:17)
l≥m≥n
2
Irreducible representations of G, the complex dual group of G, are parameterized by their
highest weights λ ∈ Λ+. Let R(G) be the representation ring of G. That is, R(G) is the
c b
C-vector space with basis consisting of the irreducible representations of G. We denote the
b b b
representation of highest weight λ by V , and consider the map
λ
b
(2) R(G) → C[Λ ] V 7→ m (µ)[µ]
c µ λ
Xµ
b
where m (µ) isthe dimension ofthe µ-weight space inV . Letting W denote the Weyl group,
λ λ
this gives an isomorphism R(G) ≃ C[Λ ]W.
c
The Hecke algebra H consists of all locally constant compactly supported K-biinvariant
G
functions f : G → C. By Prboposition 2.1, H has a basis consisting of the characteristic
G
functions of Kλ(̟)K with λ ∈ Λ+. We denote these by T .
c λ
If (σ,V) is a smooth G-module then the action of f ∈ H is given by
G
f ∗v = f(g)σ(g)vdg.
Z
G
We normalize the Haar measure dg so that vol(K) = 1. Let U be the unipotent radical of
B as above. By the Iwasawa decomposition, we can write
(3) Kλ(̟)K = utK
[u [t
for some representatives t ∈ T and u ∈ U. So if v is a K-fixed vector then
(4) T ∗v = σ(u)σ(t)v.
λ
Xu,t
If r : V → V is the natural projection, where V is the space of U-coinvariants, then
U U U
(5) r (T ∗v) = n(t)r (σ(t)v)
U λ U
Xt
where n(t) is the number of single cosets of type utK appearing in Kλ(̟)K.
2.2. The (relative) Satake transform. Let δ denote the modular character of B given
U
by
d(bub−1) = δ (b)du.
U
The measure du is normalized so that vol(K ∩U) = 1. Obviously, δ is trivial on U, and so
U
this defines a character δ : T → R×. We take δ1/2(t) to be the positive square root of this
+ U
character. Let Φ+ denote the positive roots of G (determined by B.) Then if µ ∈ X (T),
•
(6) δ1/2(µ(̟)) = qhµ,ρi, 2ρ = α.
X
α∈Φ+
The Satake transform S : H → H is given by
T G T
(7) S f(t) = δ(t)1/2 f(tu)du.
T
Z
U
ItisafactthatS isinjectiveanditsimageisequaltotheWeylgroupinvariants. SinceH =
T T
C[X (T)] = C[Λ ], it follows from our discussion above that this defines an isomorphism
• c
S : H → R(G).
G
3
b
The Satake transform can be defined analogously for any parabolic P = MN. This yields
the relative Satake transform S : H → H :
M G M
1/2
(8) S f(m) = δ (m) f(mn)dn.
M N Z
N
As above, dn is the measure which gives N ∩K volume 1, and δ : M → R× is the modular
N +
character.
We may assume P ⊃ B, so the composition of S with the Satake transform from H to
M M
H is S .
T T
2.3. Some general lemmas. In this section, we prove various simple lemmas which will
be used in our proof of Theorem 1.1.
Throughout this paper parabolic induction and the Jacquet functors will be normalized as
follows. Let P = MN be a parabolic subgroup of G. Suppose that (σ,W) is a representation
of M which we extend trivially to P. Then we define (ρ,iG(W)) to be the representation of
P
G consisting of smooth functions f : G → W which satisfy,
1/2
(9) f(mng) = δ (m)σ(m)f(g) for all m ∈ M,n ∈ N,g ∈ G
N
with right regular action ρ(g)f : h 7→ f(hg). Note that the usual induction functor is
(10) IndG(W) = iG(δ−1/2 ⊗W).
P P N
If (π,V) is a representation of G, the Jacquet functor with respect to N, (π ,r (V)), is
M N
defined as follows. As usual,
V(N) = hπ(n)v −v | n ∈ N,v ∈ Vi,
so V = V/V(N) is the space of coinvariants. Let (π ,r (V)) be a representation of M
N M N
such that r (V) = V , but the M-action is given by
N N
−1/2
(11) π (m)v = δ (m)π(m)v.
M N
Since N is normal, it is trivial to see that this is well defined.
Since the induction and the Jacquet functor are normalized, the statement of Frobenius
reciprocity is quite simple:
(12) Hom (V,iG(W)) = Hom (r (V),W).
G P M N
Lemma 2.2. Suppose that G is a split reductive group, K ⊂ G is a maximal compact
subgroup and MN = P ⊂ G is a parabolic subgroup. Let K = K ∩ M. For (π,V) any
M
smooth representation of M, the following statements hold.
(i) The map ϕ : (iG(V))K → VKM given by f 7→ f(1) is an isomorphism.
P
(ii) If T ∈ H and v ∈ (iG(V))K then ϕ(T ∗v) = S (T)∗ϕ(v).
P M
Proof. Since G = PK any element g in G can bewritten as g = mnk, where k ∈ K, mn ∈ P,
and we can define
ψ : VKM → (iG(V))K
P
by specifying that ψ(v)(mnk) = δ (m)1/2π(m)v. This is well defined precisely because
N
v ∈ VKM. Now, (i) follows by computing that ψ is the inverse of ϕ.
4
For remainder of the proof let S = S . Let T be the characteristic function of Kλ(̟)K.
M λ
Note that S(T ) is determined by its values on a set of coset representatives for M/K . We
λ M
fix such a set. Using the Iwasawa decomposition G = PK, we may write (in analogy to (3))
(13) Kλ(̟)K = m n K
i i
[i [j
with the m chosen from the given set of coset representatives. If m ∈/ Kλ(π)K, then
i
obviously S(T )(m) = 0. Otherwise, m = m for some m appearing in the decomposition
λ j0 j0
(13). Let
n(i,j) := #{n | m n K appears in (13)}.
i j i
Then
S(T )(m ) = δ1/2(m ) T (m n)dn
λ j0 j0 Z λ j0
N
= δ1/2(m ) T (m n )
j0 λ j0 i
Xi,j
= δ1/2(m )n(i,j ),
j0 0
since vol(N/(K ∩N)) = 1.
Let f ∈ (IndGV)K. By (i), ϕ(f) = f(1) = v ∈ VKM. So, by the previous calculation,
P
S(T )∗v = S(T )(m)π(m)vdm
λ λ
Z
M
= S(T )(m )π(m )v
λ j j
Xj
= δ1/2(m )n(i,j)π(m )v.
j j
Xj
On the other hand, since f is fixed by K,
ϕ(T ∗f) = (T ∗f)(1) = T (g)σ(g)f(1)dg
λ λ λ
Z
G
= f(1m n )
j i
Xi,j
= n(i,j)δ1/2(m )π(m )v.
j j
Xj
By Proposition 2.1, {T | λ ∈ Λ+} forms a basis of H. Therefore, (ii) is proved. (cid:3)
λ c
Lemma 2.3. Let G be a reductive group with P = MN the Levi decomposition of a parabolic.
If V is any G-module, the map VK → VKM is injective.
N
Proof. For I ⊂ G the Iwahori subgroup, Borel (see [1]) proved that VI ֒→ VIM. As the
N
Jacquet functor is intertwining for the action of P, the image of VK is clearly fixed by K .
M
Since VK ⊂ VI, this gives the desired result. (cid:3)
Let G be a split reductive group defined over F. If χ : G → GL is a character then
1
χ∗ : C× → G will denote the corresponding co-character. The group χ∗(C×) is in the center
of G. For example, if G = GL and χ = det, then χ∗(z) is the scalar matrix in G = GL (C).
b n n
5
b b
Lemma 2.4. Let χ : G → GL be a character. Let V be a finite dimensional irreducible
1
representation of G (i.e. a Hecke operator for G). Let m be a half integer. Then χ∗(qm) acts
on V as qn for some half integer n. Let π be a representation of G. Then V acts on π⊗|χ|m
b
as qn ·V acts on π.
Lemma 2.5. Let π be a representation of G = G′ × G′′ obtained as the pullback of π′, a
representation of G′. Let s ∈ G′′ be the image of q1/2 0 under the principal SL → G′′.
0 q−1/2 2
(cid:16) (cid:17)
For V a finite dimensional rebpresentation of G (i.e. a Hecke operator for G), write Vb=
V′ ⊗V′′, the restriction of V to G′ ×G′′. Thben V acts on π as TrV′′(s)V′ acts on π′.
P P
Proof. Since the Satake parameter obf thebtrivial representation of a group G′′ is s, the result
(cid:3)
is clear.
Lemma 2.6. Let χ : G → GL be a character, and let C = χ∗(C×) ⊆ G. Let π′ be a
1
representation of GL . Then π = π′ ◦χ is a representation of G. Let s ∈ G be the image of
1 b
q1/2 0 under the principal SL → G. For V a finite dimensional representation of G
0 q−1/2 2 b
(cid:16) (cid:17)
(i.e. a Hecke operator for G), write V = V′ ⊗V′′, the restriction of V to C ×SL . Then
b 2 b
V acts on π as TrV′′(s)V′ acts on π′. P
P
Proof. We prove this for G = GL and χ = det is the determinant. In this case C is the
n
center. It suffices to prove the statement for the fundamental representations V = V =
λi i
∧iCn where
λ = (1,...,1,0,...,0).
i
i times
Since V is miniscule, the Satake isomo|rp{hzism} gives S(T ) = qi(n−i)/2V . Therefore, the
i λi i
action of π(V ) is
i
qi(i−n)/2 π(g)dg =qi(i−n)/2vol(Kλ (̟)K)π′(det(λ (̟)))
i i
Z
Kλi(̟)K
=qi(i−n)/2vol(Kλ (̟)K)π′(̟)i.
i
The center C acts on V by the character z 7→ zi. Note that
i
q(n−1)/2
q(n−3)/2
s = ... ,
q(1−n)/2
is the image of q1/2 under the principal SL → G. So, to complete the proof we just
q−1/2 2
(cid:16) (cid:17)
need to show that
b
qi(i−n)/2vol(Kλ (̟)K) = Tr (s).
i Vi
This is immediate from the discussion in [2, Section 3] and the fact that V is miniscule. (cid:3)
i
Lemma 2.7. Let G be a reductive group. Let C∞(G) denote the space of smooth, compactly
c
supported functions on G. This is a G×G module for the left and right action of G called
the regular representation. On C∞(G)K×K we have a matching of the Hecke algebras for the
c
left and right action.
6
Proof. This is obvious since C∞(G)K×K is nothing else but the Hecke algebra itself. To be
c
precise, since the left action on f in C∞(G) is by λ (f)(x) = f(g−1x), a Hecke operator R,
c g
acting from the right, is matched with R∗ defined by R∗(x) = R(x−1). (cid:3)
Remark. Notice that this matching of Hecke operators, when considered as a matching of
virtual representations, matches V ∈ R(G) with its dual V. In particular, if V is self-dual
then it is matched with itself.
e
3. Our groups
3.1. Octonions. Let O denote the non-associative division algebra of rank 8 over F. There
is an F-linear anti-involution x 7→ x¯ on O, hence norm and trace maps
N : O → F x 7→ xx¯ = x¯x, Tr : O → F, x 7→ x+x¯
satisfying
N(x·y) = N(x)N(y), Tr(x·y) = Tr(y ·x), Tr(x·(y ·z)) = Tr((x·y)·z).
On the set O0 of trace zero elements, we have x¯ = −x. The group G is the automorphism
2
group of O.
The quadratic form N : O → F has signature (4,4), which means that O has a basis
{1,i,j,k,l,li,lj,lk}. (Note that l2 = 1.) The following basis is particularly useful.
s = 1(i+li), s = 1(j +lj), s = 1(k +lk), s = 1(1+l),
(14) 1 2 2 2 3 2 4 2
t = 1(i−li), t = 1(j −lj), t = 1(k −lk), t = 1(1−l).
1 2 2 2 3 2 4 2
The multiplication table for this basis is given in Table 2.
s s s t t t s t
1 2 3 1 2 3 4 4
s 0 −t t s 0 0 0 s
1 3 2 4 1
s t 0 −t 0 s 0 0 s
2 3 1 4 2
s −t t 0 0 0 s 0 s
3 2 1 4 3
t t 0 0 0 s −s t 0
1 4 3 2 1
t 0 t 0 −s 0 s t 0
2 4 3 1 2
t 0 0 t s −s 0 t 0
3 4 2 1 3
s s s s 0 0 0 s 0
4 1 2 3 4
t 0 0 0 t t t 0 t
4 1 2 3 4
Table 2. Multiplication Table for Octonions
Remark. From this basis it is evident that a subspace V ⊂ O0 on which multiplication
is trivial is at most 2-dimensional. (We call such a subspace a null space or a null sub-
space.) Indeed, let {i,j,k} = {1,2,3}. Then from the multiplication table we see that
s⊥ = hs ,t ,t i, and the null spaces of O0 which contain s are all of the form hs ,at +bt i
i i j k i i j k
for fixed a,b ∈ F. Since G acts transitively on (nonzero) elements of trace zero and norm
2
zero, this phenomenon is generic.
7
3.1.1. Maximal parabolic subgroups in G . They are described as the stabilizers of null sub-
2
spaces V ⊂ O0. Let V be spanned by s and V by s and t . Then V = V⊥ is spanned by
1 1 2 1 2 3 1
s ,t and t . Let P = M N and P = M N be the stabilizers of V and V , respectively.
1 2 3 1 1 1 2 2 2 1 2
∼
TheLevi factorM actsonV . Thechoice ofthebasisinV gives anisomorphism M = GL .
2 2 2 2 2
∼
The Levi factor M acts on V /V and we have an isomorphism M = GL . It is not difficult
2 3 1 1 2
∼
to see that g ∈ GL = M acts on V by det(g). The set of all g in G such that all s and
2 1 1 2 i
t are eigenvectors is a maximal split torus T in G . The modular characters are
i 2
(15) δ (g) = |det(g)|5 and δ (g) = |det(g)|3.
N1 N2
The stabilizer of s −t in G is a group isomorphic to SL . Under the action of this group
4 4 2 3
we have a decomposition
O0 = hs ,s ,s i⊕ht ,t ,t i⊕hs −t i.
1 2 3 1 2 3 4 4
We can identify the stabilizer of s −t with SL so that the action on hs ,s ,s i is standard.
4 4 3 1 2 3
The torus T of G sits in SL . In this way, we can represent elements in T by 3×3 matrices.
2 3
For example, if α is a long root and α a short root perpendicular to α then, up to Weyl
l s l
group conjugation,
t t
(16) α∗(t) = 1 and α∗(t) = t−2 .
l s
t−1 t
3.2. Description of groups. Let P = MN be a maximal parabolic subgroup of G as in
thetable below. The groupN is abelian, except inthe case E , where N has one-dimensional
8
center Z. In order to give a uniform notation, let Z be trivial if N is abelian. Let d denote
∼
the dimension of N/Z. Let C = GL be the center of M. Fix an isomorphism λ : GL → C
1 ∗ 1
such that the adjoint action of λ (z) on N/Z is given by multiplication by z. We have a
∗
dual pair G × H ⊂ G such that Q = LU = H ∩ P is a maximal parabolic of H. Let
2
N ⊆ N/Z be the complement of U¯/Z¯ under the invariant pairing induced by the Killing
0
form. We fix an isomorphism of L with a classical group so that the action of G × L on
2
N is isomorphic to O ⊗ Fn, the space of n-tuples in O , and h ∈ L acts on an n-tuple
0 0 0
(x ,...,x ) by (x ,...,x )h−1. Since the scalar matrix z−1 in L acts on N/Z as z, the center
1 n 1 n
of L coincides with the center of M. In the case of G = E , let i be the isogeny character of
8
GSp .
6
G D E E E
5 6 7 8
M D D E E
4 5 6 7
d 8 16 27 56
L GL GL GL GSp
1 2 3 6
δ |det| |det| |det|2 |i|8
U¯
δ |det|8 |det|8 |det|9 |i|29
N¯
Table 3. Maximal parabolic subgroups
The last row of the table is the restriction of the character δ to L. The group GSp acts
N¯ 6
by the isogeny character i on Z¯.
8
3.2.1. Maximal parabolic subgroups in L. Assume first that L = GL . Recall that g ∈ GL
n n
acts on Fn, the space of n-tuples (x ,...,x ) by (x ,...,x )g−1. For m ≤ n let Q be
1 n 1 n m
the stabilizer of the subspace consisting of the n-tuples whose last n−m entries are 0. We
have a Levi decomposition Q = L U where L = GL ×GL is the stabilizer, in Q ,
m m m m m n−m m
of the subspace consisting of the n-tuples whose first m entries are 0. Let g = (g ,g ) ∈
1 2
GL ×GL . The modular character δ is
n−m m Um
(17) δ (g) = |det(g )|m−n ·|det(g )|m.
Um 1 2
Assume now that L = GSp . This is a group of isogenies of a symplectic form (·,·) on a
2n
2n dimensional space. Let e ,...,e ,f ,...,f be a symplectic basis, i.e.
1 n 1 n
(e ,f ) = −(f ,e ) = δ and (e ,e ) = (f ,f ) = 0.
i j j i ij i j i j
Using this basis we identify the symplectic space with F2n, the space of 2n-tuples. We
identify GSp with the group of 2n × 2n matrices g acting on F2n from the right, and
2n
preserving the symplectic form, up to an isogeny character :
(vg−1,ug−1) = i(g)−1(v,u).
For every m ≤ n, let Q be the subgroup of GSp preserving the subspace spanned by
m 2n
e ,...,e . We have a Levi decomposition Q = L U where L is the stabilizer, in Q
1 m m m m m m
∼
of the subspace spanned by f ,...,f . Then Q = GL ×GSp where g = (g ,g ) ∈
1 m m m 2(n−m) 1 2
GL ×GSp acts as follows: e 7→ e g−1, for 1 ≤ i ≤ m, f 7→ i(g )−1f g⊤ for 1 ≤ i ≤ m
m 2(n−m) i i 1 i 2 i 1
(here g⊤ is the transpose of g ) and as g−1 on the remaining 2(n−m) basis elements. The
1 1 2
modular character δ is
Um
(18) δUm(g) = |det(g1)|−(2n−m+1)|i(g2)|m(2n−2m+1).
3.2.2. Example: G = E . Let J be the 27-dimensional Jordan algebra over F given by
7 27
a z y¯
(19) J = A = z¯ b x (cid:12)a,b,c ∈ F, x,y,z ∈ O.
27
(cid:12)
y x¯ c (cid:12)
(cid:12)
(cid:12)
The determinant on J gives an F-valued cubic form on J . The adjoint group G has a
27 (cid:12) 27
maximal parabolic P = MN such that
(20) M ∼= {g ∈ GL(J ) | det(g(A)) = λ(g)det(A) for some similitude λ(g) ∈ F×},
27
a reductive group of type E , and N ∼= J as M-modules. Moreover, N¯ ∼= J , and the
6 27 27
¯
natural pairing of N and N can be identified with the trace on J .
27
Evidently, G ⊂ M acts term by term on the elements of A, and since g ∈ GL acts via
2 3
(21) g ·A = (detg)−1gAgt,
the action of G and GL obviously commute, hence G × GL ⊂ M. As described in [4,
2 3 2 3
Section 5], G × PGSp ⊂ G is a dual pair, and Q = LU = PGSp ∩ P where U can be
2 6 6
identified with J , the subalgebra of J consisting of (symmetric) matrices with entries in
6 27
the field F, and L ≃ GL with action on J ≃ N given by (21). Thus, the orthogonal
3 27
¯
complement N of U in N is identified with the subspace of J consisting of matrices with
0 27
0 on the diagonal and traceless octonions off the diagonal, that is, N is identified with the
0
set of triples (x,y,z) of traceless octonions. Moreover, (g,h) ∈ G ×GL acts on (x,y,z) by
2 3
(gx,gy,gz)h−1. (The action of h follows from Cramer’s rule.)
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4. The proof
4.1. The base case. Let V be the minimal representation of D . Let G′ = S be the group
4 3
of permutations of 3 letters. Then S acts on D , by outer automorphisms, fixing G . Since
3 4 2
V can be extended to a representation of a semi-direct product of D and S , we have a dual
4 3
pair G×G′ = G ×S acting on V. We let G′ = S and
2 3 3
r : S3 →b G2(C)
such that the centralizer C of r(S ) in G (C) is SO(3) ⊂ SL (C) ⊂ G (C). (The group
3 2 3 2
SO(3) ≃ PGL (C) corresponds to the subregular unipotent orbit by the Jacobson-Morozov
2
theorem.)
Let K′ = G′. Then the Hecke algebra H′ is one-dimensional. With these choices, Theo-
rem 1.1 asserts that the S -invariants of the minimal representation of D is the unramified
3 4
representation π of G whose Satake parameter corresponds to the subregular orbit. This
sr 2
is proved in [3].
4.2. Thegeneral case. AssumethatG 6= D . ThenwehaveamaximalparabolicP = MN
4
in G and the corresponding maximal parabolic Q = LU in H as in Table 3. For simplicity,
let K be the maximal compact subgroup of G , and K′ that of H. Assume that we want to
2
show matching of two operators T and T′. By Lemma 2.3,
VK×K′ ֒→ rU¯(V)K×KL′
where K′ = K′ ∩ L. Thus Theorem 1.1 holds if we can show matching of T and S (T′)
L L
on rU¯(V)K×KL′. If G 6= E8, the unnormalized Jacquet functor VU¯ was computed in [4]. In
the context of the present work, we find it convenient to describe these results in terms of
∼
maximal parabolic subgroups Q = L U of L = GL and maximal parabolic subgroups
m m m n
P = M N of G (as defined in Sections 3.2.1 and 3.1.1, respectively). In particular, we
m m m 2
∼ ∼
have fixed isomorphisms L = GL ×GL and M = GL .
m m n−m m 2
Let s,t be a pair of real numbers. For m = 1,2, let C∞(GL )[s,t] be the vector space
c m
C∞(GL ) with an M ×L -module structure defined by
c m m m
(g ,g )·f(h) = |detg |s|detg |tf(hg )
1 2 1 2 1
for any (g ,g ) ∈ L , and by
1 2 m
f(detg−1h) if m = 1,
(22) g ·f(h) =
(cid:26) f(g−1h) if m = 2,
for any g ∈ M .
m
We shall omit [s,t] in the notation if s = t = 0. With this notation in hand, we now
describe the Jacquet module r (V) for each of our cases.
U¯
4.2.1. G = D .
5
Proposition 4.1. As a G × GL -module, r (V) has a filtration with two successive sub
2 1 U¯
quotients
(1) iG2(C∞(GL )).
P1 c 1
1 5
(2) V(M)⊗|det| ⊕1⊗|det| .
2 2
Here V(M) is the minimal representation of M/C.
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