Contemporary Mathematics
Distinguished Supercuspidal Representations of SL2
Jeffrey Hakim and Joshua M. Lansky
Abstract. We compute distinguished tame supercuspidal representations of
SL2 (F ) following the methods of [HM].
Contents
1. Introduction
2. Elliptic tori
3. Involutions of SL2
4. Multiplicity constants
5. Supercuspidal representations
6. Distinguished toral supercuspidal representations
7. Distinguished depth-zero supercuspidal representations
Appendix A. The building and the Moy-Prasad groups
References
1
3
4
9
10
13
15
19
31
1. Introduction
Let G be the F -group SL2 , where F is a nonarchimedian field whose residue
field has characteristic p 6= 2. The purpose of this paper is to consider the theory
of distinguished supercuspidal representations for G = G(F ). More precisely, if
θ is an involution of G, that is, an F -automorphism of G of order two, and if
π is an irreducible supercuspidal representation of G then we compute the space
HomGθ (π, 1), where Gθ is the group of fixed points of θ in G. When this space is
nonzero, we say that π is Gθ -distinguished.
The main result says that π is distinguished with respect to some Gθ precisely
when its central character is trivial (or, equivalently, when it is trivial at −1). There
are seven conjugacy classes of subgroups Gθ of G. Each conjugate gGθ g −1 of a given
0
Gθ has the form Gθ for some θ0 and it is easy to see that π is Gθ -distinguished
0
exactly when it is Gθ -distinguished. If π is Gθ -distinguished for some θ then
2010 Mathematics Subject Classification. Primary 22E50, 11F70.
Key words and phrases. supercuspidal representation, involution, distinguished representation, special linear group.
The authors of this paper were partially supported by NSF grant DMS-0854844.
c
0000
(copyright holder)
1
2
JEFFREY HAKIM AND JOSHUA M. LANSKY
we show that it is distinguished with respect to exactly two conjugacy classes of
0
Gθ ’s. One of theseconjugacy classes always includes the group Gθ0 associated
1 0
to θ0 = Int
. For general θ, if π is Gθ -distinguished then we show that
0 −1
HomGθ (π, 1) almost always has dimension two.
This work develops the methods of [HM] and [L] in a class of examples that is
comparatively simple, yet is still rich enough to convey some of the subtleties that
one encounters in general. For example, the pair (G, θ) is not “multiplcity-free,”
or, in other words, the dimension of the spaces HomGθ (π, 1) can exceed one. In
fact, we explain two different reasons why the latter dimensions can exceed one in
the examples we consider.
The methods of [HM] apply to irreducible supercuspidal representations that
are tame in the sense of [Y]. However, it is known that all irreducible supercuspidal
representations of G = SL2 (F ) are tame. (See [ADSS], as well as [M], [MS],
[K].) Therefore, we do not need to impose any tameness assumptions on our
representations.
Since the theory in [HM] is rather complicated, and since the present class
of examples are so accessible, we have made an extra effort to convey in explicit
detail the mathematical structures that underly [HM] in our simple setting. In
other words, this paper has been designed to serve as an expository introduction or
companion to [HM]. We also hope that it complements the paper [ADSS] by providing the symmetric space theory for the supercuspidal representations considered
there.
The results of [L] on distinguished Deligne-Lusztig characters can be phrased
in a form that is analogous to those in [HM]. In this paper, we give a statement
of the main result of [L] in this form (see [HL] or [M] for a complete treatment),
and demonstrate its usage in the particular cases under consideration.
This paper draws on some refinements to [HM] in [HL]. The examples considered are also closely related to those in [HL], namely representations of GLn (F )
distinguished with respect to odd orthogonal groups. In both cases, the distinguished representations have the property that their central character is trivial at
−1. In addition, they are the representations that lie in the image of a certain
metaplectic correspondence (generalizing the Shimura correspondence). In [HL],
one considers supercuspidal representations of GLn (F ), with n odd, and involutions
of the latter group that give rise to orthogonal groups. We hope to extend these
results to the case in which n is even and to better explain the connection between
the distinguished representations in this paper and the representations of GL2 (F )
that are distinguished by an orthogonal group.
The structure of this paper is as follows. Sections 2 through 4 involve algebraic
preliminaries. In §2, we classify the elliptic tori in G, since characters of these tori
are used to construct supercuspidal representations. We show that such a torus
determines a point in the Bruhat-Tits building of G. The latter point is used in Yu’s
construction of supercuspidal representations. In §3, we classify the involutions of G
as well as their orbits with respect to both G and an elliptic torus, since these orbit
space structures are needed in the study of distinguished representations. Then we
isolate the orbits that are relevant to the study of HomGθ (π, 1) for a specific π. In §4,
we compute certain constants mT (Θ0 ) that can cause the dimension of HomGθ (π, 1)
to exceed one. In §5, §6, and §7, we give simplified treatments for our examples
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
3
of: the construction of supercuspidal representations (based on [Y]), the theory of
distinguished representations of positive depth (based on [HM]), and the theory of
distinguished supercuspidal representations of depth zero (based on [L]). Finally,
we include an extensive appendix that illustrates in concrete and unsophisticated
terms some of the basic structures used in the theory of distinguished supercuspidal
representations.
Special thanks are due to Jeffrey Adler who was consulted frequently during
the writing of this paper.
2. Elliptic tori
We will adopt the following notation throughout this paper. Let F , p, G and
G be as in the introduction. Let OF denote the ring of integers of F , and let PF
denote the unique prime ideal of OF . Subgroups of G that are defined over F will
be denoted with boldface capital letters. Taking F -rational points corresponds to
removal of the boldface.
2.1. Classification.
Fix a nonsquare τ in F . There is an associated quadratic
√
extension E = F [ τ ] of F and an elliptic torus T = Tτ whose group of F -points
a bτ
T = Tτ =
∈ G : a, b ∈ F, a2 − b2 τ = 1 ,
b a
that is, T is isomorphic to the subgroup of E × consisting of elements of norm one
(with respect to NE/F ). (Any elliptic maximal F -torus in G is GL2 (F )-conjugate
to a torus of this form.) The F -rational Lie algebra of T is
0 bτ
t=
: b∈F .
b 0
We let
γ=
1
1
√ √τ ,
− τ
and observe that γT γ −1 consists of diagonal matrices and
√
0 τ
τ
0
√
γ
γ −1 =
.
1 0
0 − τ
Note that γ is analogous to the “Cayley transform” matrix that arises when mapping the complex upper half-plane to the complex unit disk when studying Möbius
transformations.
2.2. The point in the building determined by an elliptic torus. We
introduce the following notations:
• g = sl2 is the Lie algebra of G, g = g(F ),
• S is the group of diagonal matrices in G,
• B(G, L) is the Bruhat-Tits building of G over L when L is an extension
of L,
• A(G, S0 , L) is the apartment in B(G, L) associated to S0 whenever S0 is
a maximal L-split torus in G,
• A(G, T, F ) = A(G, T, E) ∩ B(G, F ).
4
JEFFREY HAKIM AND JOSHUA M. LANSKY
In general, if α is an E-rational automorphism of G then there is a natural
action of α on B(G, E). In particular, when g ∈ GL2 (E) then there is an associated
E-automorphism Int(g) of G and thus GL2 (E) acts on B(G, E).
Let σ be the nontrivial element of Gal(E/F ). We also view σ as an automorphism of G(E), letting it act entrywise on matrices. There is a corresponding
action of σ on B(G, E).
Suppose x ∈ A(G, S, E). By inspection of the explicit description of g(E)x,0 ,
we see that g(E)σ(x),0 = g(E)x,0 . This implies σ(x) = x or, in other words,
Gal(E/F ) fixes A(G, S, E) pointwise. We can identify both A(G, S, E) and A(G, S, F )
with R (see A.2 in the appendix) and we use these identifications to identify
A(G, S, E) with A(G, S, F ). The action of G then yields an embedding of B(G, F )
in B(G, E) such that if x ∈ B(G, F ) then
gx,0 = g(E)x,0 ∩ g.
The image of this embedding is B(G, E)Gal(E/F ) .
We observe that T = γ −1 Sγ. Therefore, the apartment A(G, T, E) in B(G, E)
associated to T is γ −1 A(G, S, E). We now compute
A(G, T, F ) = (γ −1 A(G, S, E))Gal(E/F ) = γ −1 A(G, S, E) ∩ B(G, F ).
Suppose x ∈ A(G, S, F ) and σ(γ −1 x) = γ −1 x. The condition σ(γ −1 x) = γ −1 x
reduces to
0 1
−1
x = γσ(γ) =
x = −x.
1 0
Therefore, x is the origin λ0 in A(G, S, F ).
We now have
A(G, T, F ) = {γ −1 λ0 }.
Let
yτ = γ −1 λ0 .
In this way, we can associate to each elliptic maximal F -torus in G a canonical
point in B(G, F ).
The definition yτ = γ −1 λ0 must be interpreted with care, since Int(γ −1 ) does
not stabilize G. We refer to the appendix for more details.
3. Involutions of SL2
3.1. Classification. Let G = SL2 viewed as an algebraic group over F . As
in [HM], we use the terminology “involution of G” to mean “F -automorphism of
G of order two.” Note that this is a slight abuse of terminology since it depends
on the algebraic group structure of G and not just the group of F -rational points.
For example, if F/F 0 is a quadratic extension then, using restriction of scalars, we
can construct an F 0 -group G0 such that G0 (F 0 ) = SL2 (F ). Relative to G0 , the
notion of “involution of SL2 (F )” is different than for us. (The latter involutions
are considered in [AP].)
Every automorphism of G is inner. For example, if g ∈ G then
t −1
g
= jgj −1 ,
where
j=
0
−1
1
.
0
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
5
An F -automorphism of G is not necessarily of the form Int(g) with g ∈ G. In fact,
the F -automorphisms of G are precisely the automorphisms of the form Int(g) with
g ∈ GL2 (F ). So the involutions of G are precisely the automorphisms of the form
Int(g) with g ∈ GL2 (F ) such that g 2 is a scalar matrix, while g is not a scalar
matrix. It is easy to see that this means precisely that g must lie in the set
I = GL2 (F ) ∩ sl2 (F )
of trace zero matrices in GL2 (F ). Thus g 7→ Int(g) gives a bijection between I/F ×
and the set of involutions of G.
3.2. G-orbits of involutions. Given
a b
g=
∈I
c −a
then there is an associated scalar δ = δ(g) such that g 2 = δ. We observe that
δ(g) = g 2 = a2 + bc = − det g.
Since
0 b
= b,
1 0
for all b ∈ F × , we see that δ defines a surjective map from I onto F × .
Let δ̄ = δ̄(g) be the image of δ(g) in F × /(F × )2 . We also view δ̄ as defining a surjective map from I/F × (or, equivalently, the set of involutions of G) to
F × /(F × )2 .
Let G act on the set of involutions of G by:
δ
g · θ = Int(g) ◦ θ ◦ Int(g)−1 .
This corresponds to the action of G on I/F × by conjugation. The problem of classifying the G-orbits of involutions of G is equivalent to classifying the G-conjugacy
classes of orthogonal groups in GL2 (F ). Indeed, we have a bijection between I and
the set of symmetric matrices in GL2 (F ) given by
a b
−b a
x=
7→ xj =
c −a
a c
and we also have a matrix identity
gxg −1 = g (xj) t g j −1 ,
for g ∈ G and x ∈ I.
We now classify the G-orbits of involutions of G via the map
δ̄ : I/F × → F × /(F × )2 .
The fibers of δ̄ are unions of G-orbits. Our classification thereby reduces to determining how the fibers of δ̄ decompose into G-orbits.
Lemma 3.1. The fiber δ̄ −1 ((F × )2 ) of the identity coset consists of a single
G-orbit in I/F × .
Proof. It suffices to show that if x ∈ I and δ(x) = 1 then x is G-conjugate to
1 0
w=
.
0 −1
Given x ∈ I, we observe that {1, −1} must be the set of eigenvalues of x, since
x2 = 1 and x 6= ±1. So there exists g ∈ GL2 (F ) such that gxg −1 = w. Now let d
6
JEFFREY HAKIM AND JOSHUA M. LANSKY
be a diagonal matrix in GL2 (F ) such that g 0 = dg ∈ G. Since g 0 xg 0−1 = w, we are
done.
2
−1
× 2
Lemma 3.2. If τ ∈ F× − (F ×
) then the fiber δ̄ (τ (F ) ) consists of two
a b
G-orbits in I/F × . If g =
∈ I and δ(g) = τ then the G-orbit of gF × is
c −a
√
determined by the class of c in F × /NE/F (E × ), where E = F [ τ ].
Proof. Fix c0 ∈ F × and let
h0 =
c−1
0 τ
0
0
c0
.
Then h0 is an element of I with δ(h0 ) = τ .
Suppose
w x
g=
∈ G.
y z
Then
gh0 g −1 =
a
c
b
,
−a
2
×
for some a, b, c ∈ F with c = c0 (z 2 − (yc−1
0 ) τ ) ∈ c0 NE/F (E ).
It now suffices to show that if
a b
h=
c −a
is an arbitrary element of I with δ(h) = τ and c ∈ c0 NE/F (E × ) then there exists
w x
g=
∈G
y z
such that gh0 g −1 = h. Since we assume c ∈ c0 NE/F (E × ), we may choose y, z ∈ F
2
such that c = c0 (z 2 − (yc−1
0 ) τ ). Having chosen y and z, it is easy to verify that
there exist unique w, x ∈ F such that
−c−1
0 ywτ + c0 zx = a
zw − yx =
1.
(The first equation here is just the requirement that (gh0 g −1 )11 = h11 .) Then for
these values of w, x, y, z, one obtains g ∈ G such that gh0 g −1 = h. Our claim
follows.
The previous two lemmas immediately imply:
Proposition 3.3. There are seven G-orbits of involutions of G. Let {b1 , b2 , b3 }
be
a
set
of
representatives
for
the
nontrivial
cosets
in
F ×√/(F × )2 . Choose c1 , c2 , c3 ∈ F so that ci is not the norm of an element of
F [ bi ] or, equivalently, the Hilbert symbol (bi , ci ) = −1. Then the set
1 0
0 bi
0 bi c−1
i
,
,
, i = 1, 2, 3
0 −1
1 0
ci
0
is a set of representatives for the G-orbits of in I/F × .
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
7
3.3. Involutions stabilizing an elliptic torus. We are interested in the
involutions of G that stabilize T or, equivalently, the corresponding F -torus T.
0 τ
Lemma 3.4. The involution Int
is the unique involution of G that
1 0
fixes
T pointwise.
The remaining involutions of G that stabilize T have the form
a
bτ
Int
, for a, b ∈ F such that a2 6= b2 τ . If θ is one of the latter
−b −a
involutions, then T is θ-split in the sense that θ(t) = t−1 for all t ∈ T.
Proof. Let β be the square root of τ given by
0 τ
β=
.
1 0
Then T consists of the matrices x+yβ with x, y ∈ F and x2 −y 2 τ = 1. Equivalently,
it is the centralizer of β in G. Let G0 = GL2 (F ) and let T 0 be the centralizer of β
in G0 . We observe that T 0 is the elliptic maximal torus in G0 consisting of elements
of the form x + yβ with x, y ∈ F × and x2 6= y 2 τ . It can also be described as the
centralizer in G0 of any element of T other than ±1 or simply as the centralizer in
G0 of T .
Suppose θ is an involution of G that preserves T. Choose g ∈ G0 such that
θ = Int(g). Let θ0 be the involution of G0 defined by Int(g). Then θ0 (T 0 ) = T 0 .
Since the only square roots of τ in T 0 are ±β, it must be the case that gβg −1 = ±β.
Suppose gβg −1 = β. The latter condition is equivalent to g ∈ T 0 . In order for g
to yield an involution of G, we need for g 2 to be a scalar matrix while g itself cannot
be a scalar matrix (since θ must have order two). This implies that g must be a
scalar multiple of β. So Int(β) is the unique involution of G that fixes T pointwise.
The elements g ∈ G0 such that gβg −1 = −β clearly form a single coset in G0 /T 0
and that coset is represented by the matrix
1 0
α=
.
0 −1
Moreover, if t0 ∈ T 0 then Int(αt0 ) defines an involution θ of G such that T is
θ-split.
3.4. The T-relevant T -orbits of involutions. Below, we will associate supercuspidal representations of G to certain characters of elliptic maximal tori T in
G. Then a formula for the dimension of the space of invariant linear forms on the
representation space will be given. The formula will involve certain T -orbits Θ0 of
involutions of G that lie in a given G-orbit Θ of involutions. The T -orbits that
contribute to the latter formula are identified in the following definition:
Definition 3.5. If T is a torus in G and θ is an involution of G then θ is
T-relevant if T is θ-split. If θ is T-relevant then so are all of the elements in the
T -orbit Θ0 of T and, in this case, we will say that Θ0 is T-relevant.
We will see that a necessary condition for a supercuspidal representation associated to a character of T to be Gθ -distinguished is that the G-orbit Θ of θ contains
at least one T-relevant orbit. This motivates the following definition:
Definition 3.6. A G-orbit of involutions of G is T-relevant if it contains a
T-relevant T -orbit of involutions.
8
JEFFREY HAKIM AND JOSHUA M. LANSKY
The following lemma is the key to classifying the T-relevant orbits:
√
Lemma 3.7. Let E = F [ τ ] be a quadratic extension of F , where τ ∈ F × −
(F × )2 . Then
(
× 2
×
E /(F × (E 1 )2 ) = 4, if −τ ∈ (F ) ,
2, if −τ 6∈ (F × )2 .
If −τ ∈ (F × )2 then {1, ε1E , εE , ε1E εE } is a set of representatives for E × /(F × (E 1 )2 ),
where εE 1 and εE are roots of unity of order q + 1 and q 2 − 1, respectively,
and
√
q is the order of the residue field of F . If −τ 6∈ (F × )2 then {1, τ } is a set of
representatives for E × /(F × (E 1 )2 ).
Proof. We consider the exact sequence
1 → F × E 1 /F × (E 1 )2 → E × /F × (E 1 )2 → E × /F × E 1 → 1.
Note that the norm map NE/F yields an isomorphism
E × /F × E 1 ∼
= NE/F E × /(F × )2
of groups of order two. We also have isomorphisms
F × E 1 /F × (E 1 )2 ∼
= E 1 /(F × ∩ E1 )(E 1 )2 = E 1 /{±1}(E 1 )2 .
To analyze the latter quotient, we note that x 7→ x2 yields an isomorphism
E 1 ∩ (1 + PE ) ∼
= (E 1 )2 ∩ (1 + PE ).
Let kE and kF be the residue fields of E and F and let q be the order of kF . Let
×
K be the kernel of the homomorphism kE
→ kF× given by x 7→ xq+1 . Then
E 1 /{±1}(E 1 )2 ∼
= K/K2 {±1}.
×
If E/F is ramified then the map kE
→ kF× is just x 7→ x2 and, consequently, K =
2
{±1}. It follows that K/K {±1} is trivial and hence NE/F induces an isomorphism
E × /F × (E 1 )2 ∼
= NE/F (E × )/(F × )2 .
√
Moreover, {1, τ } is a set of representatives for E × /F × (E 1 )2 .
×
Assume now that E/F is unramified. In this case, the map kE
→ kF× is the
norm map NkE /kF . It is surjective and its kernel K has order q + 1 and is the group
×
of (q + 1)-roots of 1 in kE
. The group K2 has order (q + 1)/2 and is the set of
×
(q + 1)/2-roots of 1 in kE . So the quotient K/K2 has order two. We observe that
−1 ∈ K2 precisely when (−1)(q+1)/2 = 1 or, in other words, when q ≡ −1 (mod
4). On the other hand, −1 is a square in F × if and only if it is a square in O×
F,
but, by Hensel’s Lemma, the latter condition is equivalent to −1 being a square in
kF× . Since, (kF× )2 is the group of (q − 1)/2-roots of 1 in kF× , we see that −1 ∈ (kF× )2
precisely when q ≡ 1 (mod 4).
We have just seen that: K/(K2 {±1}) is trivial ⇔ −1 6∈ K2 ⇔ q ≡ 1 (mod 4)
⇔ −1 ∈ (kF× )2 ⇔ −1 ∈ (F × )2 .
If q ≡ 1 (mod 4) then NE/F induces an isomorphism
E × /F × (E 1 )2 ∼
= NE/F (E × )/(F × )2
√
and {1, τ } is a set of representatives for E × /F × (E 1 )2 , as in the ramified case.
On the other hand, if E/F is unramified and q ≡ −1 (mod 4) then we have an
exact sequence
1 → K/K2 → E × /F × (E 1 )2 → NE/F (E × )/(F × )2 → 1.
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
9
One easily checks that the set {1, εE 1 , εE , εE 1 εE } in the statement of the lemma is
a set of representatives for E × /F × (E 1 )2 .
If T is a maximal elliptic torus of G, then the action of T on the set of Trelevant orbits of involutions of G is particularly simple, namely, t · θ = Int(t2 ) ◦ θ.
The next result is an immediate consequence of this, together with Lemmas 3.4 and
3.7:
Proposition 3.8. Fix τ ∈ F√× − (F × )2 and let T = Tτ be√the associated
elliptic torus in G and let E = F [ τ ]. For each element z = x + y τ ∈ E × there
is an associated involution
x
yτ
θz = Int
−y −x
of G such that θz is T-relevant.
This yields a bijection between E × /F × and the set of all T-relevant involutions
of G. It also yields a bijection between E × /(F × (E 1 )2 ) and the set of T-relevant
T -orbits of involutions of G, where (E 1 )2 is the group of squares of elements in the
kernel of the norm map NE/F : E × → F × .
If −τ ∈ (F × )2 then {θ1 , θεE1 , θεE , θεE1 εE } is a set of representatives for the Trelevant T -orbits of involutions of G. The involutions in each of the sets {θ1 , θεE1 }
and {θεE , θεE1 εE } lie in a common G-orbit and the G-orbits determined by these
two sets are distinct.
If −τ 6∈ (F × )2 then {θ1 , θ√τ } is a set of representatives for the T-relevant
T -orbits of involutions of G. The two involutions in the latter set lie in different
G-orbits.
4. Multiplicity constants
Throughout this chapter, we fix an F -elliptic maximal torus T in G. For
simplicity, we √
will assume T has the form Tτ for some τ ∈ F × − (F × )2 . Let
E = Eτ = F [ τ ]. There is no loss in generality in making this assumption or,
in other words, the results we obtain extend in an obvious way to all F -elliptic
maximal tori.
4.1. Gθ . Let θ be an involution of G such that T is θ-split. Thus θ has the
form
x −yτ
θ = Int(gθ ), with gθ =
∈ GL2 (F ).
y −x
Let δ = gθ2 = − det gθ = x2 − y 2 τ ∈ NE/F (E × ).
The centralizer of gθ in G is the maximal torus Gθ in G. Note that F [gθ ]∩G =
θ
G . If δ ∈ F × − (F × )2 then F [gθ ] is a quadratic extension of F and Gθ is an F elliptic torus. Otherwise, F [gθ ] is not a field and Gθ is an F -split torus.
Let
Gθ = {g ∈ G : gθ(g)−1 ∈ Z} = {g ∈ G : ggθ g −1 = ±gθ }.
Then Gθ is the normalizer of Gθ in G and it contains Gθ as a subgroup of index
two.
10
JEFFREY HAKIM AND JOSHUA M. LANSKY
4.2. mT (Θ0 ). Let us define
mT (θ) = [Gθ : (T ∩ Gθ )Gθ ].
The latter quantity is essentially defined in [HL] and it is easy to see that it only
depends on the T -orbit of θ. Accordingly, if Θ0 is the T -orbit of θ then we write
mT (Θ0 ) instead of mT (θ). (In fact, mT (θ) only depends on the K-orbit of θ, where
K is a certain compact open subgroup
√ of G, defined below, that contains T .)
Suppose −τ ∈ (F × )2 and say −τ is a square root of −τ in F × . Then it is
easy to see that
√ 0
τ / −τ
√
.
T ∩ Gθ = Z ∪ ±
0
1/ −τ
On the other hand, if −τ 6∈ (F × )2 then T ∩ Gθ = Z and hence we have:
Lemma 4.1.
(
mT (θ) =
1,
2,
if − τ ∈ (F × )2 ,
if − τ 6∈ (F × )2 .
Note that −τ ∈ (F × )2 precisely when E/F is unramified and −1 6∈ (F × )2 .
5. Supercuspidal representations
5.1. Cuspidal G-data. Let G be the F -group SL2 . A cuspidal G-datum
~ whose properties we now recall. The notion
~ y, ρ, φ)
is a certain 4-tuple Ψ = (G,
was introduced in [Y] though we follow the presentation in [HM]. Cuspidal Gdata (modulo the equivalence relation given in [HM]) are the basic objects that
parametrize Yu’s tame supercuspidal representations.
For G = SL2 (F ), cuspidal G-data come in two types: the depth zero data and
the toral data.
Definition 5.1. Ψ is a depth-zero cuspidal G-datum if
~ = G.
• G
• y is a point in A(G, T, F ) = A(G, T, E) ∩ B(G, F ), where T is an elliptic
maximal F -torus of G and E is a quadratic unramified extension of F
such that T(E) ∼
= E×.
• ρ is an irreducible representation of K = Gy,0 such that ρ | Gy,0+ is 1isotypic and the compactly induced representation indG
K ρ is irreducible
(hence supercuspidal). (Hence y must be a vertex in B.)
~ = 1 is the trivial character of G.
• φ
If Ψ is a depth-zero cuspidal-G datum then indG
K ρ is the supercuspidal representation π(Ψ) associated to Ψ.
Definition 5.2. Ψ is a toral cuspidal G-datum if
~ = (T, G), where T is an elliptic maximal F -torus in G. In this case,
• G
T = T(F ) is isomorphic to the group of elements of norm 1 in a quadratic
extension E of F .
• y is a point in A(G, T, F ), where A(G, T, F ) = A(G, T, E) ∩ B(G, F )
and A(G, T, E) denotes the apartment in B(G, E) corresponding to T.
• ρ is a quasicharacter of T such that ρ | T0+ is trivial.
~ = (φ0 , 1), where φ0 is a quasicharacter of T and 1 denotes the trivial
• φ
character of G (which is, in fact, the only character of G). We assume
φ0 has depth r > 0, that is, φ0 is trivial on Tr+ but nontrivial on Tr .
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
11
5.2. Refactorizations and toral data. Assume Ψ = ((T, G), y, ρ, (φ0 , 1))
is a toral cuspidal G-datum. A refactorization of Ψ is another toral cuspidal Gdatum Ψ̇ of the form ((T, G), y, ρ̇, (φ̇0 , 1)) such that φ̇0 |T0+ = φ0 |T0+ and ρ̇ ⊗ φ̇0 =
ρ ⊗ φ0 . The supercuspidal representations associated to Ψ and a refactorization Ψ̇
are easily seen to be equivalent.
Our given datum Ψ has a canonical refactorization
Ψ∗ = ((T, G), y, 1, (ρ ⊗ φ0 , 1)).
This says that we may as well assume that we start with a datum of the form
Ψ = ((T, G), y, 1, (φ0 , 1)).
Indeed, since y is determined by T, in the present setting, the notion of a toral
cuspidal G-datum can be simplified as follows.
Definition 5.3. A toral datum is a pair (T, φ), where T is an F -elliptic
maximal torus in G and φ is a G-generic character of T of positive depth.
Given a toral cuspidal G-datum Ψ and a refactorization Ψ̇, it is easy to see that
Yu’s construction maps Ψ and Ψ̇ to isomorphic supercuspidal representations. (See
[HM].) Thus it may seem that it would be sufficient for us to study toral cuspidal
G-data with ρ = 1. However, the theory in [HM] requires that given Ψ, we must
consider all of its refactorizations and K-conjugates. For most symmetric spaces,
one expects that certain refactorizations of Ψ behave better than others. We will
see that for SL2 all refactorizations are equally favorable and thus we may as well
deal with data for which ρ = 1.
5.3. Heisenberg groups. Fix τ ∈ F × − (F × )2 , let T = Tτ , and let E =
√
~ y, ρ, φ) be a toral cuspidal G-datum with G
~ = (T, G) and
F [ τ ]. Let Ψ = (G,
~
φ = (φ0 , 1). Suppose that ψ0 has depth r and let s = r/2. We now discuss how
to associate to Ψ a Heisenberg group associated to a (possibly trivial) symplectic
space over Fp . In the next section, we describe how this Heisenberg group arises in
Yu’s construction of tame supercuspidal representations.
The Moy-Prasad filtrations we use below will always be defined relative to
dte
the standard valuation on F . For example, if t is a real number then Ft = PF ,
×
×
×
F0 = OF and Ft = 1 + Ft , for t > 0. We let Ft+ = ∪t0 >t Ft0 and, when t ≥ 0,
Ft×+ = ∪t0 >t Ft×0 . For E (and similarly for other extensions of F ), we define Et so
that Et ∩ F = Ft and, when t ≥ 0, Et× ∩ F × = Ft× .
We let t denote the Lie algebra of T . More explicitly, we have
0 τ
t=F·
.
1 0
To define Tr , embed it in E × ∼
= T(E) and transfer filtrations. Define tr by similarly
embedding it in E.
For Yu’s construction of tame supercuspidal representations, we assume we
have a quasicharacter φ of T that is G-generic of depth r. This means there is an
element X ∈ t−r − t(−r)+ such that
φ(e(Y + tr+ )) = ψ(tr(XY )),
∀Y ∈ tr ,
where e is the Cayley transform defined in the appendix.
Let JE be the group generated by T(E)r and the groups Uyτ ,a,s with a ∈
Φ(G, T, E) defined in the appendix (with F replaced by E). Let J = JE ∩ G.
12
JEFFREY HAKIM AND JOSHUA M. LANSKY
Define (JE )+ and J+ similarly, replacing (r, s) by (r, s+ ) and define (JE )++ and
J++ using (r+ , s+ ) instead of (r, s).
We now observe that [J, J] ⊂ J+ . Define a character ζ of J+ by the conditions:
(1) ζ|J++ ≡ 1, and (2) ζ|Tr = φ|Tr . Let N = ker ζ. Then H = J/N is a Heisenberg
p-group with center Z = J+ /N . Then W = J/J+ = H/Z is a multiplicative
Fp -symplectic space with symplectic form
hu, vi = ζ(uvu−1 v −1 ),
for u, v ∈ J.
We note that, as shown in §A.9, the above symplectic spaces and Weil representations will be nontrivial precisely when E/F is unramified and r is a positive
even integer.
5.4. Yu’s construction. In the toral case, the compact open subgroup K
of G is defined to be T J. This is the inducing subgroup for the supercuspidal
representation π(Ψ) associated to Ψ, which we will describe below.
5.4.1. Toral representations not involving a Weil representation. Assume we
have the following:
• an element τ that is either a prime element or a nonsquare unit in F ,
• a positive integer r that is required to be odd when τ is a unit,
• a character ρ of T = Tτ that is trivial on T0+ ,
• a character φ0 of T that is G-generic of depth r.
~ y, ρ, φ) with G
~ = (T, G) and
We consider the toral cuspidal G-datum Ψ = (G,
~ = (φ0 , 1).
φ
√
In the present setting, E is the quadratic extension F [ τ ] of F . Our J groups
are defined with respect to y = yτ and r and, as verified in the appendix, J = J+ ,
J = J+ , and W = J/J+ is trivial. Note that ρ may be regarded as a character of
T /T0+ = T0:0+ ∼
= E 1 /(E 1 ∩ (1 + PE )).
Our assumptions regarding φ0 imply that there exists X ∈ t−r − t(−r)+ such that
φ0 (e(Y )) = ψ(tr(XY )),
for all Y ∈ tr . Note that φ0 is trivial on Tr+ but nontrivial on Tr .
Let κ−1 be the character of K that is trivial on J and agrees with ρ on T . Let
κ0 be the character of T J = T J++ that agrees with φ0 on T and is trivial on J++ .
The tame supercuspidal representation associated to Ψ is π = indG
K (κ), where κ is
the character κ = κ−1 ⊗ κ0 of K.
5.4.2. Toral representations involving a Weil representation. Assume we are
given:
• an nonsquare unit τ in F ,
• a positive even integer r,
• a character ρ of T = Tτ that is trivial on T0+ ,
• a character φ0 of T that is G-generic of depth r.
~ y, ρ, φ) with G
~ = (T, G)
Again, we consider the toral cuspidal G-datum Ψ = (G,
√
~ = (φ0 , 1). In the present case, E = F [ τ ] is an unramified quadratic
and φ
extension of F and the J groups defined with respect to y = yτ and r yield a
nontrivial symplectic space W = J/J+ . We make the following definitions:
• κ−1 is the character of K that is trivial on J and agrees with ρ on T .
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
13
• ζ is the character of J+ that is trivial on J++ and agrees with φ0 on Tr .
• N = ker ζ, H = J/N , Z = J+ /N , W = J/J+ , S = Sp(W ).
• W ] = W × Z viewed as a Heisenberg group with multiplication rule
(w1 , z1 )(w2 , z2 ) = (w1 w2 , z1 z2 hw1 , w2 i(p+1)/2 ).
• ν : H → W ] is a relevant special isomorphism. This means that the
following diagram commutes:
1
↓
1
→
Z
↓
→ Z
→
H
↓
→ W]
→ W
↓
→ W
→
1
↓
→ 1,
where the vertical maps other than ν are identity maps, and, in addition,
the mapping k 7→ ν ◦ Int(k) ◦ ν −1 maps T into Sp(W ).
• (τ, V0 ) is a Heisenberg representation of H with central character ζ. Then
τ ] = τ ◦ ν −1 is the associated Heisenberg representation of W ] . This
extends uniquely to a representation τ̂ ] of S n W ] on V0 .
• f 0 : T → S is given by the conjugation action of T on J.
• Define κ0 on K = T J by
κ0 (kj) = φ0 (k) τ̂ ] (f 0 (k)) τ (j),
if k ∈ T and j ∈ J.
The tame supercuspidal representation associated to Ψ is π = indG
K (κ), where
κ = κ−1 ⊗ κ0 .
6. Distinguished toral supercuspidal representations
6.1. Compatibility. Fix a toral cuspidal G-datum
Ψ = ((T, G), y, ρ, (φ0 , 1))
and an involution θ of G. Following [HM], one says that Ψ is weakly θ-symmetric
if θ(T) = T and φ0 ◦ θ = φ−1
0 . If Ψ is weakly θ-symmetric and if θ(y) = y then one
says that Ψ is θ-symmetric.
Lemma 6.1. The following conditions are equivalent:
(1) Ψ is weakly θ-symmetric,
(2) Ψ is θ-symmetric,
(3) T is θ-split.
Proof. All three conditions imply that T is θ-stable.
But since
A(G, T, F ) = {y}, the condition of θ-stability implies that θ(y) = y. Therefore,
conditions (1) and (2) are equivalent. In addition, condition (3) clearly implies
condition (1). It therefore suffices to show that condition (1) implies condition (3).
Assume Ψ is weakly θ-symmetric. Then θ(T) = T implies that either T is
pointwise fixed by θ or, otherwise, T is θ-split. Suppose that T is pointwise fixed
by θ. Let r be the unique positive integer such that φ0 |Tr is nontrivial and φ0 |Tr+
2
is trivial. The condition φ0 ◦ θ = φ−1
0 implies that φ0 is trivial. Therefore, φ|Tr is
2
a nontrivial character of a pro-p-group and (φ0 |Tr ) is trivial. Since p is odd, this
is impossible. It follows that T must be θ-split.
14
JEFFREY HAKIM AND JOSHUA M. LANSKY
We observe that if Ψ is θ-symmetric then so are all its refactorizations. This
phenomenon does not occur for general symmetric spaces.
Let Θ0 be the K-orbit of θ and let ξ be the K-equivalence class of Ψ, as defined
in [HM]. We say that Θ0 and ξ are moderately compatible if for some (hence
all) θ0 ∈ Θ0 there exists a θ0 -symmetric datum in ξ. The notion of moderate
compatibility is defined differently in [HM], however, the equivalence with our
definition is demonstrated in Proposition 5.7 (2) [HM]. Lemma 6.1 implies:
Lemma 6.2. A K-orbit Θ0 of involutions of G is moderately compatible with the
equivalence class ξ of Ψ precisely when there exists θ0 ∈ Θ0 such that T is θ0 -split.
The results of [HL] imply that:
Lemma 6.3. If a K-orbit Θ0 of involutions of G is moderately compatible with
the equivalence class ξ of Ψ then the set of all θ0 ∈ Θ0 such that T is θ0 -split
comprises a single T -orbit of involutions of G.
If Θ0 is a K-orbit of involutions of G then we define
hΘ0 , ΨiK = hΘ0 , ξiK = dim HomK θ (κ(Ψ), 1),
where θ is any element of Θ0 . Following [HM], we say Θ0 and ξ are strongly compatible if hΘ0 , ξiK 6= 0. Proposition 5.20 in [HM] says that strong compatibility
implies moderate compatibility.
If Θ0 is a T -orbit of involutions of G contained in a K-orbit Θ00 then we define
(
hΘ00 , ξiK if Ψ is θ-symmetric for all θ ∈ Θ0 ,
0
hΘ , ΨiT =
0,
otherwise.
We also define
hΘ, ΨiG = hΘ, ξiG = dim HomGθ (π(Ψ), 1),
where θ is any element of the G-orbit Θ of θ. We have
X
(6.1)
hΘ, ΨiG =
mK (Θ0 )hΘ0 , ΨiK .
Θ0 ∈ΘK
The latter formula appears in [HL] (in more generality), and it is a corrected version
of a formula in [HM]. Implicit in the formula is the fact that mT (θ) is constant for
θ in Θ0 and we have let mK (Θ0 ) denote this common value. A more refined formula
that takes into account Lemma 6.3 also appears in [HL]:
X
hΘ, ΨiG =
mT (Θ0 )hΘ0 , ΨiT .
Θ0 ∈ΘT
We have already computed the values of mT (Θ0 ). The next lemma describes
the terms hΘ0 , ΨiT .
Lemma 6.4. Let Ψ = (T, φ) be a toral datum. Let Θ0 be a T -orbit of involutions
of G. Then
(
1, if φ(−1) = 1 and Θ0 is T-relevant,
0
hΘ , ΨiT =
0, otherwise.
Proof. Suppose Θ0 is not T-relevant. Then for every θ ∈ Θ0 the torus is not
θ-split and thus, according to Lemma 6.1, Ψ is not θ-symmetric. So hΘ0 , ΨiT = 0
by Proposition 5.20 of [HM].
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
15
Now suppose Θ0 is T-relevant. Then Lemma 6.1 implies that Ψ is θ-symmetric
for all θ ∈ Θ0 and hence hΘ0 , ΨiT = hΘ00 , ΨiK , where Θ00 is the K-orbit containing
Θ0 . Proposition 5.31 (1) of [HM] implies that
(
1, if φ(−1) = 1,
00
hΘ , ΨiK =
0, otherwise.
Our claim follows.
6.2. The main theorem in the toral case.
Theorem 6.5. Let Ψ = (T, φ) be a toral datum and let Θ be a G-orbit of
involutions. Then
(
2, if φ(−1) = 1 and Θ is T-relevant,
hΘ, ΨiG =
0, otherwise.
There are seven G-orbits of involutions
of G. Exactly two of these orbits are T1 0
relevant. The G-orbit of Int
is always T-relevant. The other T-relevant
0 −1
0 τ
G-orbit is the orbit of: Int
, if −τ 6∈ (F × )2 , and otherwise it is the orbit
−1 0
√
a
bτ
of Int
, where a + b τ is a root of unity that generates the multiplicative
−b −a
group of the residue field of E.
Proof. Consider the formula:
hΘ, ΨiG =
X
mT (Θ0 )hΘ0 , ΨiT .
Θ0 ∈ΘT
Lemma 4.1 provides the formula
(
1, if − τ ∈ (F × )2 ,
mT (Θ0 ) =
2, if − τ 6∈ (F × )2 ,
for mT (Θ0 ). Lemma 6.4 says that
(
1, if φ(−1) = 1 and Θ0 is T-relevant,
0
hΘ , ΨiT =
0, otherwise,
and Proposition 3.8 provides a detailed description of the T-relevant orbits of involutions.
If −τ ∈ (F × )2 then there are two T-relevant G-orbits, each of which contains
two T-relevant T -orbits. The relevant G-orbits are precisely those in the statement
of the present putative result. When φ(−1) = 1 and Θ is T-relevant, the formula
for hΘ, ΨiG yields 1 · 1 + 1 · 1 = 2.
If −τ 6∈ (F × )2 then there are again two T-relevant G-orbits, but now each of
these contains a unique T-relevant T -orbit. The relevant G-orbits are again as we
have claimed. When φ(−1) = 1 and Θ is T-relevant, the formula for hΘ, ΨiG yields
2 · 1 = 2. Our assertions have therefore been proven.
7. Distinguished depth-zero supercuspidal representations
This section describes a result of Lusztig [L], which we use to determine the
distinguished depth-zero supercuspidal representations of SL2 (F ).
16
JEFFREY HAKIM AND JOSHUA M. LANSKY
7.1. A result of Lusztig. Let q be a power p, and let Fq denote the field
with q elements. If H is a reductive Fq -group, as in [L], we define
σ(H) = (−1)Fq -rank of H .
Now fix a reductive Fq -group G. To be consistent with our conventions for
groups over p-adic fields, we refer to an Fq -automorphism of G of order two as an
“involution of G(Fq ).” The group G(Fq ) acts on the set of its involutions in the
usual way. Let θ be any involution of G(Fq ), and let Gθ denote the group of points
in G fixed by θ. Let Gθ denote the group of g ∈ G such that gθ(g)−1 = ±1.
Fix a maximal Fq -torus T of G and a complex character λ of T(Fq ). Let
λ
G
RT
= RT,λ
denote the virtual character of G(Fq ) defined by Deligne-Lusztig [DL].
Suppose S is a maximal torus in G that is defined over Fq . For a subgroup H
of G, let ZG (H) denote the centralizer of H in G, and let
Hθ = H ∩ Gθ ,
Hθ = H ∩ Gθ .
If s ∈ S(Fq ), let Zs be the identity component of the centralizer of s in G and let
εS,θ : Sθ (Fq ) → {±1} be defined by
εS,θ (s) = σ(ZG ((Sθ )◦ )) σ(ZZs ((Sθ )◦ )).
(We warn the reader that our notation Zs conflicts with the notations in [L].)
Let Θ be a G(Fq )-orbit of involutions of G(Fq ), and set
ΘT,λ = {θ ∈ Θ : θ(T) = T, λ|Tθ (Fq ) = εT,θ }.
Note that ΘT,λ is a union of T(Fq )-orbits of involutions. If Θ0 is a T(Fq )-orbit in
ΘT,λ , let
mT (Θ0 )
=
[Gθ (Fq ) : Gθ (Fq ) · Tθ (Fq )],
hΘ0 , Ti = σ(T) σ(ZG ((Tθ )◦ )),
where θ is an arbitrary element of Θ0 .
Now let θ be any element of Θ. If (·, ·) denotes the usual normalized inner
product on the space of complex-valued functions on Gθ , let
hΘ, λiG = (RT,λ , 1),
where 1 is the trivial character of Gθ (Fq ). Note that this definition makes sense
since the inner product on the right depends only on the G(Fq )-orbit of θ. Also,
note that if RT,λ is irreducible, this inner product is (up to a sign) the dimension
of the space of G(Fq )-fixed vectors in the representation whose character is (up to
a sign) equal to RT,λ .
The following theorem is a special case of a result of Lusztig [L] in a form
modified to parallel that of (6.1). This reformulation appears in [HL].
Theorem 7.1.
(7.1)
hΘ, λiG =
X
mT (Θ0 ) hΘ0 , Ti,
Θ0 ⊂ΘT,λ
where the sum is over T(Fq )-orbits Θ0 in ΘT,λ .
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
17
7.2. Distinguished cuspidal representations of SL2 (Fq ). We now specialize to G = SL2 and let T be an elliptic maximal Fq -torus of G. (Such a torus is
unique up to conjugacy in G(Fq ).) Every irreducible cuspidal character of G(Fq )
occurs within some RT,λ for a nontrivial character λ of T(Fq ). Indeed, if λ is not
quadratic, RT,λ is precisely the negative of an irreducible cuspidal character, and
all (q − 1)/2 such characters of degree q − 1 arise in this way. On the other hand,
if λ is the nontrivial quadratic character of T(Fq ), RT,λ is the negative of the sum
of the two remaining irreducible cuspidal characters of G(Fq ) (which have degree
(q − 1)/2).
Suppose Θ is a G-orbit of involutions such that hΘ, λiG 6= 0, i.e., such that the
representation with character −RT,λ is distinguished with respect to the involutions
in Θ. We now compute hΘ, λiG . Let θ0 ∈ Θ. If θ0 stabilizes T, then, as in §3.3, it
0
must either fix T pointwise or act via inversion. If the former holds, then T = Gθ ,
/ ΘT,λ since
and it follows that εT,θ0 is the trivial character of T(Fq ). Then θ0 ∈
λ is nontrivial. Hence every θ0 ∈ ΘT,λ must act by inversion on T, and hence
0
T ∩ Gθ = {±1}.
It is easily seen in this case that εT,θ0 is the trivial character of {±1}. Thus
ΘT,λ is nonempty only if λ(−1) = 1. We therefore assume now that λ satisfies this
condition since otherwise hΘ, λiG = 0.
In order to compute hΘ, λiG , we need to understand the set of T(Fq )-orbits in
ΘT,λ = {θ0 ∈ Θ : θ0 (T) = T}. This is a direct analogue of what is computed in
Proposition 3.8 for the field F . However, there is no change in the
√ argument if F
is replaced by Fq . Let τ be a non-square in Fq . Then Fq2 = Fq [ τ ] and T(Fq ) is
isomorphic to the group of elements in Fq2 of norm 1. According to Proposition 3.8,
if −τ ∈ (F × )2 , then ΘT,λ contains two T(Fq )-orbits, while if −τ ∈
/ (F × )2 , then ΘT,λ
is a single T(Fq )-orbit. Thus there are two summands in (7.1) if −τ ∈ (F × )2 , and
one summand if −τ ∈
/ (F × )2 .
According to §4.2 (which again holds for finite as well as p-adic fields), if Θ0 is
a T(Fq )-orbit in ΘT,λ , we have that
(
2
1, if − τ ∈ (F×
q ) ,
0
mT (Θ ) =
2
2, if − τ ∈
/ (F×
q ) .
Also, for θ0 ∈ Θ,
0
hΘ0 , Ti = σ(T) σ(ZG ((T ∩ Gθ )◦ )) = σ(T) σ(G) = −1.
Putting this information together yields that in all cases, if hΘ, λiG is nonzero, it
must equal −2. Thus the representation ρ with character −RT,λ satisfies
dim HomGθ (Fq ) (ρ, 1) = 2.
As explained above, if ρ is reducible, it is the sum of two irreducible cuspidal
representations ρ1 and ρ2 . Moreover, any automorphism α of G of the form Int(g)
for g ∈ GL2 (Fq ) not in G(Fq ) · Z(GL2 )(Fq ) has the property that ρ1 ◦ α = ρ2 .
Suppose that dim HomGθ (Fq ) (ρ1 , 1) 6= 0 for some θ ∈ Θ. (According to the above
discussion, this will happen if and only if the nontrivial quadratic character λ of
2
T(Fq ) is trivial on ±1, i.e., when −τ ∈ (F×
q ) .) Suppose θ = Int(g) for g ∈ S̃(Fq ),
where S̃ is a maximal torus of GL2 . There exists h ∈ S̃(Fq ) − (G(Fq ) · Z(GL2 )(Fq )).
Thus ρ1 ◦ Int(h) = ρ2 , and Int(h) commutes with θ so that α(Gθ ) = Gθ . It follows
18
JEFFREY HAKIM AND JOSHUA M. LANSKY
that dim HomGθ (Fq ) (ρ2 , 1) 6= 0. Thus for i = 1, 2,
dim HomGθ (Fq ) (ρi , 1) = 1.
We have proved the following result. (Note that the central character of ρ
agrees with λ on ±1.)
Proposition 7.2. Suppose ρ is an irreducible cuspidal representation of G(Fq ).
If the central character of ρ is nontrivial, then ρ is not distinguished with respect to
any involution of G(Fq ). Otherwise, ρ is distinguished with respect to exactly two
G(Fq )-orbits of involutions of G(Fq ). If θ is an element of one of these two orbits,
then
(
2, if ρ has degree q − 1,
dim HomGθ (Fq ) (ρ, 1) =
1, if ρ has degree (q − 1)/2.
7.3. Depth-zero supercuspidal representations. Let y ∈ B(G, F ), and
let T be a an elliptic maximal kF -torus of Gy . Let T be an unramified elliptic
maximal F -torus of G such that y ∈ A(G, T, F ) and such that T /T0+ = T(Fq ). In
this situation, we will say that T reduces to T in Gy .
Theorem 7.3. Let Ψ = (G, y, ρ, 1) be a depth-zero cuspidal G-datum, and let
Θ be a G-orbit of involutions of G. There exists an unramified elliptic maximal
F -torus of G which reduces to an elliptic maximal kF -torus of Gy . Let T be any
such torus. Then


2, if ωρ = 1, Θ is T-relevant, and deg(ρ) = q − 1,
hΘ, ΨiG = 1, if ωρ = 1, Θ is T-relevant, and deg(ρ) = (q − 1)/2,


0, otherwise.
The T-relevant G-orbits are given by Theorem 6.5.
Proof. We will view the representation ρ of K = Gy,0 also as a representation
of Gy (kF ) = Gy,0 /Gy,0+ . Then [HM, Thm. 5.26 (3)] gives
(7.2)
hΘ0 , ΨiK = dim HomGθ (π(Ψ), 1) = dim HomGθ (kF ) (ρ, 1).
Let T be an unramified elliptic maximal F -torus of G which reduces to the
elliptic maximal kF -torus T of Gy ∼
= SL2 . Such a torus T exists by [D, Lemma
2.3.1]. Suppose that Θ is T-relevant, i.e., some θ ∈ Θ acts by inversion on T. It
follows that θ fixes the unique point y ∈ A(G, T, F ), hence determines a nontrivial
kF -involution (which we will also denote by θ) of the reductive kF -group Gy attached
to y by Bruhat-Tits theory. Moreover, θ must stabilize T ⊂ Gy and act on it by
inversion (since it does so on T). There exists a complex character λ of T(kF ) such
Gy
that the character of ρ occurs in RT,λ
. It follows that the sum (7.1) has at least
one nonzero term, and must therefore be nonzero itself by the discussion in §7.2.
Hence we must have
dim HomGθ (kF ) (ρ, 1) 6= 0,
0
By (7.2), hΘ , ΨiK 6= 0 and thus hΘ, ΨiG 6= 0.
Conversely, suppose Θ is a G-orbit of involutions of G such that hΘ, ΨiG 6= 0.
Then there is some K-orbit Θ0 ⊂ Θ such that hΘ0 , ΨiK 6= 0. According to [HM,
Prop. 5.20], there exists θ ∈ Θ0 such that y is a θ-fixed vertex. Thus θ acts as a
Gy (kF )-involution. Then (7.2) implies that
(7.3)
dim HomGθ (kF ) (ρ, 1) = hΘ0 , ΨiK 6= 0.
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
19
There exists an elliptic maximal kF -torus T of Gy and a complex character λ of
Gy
T(kF ) such that the character of ρ occurs in RT,λ
. By (7.3) and the discussion
in §7.2, we have that h Θ̄, λiGy 6= 0, where Θ̄ is the Gy (kF )-orbit of θ. Thus some
Gy (kF )-conjugate of θ stabilizes T; otherwise the indexing set of the sum in the
formula (7.1) for h Θ̄, λiGy would be empty. Moreover, as in §7.2, θ must act by
inversion on T. It follows that θ stabilizes and acts by inversion on some Gy (kF )conjugate Ṫ of T. According to [HL, Prop. A.3], there exists a θ-stable unramified
elliptic maximal F -torus T of G such that A(G, T, F ) contains y and T reduces
to Ṫ in Gy . Since θ acts by inversion on Ṫ, it must also do so on T. Thus Θ is
T-relevant.
Now suppose T0 is any unramified elliptic maximal F -torus of G which reduces
to an elliptic maximal kF -torus T0 of Gy . There exists ḡ ∈ Gy such that Int(ḡ)(T0 ) =
T. Let g be any preimage of ḡ in Gy . Then Int(g)(T0 ) reduces to T in Gy . It
follows from [D, Lemma 2.2.2] that there exists h ∈ Gy,0+ such that Int(hg)(T0 ) =
Int(h)(Int(g)(T0 )) = T. Then (hg)−1 · θ stabilizes T0 and Θ is T0 -relevant.
Now assume that hΘ, ΨiG 6= 0 (and hence that Θ is T-relevant). We now
compute mK (Θ0 ) for a K-orbit Θ0 ⊂ Θ. Assume for the moment that T = Tτ
for some unit τ ∈ F . Then y is the “origin” of the standard apartment and
Gy = SL2 (OF ). Moreover, it is easily seen that
a ∓bτ
θ
G =
∈G
b ±a
a −bτ
Gθ =
∈G .
b
a
Clearly, there are representatives for Gθ /Gθ in Gy . For general T and y, the same
is true since T can be transported to Tr via Int(g) for an appropriate g ∈ GL2 (F ).
Therefore, we have
mK (Θ0 ) = |Gθ /Gθ (K ∩ Gθ )| = 1.
As given in Proposition 3.8, there are exactly two T-relevant T -orbits of involutions of G, and they are contained in distinct G-orbits of involutions. Hence each
of these G-orbits contains a unique K-orbit Θ0 containing a T-relevant involution.
It follows that the summation (6.1) contains a single summand. We have therefore
shown that
hΘ, ΨiG = hΘ0 , ΨiK = dim HomGθ (kF ) (ρ, 1).
Proposition 7.2 now implies the stated formula for hΘ, ΨiG .
Appendix A. The building and the Moy-Prasad groups
A.1. Affine roots. Let S denote the subgroup of G consisting of diagonal
matrices. The character and cocharacter groups of S are
t
0
m
X ∗ (S) =
χm : S → F × | m ∈ Z, χm
=
t
0 t−1
n
t
0
X∗ (S) =
λn : F × → S | n ∈ Z, λn (t) =
0 t−n
20
JEFFREY HAKIM AND JOSHUA M. LANSKY
and we have a Z-valued pairing given by hχm , λn i = mn. The set of roots of (G, S)
is Φ = {α, −α}, where α = χ2 and −α = χ−2 . The coroots are α̌ = λ1 and
−α̌ = λ−1 .
Let V = X∗ (S) ⊗Z R. Note that the elementary tensor λn ⊗ u, with n ∈ Z
and u ∈ R, equals the elementary tensor λ1 ⊗ (nu). So λ1 ⊗ u 7→ u determines an
R-linear isomorphism of V with R.
Given a root β ∈ Φ and an integer m ∈ Z, we obtain an affine root
βm (λ1 ⊗ u) = uhβ, λ1 i + m.
Writing β = εα with ε = ±1, we have
(εα)m (λ1 ⊗ u) = 2εu + m.
The set of affine roots of (G, S) is Ψ = {βm : β ∈ Φ, m ∈ Z}.
For each ψ ∈ Ψ, let Hψ = ψ −1 (0). So
H(εα)m = λ1 ⊗ (−εm/2).
It follows that
o
n
n
{Hψ : ψ ∈ Ψ} = λ1 ⊗ : n ∈ Z .
2
A.2. The standard apartment. The Bruhat-Tits building of G over F is
denoted B(G, F ). It will be described explicitly in the next section. In this section,
we study the apartment A(G, S, F ) associated to S. We view this as “the standard
apartment” in B(G, F ). We identify A(G, S, F ) with V , viewed as an affine space
under V . It inherits the standard metric from R.
The vertices in A(G, S, F ) are then the points Hψ or, in other words, the points
λ1 ⊗ (n/2), with n ∈ Z. For nonnegative r ∈ R, define Fr× as follows. If r = 0 then
dre
×
Fr× = O×
F , and if r > 0 then Fr = 1 + PF . If r ≥ 0, let
t
0
×
Sr =
: t ∈ Fr .
0 t−1
If m ∈ Z, let
Uαm
=
U(−α)m
=
1
0
1
c
b
: b ∈ Pm
,
F
1
0
: c ∈ Pm
.
F
1
If x = λ1 ⊗ u ∈ A(G, S, F ), a = εα ∈ Φ and r ≥ 0 then we take
Ux,a,r = U(εα)m ,
where m = dr − 2εue.
If x = λ1 ⊗ u ∈ A(G, S, F ) and r ≥ 0 then let
Gx,r
= hSr , {Uψ }ψ∈Ψ,
ψ(x)∈[r,r+1) i
= hSr , {Ux,a,r }a∈Φ i
dr−2ue
1
1 PF
=
Sr ,
,
dr+2ue
0
1
PF
0
.
1
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
21
Suppose u 6∈ 12 Z or r 6= 0 (so that Gx,r is not the parahoric subgroup associated
to the vertex x). Then we have
a b
dr−2ue
dr+2ue
×
Gx,r =
: a, d ∈ Fr , b ∈ PF
, c ∈ PF
, ad − bc = 1 .
c d
Now suppose u = n/2 for n ∈ Z (i.e., x is a vertex) and r = 0, then
−1
1 0
1 0
Gx,0 =
SL
(O
)
,
2
F
0 $n
0 $n
where $ is a prime element of OF .
e be the disjoint union of R with the set of symbols r+ parametrized by
Let R
the real numbers r. The linear ordering on R extends uniquely to a linear ordering
e such that r < t implies r < r+ < t < t+ . If x ∈ A(G, S, F ) and r is a
on R
nonnegative real number, we define
[
Gx,t .
Gx,r+ =
t>r
A.3. The building. The group G acts transitively on the set of its maximal
split tori. Given a torus gSg −1 , with g ∈ G, the associated apartment
A(G, gSg −1 , F ) = gA(G, S, F )
may be viewed as the set of symbols gx, with g ∈ G and x ∈ A(G, S, F ), subject to
certain identifications. The building B(G, F ) is the union of the apartments, again
subject to the appropriate identifications. The Moy-Prasad groups associated to
gx are defined by
Ggx,r = gGx,r g −1
and this relation may be viewed as the key to how one glues the apartments together.
The simplicial structure on A(G, S, F ) is transported to the apartment gA(G, S, F ) =
A(G, gSg −1 , F ) by the map x 7→ gx. So there is a well-defined notion of “vertex”
in B(G, F ). Given two vertices x1 and x2 in B(G, F ), we declare that the gluing
relation is such that
x1 = x2 ⇔ Gx1 ,0 = Gx2 ,0 .
More generally, two points x1 and x2 lie in the same facet of B(G, F ) exactly when
they have identical parahoric subgroups Gx1 ,0 and Gx2 ,0 .
Suppose x1 and x2 are points in different apartments that are not vertices, but
they have the same parahoric subgroup. Let y1 and y2 be the vertices of the facet
that contains x1 and x2 . Then x1 and x2 determine the same point in B precisely
when d(x1 , y1 ) = d(x2 , y1 ) and thus d(x1 , y2 ) = d(x2 , y2 ).
Consider the special case in which g ∈ S. Say g = α̌(t) and consider Ggx,r .
Since gx ∈ A(G, S, F ), it must be the case that Ggx,r is one of the Moy-Prasad
groups described explicitly above. Indeed, it is easy to check that if x = λ1 ⊗ u and
g = α̌(t) then Ggx,r = Gx0 ,r where x0 = λ1 ⊗ (u − 1). This is consistent with the
fact that S acts on its apartment A(G, S, F ) according to
t
0
(λ1 ⊗ u) = λ1 ⊗ (u − vF (t)),
0 t−1
where vF is the standard valuation on F . Similarly, one verifies the formula
0 1
(λ1 ⊗ u) = λ1 ⊗ (−u).
−1 0
22
JEFFREY HAKIM AND JOSHUA M. LANSKY
We extend our definition of the root group filtration subgroups Ux,a,r in the
previous section to arbitrary points in B(G, F ) by putting
Ug·x,g·a,r = gUx,a,r g −1 ,
where (g · a)(gsg −1 ) = a(s), when g ∈ G, x ∈ A(G, S, F ), a ∈ Φ(G, S) and r ≥ 0.
A.4. The J groups. Fix a vertex x ∈ B(G, F ) and r > 0. Let s = r/2. We
consider in this section certain subgroups J, J+ and J++ of Gx,0 that play a role in
the construction of some of the supercuspidal representations of G. In the notation
of [HM] (which slightly deviates from [Y]), these groups are given by
J = (T, G)x,(r,s) ,
J+ = (T, G)x,(r,s+ ) ,
J++ = (T, G)y,(r+ ,s+ ) .
They can be expressed more explicitly when x = λ1 ⊗u ∈ A(G, S, F ). In particular,
ds−2ue
1
0
1 PF
J =
Sr ,
,
ds+2ue
0
1
PF
1
a b
×
=
: a, d ∈ Fr , b ∈ Fs−2u , c ∈ Fs+2u , ad − bc = 1 .
c d
For arbitrary points in B(G, F ), one can use the fact that (T, G)gx,(r,s) = g(T, G)x,(r,s) g −1
to obtain an explicit expression for J. Explicit expressions for J+ and J++ are obtained in an entirely analogous fashion.
e let
A.5. Moy-Prasad algebras. Let g = g(F ) = sl2 (F ). Given r ∈ R,
t 0
sr =
: t ∈ Fr .
0 −t
If m ∈ Z, let
uαm
u(−α)m
=
=
0
0
0
u
u
0
0
0
: u∈
Pm
F
,
: u ∈ Pm
.
F
If x ∈ A(G, S, F ) and r ∈ R then
X
gx,r = sr +
uψ .
ψ∈Ψ, ψ(x)∈[r,r+1)
More explicitly, when x = λ1 ⊗ u we have
gx,r = sr +
0
dr+2ue
dr−2ue
PF
PF
!
0
.
If g ∈ G then
ggx,r = g(gx,r )g −1 .
Note that if x1 , x2 ∈ B(G, F ) then
Gx1 ,0 = Gx2 ,0 ⇔ gx1 ,0 = gx2 ,0 .
The latter conditions are sometimes simpler to work with.
If x ∈ B(G, F ) and r ∈ R, we let
[
gx,r+ =
gx,t .
t>r
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
23
Let gE = g(E). Using the above definitions with F replaced by E and S replaced
by Tτ , we find that the Moy-Prasad algebras over E associated to yτ are
√ a
b τ
√ −1
(gE )yτ ,r = (tE )r +
: a, b ∈ Er
−a
−b τ
√ −1
a
bτ
= (tE )r +
: a ∈ Er , b ∈ τ Er .
−b −a
It follows that
gyτ ,r = tr +
√ −1
a
bτ
: a ∈ Fr , b ∈ F ∩ τ Er .
−b −a
If we assume that τ is either a unit or a prime element in F then
a
bτ
gyτ ,r = tr +
: a ∈ Fr , b ∈ Fr0 ,
−b −a
where r0 = r −
e−1
2
and e is the ramification degree of E/F .
A.6. Definition of the Cayley transform. A fundamental fact about matrices X that lie in g is that X 2 is a scalar, namely, X 2 = − det X. If X ∈ g,
define
det X
X2
δX = −
=
.
4
4
One easily verifies the identities
X
X
1 + δX = det 1 +
= det 1 −
.
2
2
Given X ∈ g, the (normalized) Cayley transform
−1 −1 X
X
X
X
e(X) = 1 +
1−
= 1−
1+
∈G
2
2
2
2
is therefore defined precisely when X lies in the set
g] = {X ∈ g : δX 6= 1}.
We observe that g] may also be described as the set matrices in g whose characteristic polynomial is not λ2 − 4 or as the set of matrices whose eigenvalues are not 2
and −2.
Since we are working with 2-by-2 unimodular matrices, inverse matrices are
easily computed. This allows us to obtain the following explicit formula for the
Cayley transform:
1 + δX + X
e(X) =
.
1 − δX
A.7. The inverse transform. If g ∈ G, define
tr(g)
.
2
For our purposes, the group analogue of g] is the set
τg =
G] = {g ∈ G : τg 6= −1}.
A matrix g ∈ G fails to lie in G] precisely when one (hence both) of its eigenvalues
are −1. One can also verify that G] is the set of g ∈ G such that det(g + 1) 6= 0.
24
JEFFREY HAKIM AND JOSHUA M. LANSKY
We define the inverse Cayley transform by:
`(g) = 2(g + 1)−1 (g − 1) = 2(g − 1)(g + 1)−1 .
It is easy to verify the formula
`(g) =
g − g −1
.
1 + τg
Lemma A.1. The map e defines a G-equivariant bijection between g] and G] ,
where G acts by conjugation. It restricts to the bijection X 7→ 1 + X = exp(X)
between the nilpotent set N in g and the set of unipotent elements in G. The inverse
of e is `.
Proof. We first observe that −1 cannot be an eigenvalue of an element g =
e(X) with X ∈ g] since
X
X
v =− 1−
v
e(X)v = −v ⇔
1+
2
2
⇔ v = 0.
This proves that e(g] ) ⊂ G] .
Injectivity can be shown by computing `(e(X)) or by using the following argument
−1 −1 X
X
Y
Y
e(X) = e(Y ) ⇔
1+
1−
= 1−
1+
2
2
2
2
Y
X
Y
X
⇔
1−
1+
= 1+
1−
2
2
2
2
⇔ X = Y.
Surjectivity follows from a routine calculation which establishes that e(`(g)) = g
for all g ∈ G] .
Unfortunately, if X ∈ g (hence tr(X) = 0) then 1 + X does not necessarily have
determinant one. In fact, 1 + X is unimodular precisely when X is nilpotent. So
e(X) = 1 + X precisely when X is nilpotent.
We close this section by observing that
e(−X) = e(X)−1
for all X ∈ g.
A.8. Moy-Prasad isomorphisms. The Cayley transform is not defined on
the Moy-Prasad algebras gx,0 ; for example, e(dα̌(2)) is undefined. We now consider
the Cayley transform on gx,r and gx,r:r+ = gx,r /gx,r+ for r > 0.
e and r > 0 then the Cayley transform e
Lemma A.2. If x ∈ B(G, F ), r ∈ R
2dre
defines a bijection between gx,r and Gx,r . If X ∈ gx,r then δX = − det4 X ∈ PF .
The Cayley transform yields an isomorphism
e : gx,r:r+ → Gx:r:r+
of abelian groups.
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
25
Proof. Suppose x = λ1 ⊗ u ∈ A(G, S, F ) and
a b
X=
∈ gx,r , a ∈ Fr , b ∈ Fr−2u , c ∈ Fr+2u .
c −a
Then if λ is an eigenvalue of X we have λ2 = 4δX = a2 + bc ∈ PF . Since λ ∈ PF ,
it must be the case that λ2 6= 4. Therefore, X ∈ g] and thus gx,r ⊂ g] .
So, when x ∈ A(G, S, F ), it follows directly from our explicit expressions for
gx,r , Gx,r and e that e(gx,r ) = Gx,r . This extends to x 6∈ A(G, S, F ) using the Gequivariance of e. The remaining assertions follow similarly, first for x ∈ A(G, S, F )
and then, using equivariance for all x.
The isomorphisms in the previous lemma are called Moy-Prasad isomorphisms
in [Y].
A.9. Heisenberg algebras. We identify the dual t∗ of t with t using the trace
form. In particular, given X ∗ ∈ t∗ we define X ∈ t by X ∗ (Y ) = tr(XY ), for all
Y ∈ t. The root system for (G, S) is Φ(G, S) = {α, −α}. The corresponding root
system for (G, T) is Φ(G, T) = {αT , −αT }, where αT (t) = α(γtγ −1 ).
Assume that we have fixed a nonsquare τ ∈ F such that τ is a unit or a
prime element. Fix r > 0 and let s = r/2. Assume we have fixed an element
X ∗ ∈ t∗−r − t∗(−r)+ . Then the associated element X ∈ t−r − t(−r)+ has the form
0 τ
X=b
1 0
for some b ∈ F . (See §2.1.) Letting ω be the valuation on E that extends the
standard valuation on F , we have
√
ω(X ∗ (dα̌T (1))) = ω(tr(Xdα̌T (1))) = ω(2b τ ) = −r,
since
dα̌T (1) = γ −1 dα̌(1)γ = γ −1
1
0
0
0
γ = √ −1
−1
τ
√ τ
.
0
We now observe that
√
√
√
1
ω(2b τ ) = ω(b τ ) = ω( τ ) + ω(b) ∈ + Z.
e
So we are implicitly assuming that
1
r ∈ + Z.
e
Thus
1
1
e−1
1
1
s∈
+ Z, s0 = s −
∈
+ Z
2e 2
2
2e 2
and
e−1
ω(b) = − r +
.
2
Let
u
vτ
0
J = (t, g)yτ ,(r,s) = tr +
: u ∈ Fs , v ∈ Fs ,
−v −u
u
vτ
J+ = (t, g)yτ ,(r,s+ ) = tr +
: u ∈ Fs , v ∈ F(s0 )+ .
−v −u
26
JEFFREY HAKIM AND JOSHUA M. LANSKY
Suppose E/F is ramified. Then s ∈ 41 + 12 Z implies that s is not an integer and
hence Fs = Fs+ . Similarly, Fs0 = F(s0 )+ . Therefore, J = J+ if E/F is ramified.
Now suppose E/F is unramified. Then r ∈ Z and s = s0 ∈ 12 Z. If r is odd then
s = s0 ∈ 12 + Z and J = J+ .
Finally, we consider the case in which E/F is unramified and r is an even
integer. Then s = s0 is an integer and we have
u
vτ
J = (t, g)yτ ,(r,s) = tr +
: u, v ∈ Fs ,
−v −u
u
vτ
J+ = (t, g)yτ ,(r,s+ ) = tr +
: u, v ∈ Fs+1 .
−v −u
The Cayley transform yields a Moy-Prasad isomorphism gy,s:s+ → Gy,s:s+ which
then restricts to an isomorphism J/J+ → J/J+ .
We now define a symplectic form on J/J+ . Fix a character ψ of F that is trivial
on F0+ = PF but nontrivial on F0 = OF . Given Y, Z ∈ J, let
hY, Zi = ψ(X ∗ ([Y, Z])) = ψ(tr(X[Y, Z])).
To be more explicit, if
Y =
u
−v
vτ
,
−u
and Z =
w
−x
xτ
−w
then
0 τ
[Y, Z] = 2(ux − vw)
1 0
and
hY, Zi = ψ(4bτ (vw − ux)).
The resulting symplectic space is denoted W.
Let Fs:s+ = Fs /Fs+ . Define a µp -valued symplectic form
(Fs:s+ × Fs:s+ ) × (Fs:s+ × Fs:s+ ) → µp
by
0
h(u, v), (w, x)i = ψ(4bτ (vw − ux)) = ψ −4bτ (u, v)
−1
1
w
,
0
x
where µp is the group of complex p-roots of unity. We obtain then a symplectic
isomorphism
Fs:s+ × Fs:s+ → W
by
(u, v) 7→
u
−v
vτ
.
−u
Finally, we observe that Fs:s+ is (noncanonically) isomorphic to additive group of
the residue field kF of F .
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
27
A.10. Yu’s polarization. The next two sections are not needed for our main
results, but they are included to illustrate some structural aspects of the symplectic
spaces in [Y] and [HM]. The construction of Heisenberg and Weil representations
requires that certain symplectic spaces over Fp are polarized. In this section, we
explicitly exhibit Yu’s polarization. This polarization is “defined over E” in the
sense that it is constructed using subgroups of G(E) and roots that are defined
over E. We show that the polarization degenerates over F . Yu’s polarization is not
suitable for the theory in [HM], because it is not well-behaved with respect to the
given involutions of G. In the next section, we consider the polarizations given in
[HM].
The following objects are the Lie algebra analogues of the groups JE and (JE )+
defined earlier:
u
vτ
JE = (tE )r +
: u, v ∈ Es ,
−v −u
u
vτ
(JE )+ = (tE )r +
: u, v ∈ Es+1 .
−v −u
Let ψE be an extension of ψ to a character of E that is trivial on E0+ . Define a
symplectic form on JE /(JE )+ by
hY, Zi = ψE (X ∗ ([Y, Z])) = ψE (tr(X[Y, Z])),
for Y, Z ∈ JE .
Yu’s polarization of the symplectic space WE = JE /(JE )+ is given by considering the images of the following sets in WE :
0 ∗
−1
JE (+) = JE ∩ γ
γ
0 0
√ 1
− τ
: u ∈ Es ,
=
u √ −1
τ
−1
JE (−)
0 0
JE ∩ γ −1
γ,
∗ 0
√ τ
1
√ −1
: u ∈ Es .
=
u
− τ
−1
=
Note that
JE (+) ∩ g = {0} = JE (−) ∩ g.
So Yu’s polarization does not yield a polarization of W.
A.11. The polarization associated to θ. In this section, we exhibit the
polarizations constructed in [HM] for a specific involution. This is not needed for
our main results, so we do not consider general involutions.
Fix a positive integer r. We consider the involution of G defined by
0 1
1 0
θ = Int γ −1
γ = Int
.
1 0
0 −1
The group Gθ is the group of diagonal matrices in G. We also note that
a b
θ −1
2
2
γG γ =
: a, b ∈ F, a − b = 1 .
b a
28
JEFFREY HAKIM AND JOSHUA M. LANSKY
Let gE = sl2 (E). The Lie algebra hE of fixed points of dθ in gE is given
explicitly by
0 1
hE =
aγ −1
γ:a∈E
1 0
a 0
=
:a∈E .
0 −a
Let sE be the −1 eigenspace of dθ in gE . Then we have
−1 0 −1
sE = tE + bγ
γ:b∈E
1 0
0 τ
= tE + b
:b∈E .
−1 0
The corresponding objects over F are:
1 0
h =
a
:a∈F ,
0 −1
0 τ
s = t+ b
:b∈F .
−1 0
Let
1 0
: a ∈ Fs ,
a
0 −1
0 τ
J ∩ s = tr + b
: b ∈ Fs ,
−1 0
1 0
J+ ∩ h =
a
: a ∈ Fs+1 ,
0 −1
0 τ
J+ ∩ s = tr + b
: b ∈ Fs+1
−1 0
J∩h =
W+
=
(J ∩ h)/(J+ ∩ h)
−
=
(J ∩ s)/(J+ ∩ s).
W
Then W = W+ ⊕ W− is a polarization of the symplectic space W.
A.12. The character ηθ0 . For a general reductive G and an involution θ of G,
a certain complex character ηθ0 of K 0,θ arises in analyzing the space of Gθ -invariant
linear forms on a tame supercuspidal representation. Here K 0,θ is the group of
θ-fixed points in K 0 , a certain open compact-mod-center subgroup of G that arises
in Yu’s construction.
In our case, ηθ0 is only relevant in the toral case in which there is a nontrivial
symplectic space. In this case, K 0,θ = T θ = {±1} = Z. In other words, K 0,θ is the
center Z of G. But ηθ0 is defined in terms of the action of K 0,θ on J by conjugation,
and, since the center acts trivially, ηθ0 must be trivial for our examples. Thus this
paper does not provide useful examples for how to compute ηθ0 in general. We refer
the reader to [HL] where more complicated examples are treated.
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
29
A.13. The irrelevant involutions. Consider now the involution of G defined
by
θ = Int
0
1
τ
0
1
= Int γ −1
0
0
γ .
−1
This involution fixed T pointwise. Since T is not θ-split, we have seen that θ is
essentially irrelevant to our study of distinguished toral supercuspidal representations coming from characters of T. We show in this section that the basic structures
needed to construct distinguished toral supercuspidal representations break down.
In particular, the polarizations of the relevant symplectic spaces are undefined.
Assume E/F is unramified and τ is a nonsquare unit in F . Let r be a positive
integer. It may be of some interest to consider the symplectic space in this case,
even though it is not relevant to the construction of distinguished representations.
We first observe that
0 τ
dθ = Ad
.
1 0
It follows that t is the space of fixed points of dθ in g. The −1 eigenspace of dθ is
a
bτ
s=
: a, b ∈ F .
−b −a
We have:
J
J+
a
bτ
: a, b ∈ Fs ,
−b −a
a
bτ
= tr +
: a, b ∈ Fs+1 ,
−b −a
= tr + ss = tr +
= tr + ss+
where {st }t is the obvious filtration of s. But this means Jθ = tr . So θ does not
yield a polarization of W (or WE ) as the θ-fixed space in J/J+ is trivial.
A.14. An alternate approach to counting multiplicities. In this section,
we provide an alternate method for computing the dimension of HomGθ (π, 1) for
toral supercuspidal representations π and for θ in
G-orbit of involu one specific
1 0
tions, namely the G-orbit Θ of the involution Int
. Since we have already
0 −1
computed the dimension of HomGθ (π, 1) by other methods, we do not carry out
the approach of this section for the other G-orbits.
Proposition 5.31 (4) of [HM] implies that if π is a Gθ -distinguished supercuspidal representation associated to a toral datum (T, φ) then the dimension of
HomGθ (π, 1) is the cardinality of T \S/Gθ , where
S = {g ∈ G : gθ(g)−1 ∈ T }
and T \S/Gθ denotes the set of double cosets in T \G/Gθ that have a representative
1 0
in S. We directly compute the latter cardinality. Let θ = Int
. If
0 −1
a b
g=
∈G
c d
30
JEFFREY HAKIM AND JOSHUA M. LANSKY
then
θ(g) =
a −b
d
−1
, θ(g) =
−c d
c
gθ(g)
−1
=
ad + bc
2cd
b
,
a
2ab
,
ad + bc
and S is the set of elements
g=
a
c
b
d
with a, b, c, d ∈ F , ad − bc = 1 and ab = cdτ . The set S is a union of double cosets
in T \G/Gθ . a b
Given g =
∈ S, it is straightforward to verify the following statements
c d
sequentially:
(1) The conditions defining S immediate imply that a and d are either both
zero or both nonzero. The same is true of b and c.
(2) Every double coset in T \S/Gθ contains elements g whose entries a, b, c, d
are all nonzero. (Indeed, if g ∈ S has some zero entries and if t ∈ T has
all nonzero entries, then the entries of tg are all nonzero.)
(3) If a and d are nonzero then a/d = a2 − c2 τ . (The conditions ad − cb = 1
and ab = cdτ imply that a(ad − 1)c−1 = cdτ . Rearranging terms gives
the desired identity.)
(4) If a and d are nonzero then the quantity a/d is an invariant of the coset T g.
The class of a/d in F × /(F × )2 is an invariant of the double coset T gGθ .
Define ι(T gGθ ) to be the square class of a/d. We obtain a well-defined
map
ι : T \S/Gθ → NE/F (E × )/(F × )2
since every double coset contains g with ad 6= 0.
(5) ι(T Gθ ) = (F × )2 .
(6) For g with a = d = 0, we have ι(T gGθ ) = −τ (F × )2 .
Lemma A.3. The map ι is a bijection.
Proof. Suppose ϑ1 , ϑ2 ∈ T \S/Gθ and ι(ϑ1 ) = ι(ϑ2 ). Then we can choose
g1 ∈ ϑ1 and g2 ∈ ϑ2 that have entries a1 , b1 , c1 , d1 and a2 , b2 , c2 , d2 which are all
nonzero and which are such that a1 /d1 and a2 /d2 are in the same square class. In
fact, after multiplying one of the matrices on the right by a suitable element of Gθ
we can, and will, assume that a1 /d1 = a2 /d2 .
Now let x = c2 d1 − c1 d2 and w = a−1
1 (a2 − c1 τ x). Let
t=
We will show that t ∈ T and tg1 = g2 .
w
x
xτ
w
.
DISTINGUISHED SUPERCUSPIDAL REPRESENTATIONS OF SL2
31
The upper left entries of tg1 and g2 agree according to the calculation: wa1 +
xτ c1 = (a2 − c1 τ x) + xτ c1 = a2 . For the upper right entries:
wb1 + xτ d1
= a−1
1 (a2 − c1 τ x)b1 + xτ d1
−1
= a−1
1 a2 b1 + xτ (d1 − a1 b1 c1 )
= a−1
1 (a2 b1 + xτ )
= a−1
1 (a2 b1 + τ (c2 d1 − c1 d2 ))
−1
−1
= a−1
1 (a2 b1 + d1 a2 b2 d2 − d2 a1 b1 d1 )
= a−1
1 (a2 b1 + a1 b2 − a2 b1 )
= b2 .
For the lower left entries:
xa1 + wc1
= xa1 + a−1
1 (a2 − c1 τ x)c1
−1
2
2
= xa−1
1 (a1 − c1 τ ) + a1 a2 c1
−1
= xd−1
1 + a1 a2 c1
=
−1
(c2 − c1 d2 d−1
1 ) + a1 a2 c1
−1
= c2 + c1 (a−1
1 a2 − d1 d2 )
= c2 .
For the lower right entries:
xb1 + wd1
= xb1 + a−1
1 (a2 − c1 τ x)d1
−1
= a−1
1 a2 d1 + xa1 (a1 b1 − c1 d1 τ )
= a−1
1 a2 d1
= d2 .
We have now verified that tg1 = g2 . Since g1 and g2 have determinant one, the
matrix t must lie in T . This proves that ι is injective.
We have already observed that if g is given with ad 6= 0 then a/d = a2 − c2 τ .
This shows that the image of ι is contained in the set of elements a2 − c2 τ with
a ∈ F × and c ∈ F (or rather the image of this set in F × /(F × )2 ). In fact, the image
equals the latter set since the matrix


cτ
a a2 −c
2τ


a
c a2 −c2 τ
lies in S for all a ∈ F × , c ∈ F .
Proving surjectivity now reduces to showing that there exist nonsquare norms
a2 − c2 τ with a 6= 0. If −τ is a square, this follows from the well known fact that
there exist sums of two squares in F that are not themselves squares. If −τ is
not a square, it follows from the identity −τ = a2 − c2 τ with a = 2τ /(τ − 1) and
c = (τ + 1)/(τ − 1).
References
[A]
J. Adler, Refined anisotropic K-types and supercuspidal representations, Pacific J. Math.
185 (1998), no. 1, 1–32.
[ADSS] J. Adler, S. DeBacker, P. Sally and L. Spice, “Supercuspidal characters of SL2 over a
p-adic field,” in these proceedings.
32
[AR]
[AP]
[BT1]
[D]
[DL]
[HL]
[HM]
[K]
[L]
[M]
[MP1]
[MP2]
[MS]
[M]
[Y]
JEFFREY HAKIM AND JOSHUA M. LANSKY
J. Adler and A. Roche, An intertwining result for p-adic groups, Canad. J. Math. 52
(2000), no. 3, 449–467.
U. Anandavardhanan, D. Prasad, Distinguished representations for SL(2), Math. Res.
Lett. 10 (2003), no. 5–6, 867–878.
F. Bruhat, J. Tits, Groupes réductifs sur un corps local, Chapitre I, Publ. Math. Inst.
Hautes Études Sci. 41 (1972), 5–251.
S. DeBacker, “Parameterizing conjugacy classes of maximal unramified tori via BruhatTits theory,” Michigan Math. J. 54 (2006), no. 1, 157–178.
P. Deligne and G. Lusztig, “Representations of reductive groups over finite fields,” Ann.
of Math. (2) 103 (1976), no. 1, 103–161.
J. Hakim and J. Lansky, “Orthogonal periods of tame supercuspidal representations of
GLn with n odd, preprint.
J. Hakim and F. Murnaghan, “Distinguished tame supercuspidal representations,” Int.
Math. Res. Pap., IMRP 2008, no. 2, Art. ID rpn005, 166 pp.
J.-L. Kim, An exhaustion theorem: supercuspidal representations, J. Amer. Math. Soc.
20, no. 2 (2007), 273–320.
G. Lusztig, Symmetric spaces over a finite field, in “The Grothendieck Festschrift”
(P. Cartier et. al., Eds.), Vol. III, pp. 57–81, Birkhäuser, Boston/Basel/Berlin, 1990.
A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 64
(1991), 863–930.
A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116
(1994), 393–408.
A. Moy and G. Prasad, Jacquet functors and unrefined minimal K-types, Comment.
Math. Helv. 71 (1996), 98–121.
A. Moy and P. Sally, Supercuspidal representations of SLn over a p-adic field: the tame
case, Duke Math. J. 51 (1984), no. 1, 149–161.
F. Murnaghan, “Regularity and distinction of supercuspidal representations,” in these
proceedings.
J.-K. Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14
(2001), 579–622.
Department of Mathematics and Statistics, American University, Washington, DC
20016-8050
E-mail address: jhakim@american.edu
Department of Mathematics and Statistics, American University, Washington, DC
20016-8050
E-mail address: lansky@american.edu