MOD p REPRESENTATIONS OF SL2 (Qp )
CHUANGXUN CHENG
Abstract. In [3], the authors give a rough classification of mod p representations of
GL2 (F ) where F is a finite extension of Qp . In [4], the author gives a complete classification of mod p representations of GL2 (Qp ). In this paper, using the classification in
those two papers, we classify mod p representations of SL2 (Qp ). We also define conductors and newforms for these representations and compute the conductors and the
dimensions of spaces of newforms.
1. Notation and introduction
Let F be a finite extension of Qp . In this paper, we classify the mod p principal series
of SL2 (F ). We use the results in [4] to classify all mod p representations of SL2 (Qp ). We
also develop a theory of mod p conductors and newforms for these representations.
First, we introduce some notation. Let p > 2 be a prime number, F be a finite extension
of Qp with ring of integers O and residue field κ. Let p be the maximal ideal of O and $
be a uniformizer of O. Let E be an algebraic closure of Fp .
a b
G = SL2 (F ) = {
|a, b, c, d ∈ F and ad − bc = 1}
c d
a b
P ={
∈ G|c = 0}
c d
a 0
a b
A={
∈ G|b = c = 0} = {
∈ G|a ∈ F × }
c d
0 a−1
a 0
−
A ={
∈ G|a ∈ O\0}
0 a−1
0 −1
W = NG (A)/A = {1, ω =
}
1 0
1 x
N ={
|x ∈ F }
0 1
a b
K = SL2 (O) = {
∈ G|a, b, c, d ∈ O}
c d
a b
I={
∈ K|c ≡ 0(mod p)}
c d
a b
I1 = {
∈ I|a ≡ d ≡ 1(mod p)}
c d
P (O) = P ∩ K
N (O) = N ∩ K
1
2
CHUANGXUN CHENG
$
0
0 −$−1
α=
β=
0 $−1
$
0
×
∼
Since A = F , the characters of A may (and will) be identified with the characters of
F × . Let χ : F × → E × be a character of F × . We may extend it to a character of P = N A.
We denote the induction of χ by (IndG
P χ, S(G, χ)). Here
S(G, χ) := {f ∈ F|f (pg) = χ(p)f (g), ∀p ∈ P, g ∈ G},
F is the space of locally constant functions on G with values in E. G acts on S(G, χ) by
right translation: (gf )(g 0 ) = f (g 0 g).
One of the main results of this paper is the following.
Theorem 1.1. If F = Qp , the smooth irreducible representations of SL2 (Qp ) over E are
the following:
(1) the trivial 1 dimensional representation
(2) the principal series IndG
P χ with χ 6= 1
(3) the special representation Sp := IndG
P 1/hconsti
(4) the supersingular representations Vr0 and Vr1 (0 ≤ r ≤ p − 1), such that
GL (Q )
Vr0 ⊕ Vr1 = (c-IndGL22 (Zpp)Z Symr E 2 /T )|SL2 (Qp ) .
Remark 1.2. Ramla Abdellatif also proved this result independently in her thesis. See for
example Theorem 1 of [1].
Now we explain the idea to prove Theorem 1.1. We write G̃ = GL2 (F ), Z the center of
G̃, P̃ the subset of upper triangular matrices of G̃, I˜ the Iwahori subgroup of G̃, etc. For
simplicity, by a representation of G̃ or G, we always mean a smooth representation with
central character. The following two lemmas are similar to Lemma 2.4, 2.5 in [10]. The
difference is that here we consider representations over E-vector spaces. The proof is the
same as the proof in [10].
Lemma 1.3. Let π̃ be an irreducible representation of G̃, then the restriction of π̃ to G
is an extension of finitely many irreducible representations of G.
Lemma 1.4. Let π be an irreducible representation of G, then there exists an irreducible
representation π̃ of G̃ which contains π as a subquotient.
We can prove the following proposition.
Proposition 1.5. Let χ 6= 1 be a character of F × , then IndG
P χ is irreducible as a representation of G. Furthermore, the following are irreducible representations of SL2 (F ):
(1) the trivial 1 dimensional representation,
(2) IndG
P χ for χ 6= 1,
G
(3) the unique irreducible quotient IndG
P 1/hconsti of IndP 1. We call this representation
Sp.
GL (Qp )
χ1
p)
GL (Q )
IndP̃ (Q2 ) p χ1 ⊗ χ2
p
Remark 1.6. (1) Let IndP̃ (Q2
⊗ χ2 (χ1 6= χ2 ) be a principal series representation of
GL (Q )
GL2 (Qp ). Since
' (IndP̃ (Q2 ) p χ1 χ−1
2 ⊗ 1) ⊗ χ2 and χ2 |SL2 (Qp ) is trivial,
p
we may assume that χ2 = 1 and χ1 6= 1. Then
GL (Qp )
χ1
p)
(IndP̃ (Q2
SL (Qp )
(χ1
p )∩SL2 (Qp )
⊗ 1)|SL2 (Qp ) ' IndP̃ (Q2
SL2 (Qp )
⊗ 1)|P̃ (Qp )∩SL2 (Qp ) ' IndP
χ1 .
MOD p REPRESENTATIONS OF SL2 (Qp )
3
SL (Q )
By Proposition 1.5, since χ1 6= 1, IndP 2 p χ1 is irreducible.
(2) Let St be the Steinberg representation of GL2 (Qp ). (See Remark 3.5 for definition.)
By Section 3.4 of [2], we can realize St on C(P)/hφ0 i, where C(P) is the space of continuous
and compactly supported functions on P1 (Qp ) and φ0 is a certain element in that space.
In Section 2, we will see that the special representation Sp of SL2 (Qp ) is also realized on
this space. Therefore
SL2 (Qp )
(St)|SL2 (Qp ) ' Sp = (IndP
1)/hconsti
which is also irreducible.
(3) This proposition is generalized to any split group in [9].
We know that to understand all representations of SL2 (F ) is the same to understand all
restrictions of representations of GL2 (F ). We see that principal series representations and
special representations of GL2 (F ) give the representations of SL2 (F ) in Proposition 1.5.
GL (Q )
In the case F = Qp , let π̃r = (IndQ× 2GL p(Z ) Symr E 2 )/T be a supersingular representation
p
2
p
of GL2 (Qp ) (see Section 3 for definition), then we have the following proposition, which
finishes the classification of mod p irreducible representation of SL2 (Qp ).
Proposition 1.7. π̃r |SL2 (Qp ) ' πr0 ⊕ πr1 , where πr0 and πr1 are irreducible representations
of SL2 (Qp ) and πr0 ' πr1 if and only if r = (p − 1)/2.
$ 0
0
−1
Now let K = θKθ where θ =
. Note K and K 0 are both maximal compact
0 1
subgroups of SL2 (F ). For m a positive integer, let
a b
0
Km = {
∈ SL2 (O)| c ≡ 0( mod pm )} and Km
= θKm θ−1 .
c d
Let (π, V ) be an irreducible infinite-dimensional representation of G. Let be the central
character of π. Let η be any character of F × such that η(−1) = (−1). Let c(η) denote
0 defined by
the conductor
of η. For any m ≥ c(η), η gives a character of Km and Km
a b
η(
) = η(a). We define for m ≥ 0
c d
a b
a b
πη,m = {v ∈ V | π(
)v = η(a)v ∀
∈ Km }
c d
c d
and
a b
a b
0
= {v ∈ V | π(
)v = η(a)v ∀
∈ Km
}.
c d
c d
0
= πη,m
= 0 if m < c(η). We make the following definition as in [11].
0
πη,m
Note that πη,m
Definition 1.8. The η-conductor cη (π) of π is
0
cη (π) = min{m ≥ 0 | πη,m 6= (0) or πη,m
6= (0)}.
The conductor c(π) of π is
c(π) = min{cη (π) | η is a characters of F × such that η(−1) = (−1)}.
0 ) for the space π
0
If c(π) = cη (π), write πm (resp. πm
η,m (resp. πη,m ). Write π(χ) for the
principal series IndG
P χ. Another theorem we prove in this paper is the following one.
4
CHUANGXUN CHENG
Theorem 1.9. (1) (unramified principal series) If π = π(χ) for an unramified χ and
χ 6= 1, then c(π) = 0. And cη (π) = c(π) only when η is trivial. We also have
(
1
if m = 0
dim π(χ)m =
2m if m ≥ 1.
If π = Sp is the special representation Sp, then c(Sp) = 1. cη (Sp) = c(Sp) only when η is
trivial. We have the following formula,
(
0
if m = 0
dim Spm =
2m − 1 if m ≥ 1.
(2) (ramified principal series) If π = π(χ) for a ramified χ, then c(π) = c(χ) = 1.
cη (π) = c(π) only for those η such that η = χ± on the group of units O× . In this case, we
have the following formulas.
• If χ 6= χ−1 , then
(
0
if m = 0
dim π(χ)m =
2(m − 1) + 1 if m ≥ 1.
• If χ = χ−1 , then
dim π(χ)m
(
0
=
2m
if m = 0
if m ≥ 1.
If F = Qp , we have the following.
(3) (unramified supersingular representation) If π = π00 or π01 is a supersingular representation, then c(π) = 0, and dim πc(π) = 1.
(4) (ramified supersingular representation) If r 6= 0, π = πr0 or πr1 is a supersingular
representation, then c(π) = 1, and dim πc(π) = 1.
This paper is organized as follows. In Section 2, we prove Proposition 1.5. In Section
3, we prove Proposition 1.7. In Section 4, we prove Theorem 1.9.
2. Principal series
The proofs in this section are almost the same as the proofs in [2] and [3]. We give
the detail proof for the unramified principal series. For ramified principal series and the
special representation, we just sketch the proof by indicating the difference between our
case and the case in [2] and [3].
2.1. Some basic facts. We have various decompositions (see for example [7]):
(1) Iwasawa decomposition G = N AK
(2) Cartan decomposition G = KA− K
(3) Bruhat decomposition G = P ωP ∪ P = P ωN ∪ P
(4) Iwahori decomposition K = IωI ∪ I.
If c 6= 0, then (with ad − bc = 1) we have
−1 a b
0 −1
1 ac−1
c
0
1 dc−1
=
,
c d
1 0
0
1
0 c
0
1
therefore
(5) K = P (O)ωN (O) ∪ I.
MOD p REPRESENTATIONS OF SL2 (Qp )
5
We know that P \G is isomorphic to the projective line P1 (F ). Group G acts by right
translation on P1 (F ). Let ∞ be the point represented by (0 : 1) and 0 the point represented
by (1 : 0) in P1 (F ), then we have the following lemmas.
Lemma 2.1. I1 has two orbits on P1 (F ): the orbits O0 and O∞ of 0 and ∞ respectively.
More precisely, we have
O0 = {(1 : x)|x ∈ O}
O∞ = {($x : 1)|x ∈ O}.
Proof. Since P1 (F ) = {(1 : x)|x ∈ O} ∪ {($x : 1)|x ∈ O}, it suffices to show that O0 and
O∞ are the
sets
described in the statement.
a b
Let g =
∈ I1 , then (1 : 0)g = (a : b) = (1, a−1 b) and a−1 b can be any element of
c d
O. This shows O0 = {(1 : x)|x ∈ O}. Similarly , we can show that O∞ = {($x : 1)|x ∈
O}.
Lemma 2.2. The stabilizer of 0 (resp. ∞) in I1 acts transitively on O∞ (resp. O0 ).
Therefore, I1 acts transitively on O0 × O∞ .
a b
Proof. Since (1 : 0)
= (1 : a−1 b), the stabilizer of 0 in I1 is the subgroup of lower
c d
1 0
triangular matrices in I1 . But ($x : 1) = (0 : 1)
, hence the stabilizer of 0 acts
$x 1
transitively on O
∞.
a b
Similarly, (0 : 1)
= (cd−1 : 1), the stabilizer of ∞ in I1 is the subgroup of upper
c d
triangular matrices in I1 , and it is easy to see that this subgroup acts transitively on
O0 .
Let χ : F × → E × be a character of F × with χ($) = λ−1 . Denote also χ : P → E × the
character defined by
a b
7→ χ(a).
0 a−1
We shall give another model for IndG
P χ using the Bruhat decomposition G = P ωN ∪ P .
This decomposition implies that f ∈ S(G, χ) is determined by its values on Id
andωN .
1 x
Given f ∈ S(G, χ), define φ = j(f ) : F → E by x 7→ f (ωn(x)), where n(x) =
.
0 1
Lemma 2.3. φ is locally constant. Furthermore, if val(x) 0,
φ(x) = const · χ(x−1 ).
Proof. φ is locally constant since f is locally constant. We also have
−1
1 0
x
−1
ωn(x) =
,
0
x
x−1 1
1 0
−1
thus, φ(x) = χ(x )f ( −1
).
x
1
6
CHUANGXUN CHENG
Let G be the set of functions φ : F → E that are locally constant and such that
φ(x) = const·χ(x−1 ) for val(x) 0. Then S(G, χ) and G are isomorphic via f 7→ φ = j(f )
as vector spaces. We obtain a G-module structure on G by transporting the G-module
structure on S(G, χ).
Lemma
2.4. (1) n(y)φ(x) = φ(x + y).
a 0
(2)
φ(x) = χ(a−1 )φ(a−2 x). In particular,
0 a−1
$
0
φ(x) = λφ($−2 x),
0 $−1
−1
$
0
φ(x) = λ−1 φ($2 x).
0
$
Proof. (1) n(y)φ(x) = n(y)f (ωn(x)) = f (ωn(x)n(y)) = f (ωn(x + y)) = φ(x + y).
(2)
a 0
a 0
φ(x) = f (ωn(x)
)
0 a−1
0 a−1
−1
1 0
a 0
x
−1
= f(
)
0
x
x−1 1
0 a−1
a
0
= χ(x−1 )f (
)
ax−1 a−1
1 x
0
−a−1 x
0
−a−1 x
−1
−1
= χ(x )f (
) = χ(x )f (
)
0 1
ax−1
a−1
ax−1
a−1
−1
a x
0
−1
= χ(x )f (
ωn(a−2 x))
0
ax−1
= χ(a−1 )φ(a−2 x),
2.2. Unramified principal series. In this subsection, we consider the case that χ is
unramified, i.e. χ|O× is trivial.
K
∼
Lemma 2.5. (IndG
P χ)|K = IndK∩P (χ|K∩P ) as K-representations.
Proof. The K-isomorphism is given by the restriction of an element in IndG
P χ to K. It is
easy to see that this is surjective. For injectivity, if f1 , f2 ∈ IndG
χ
and
f
|
1 K = f2 |K , then
P
since G = P K, ∀g ∈ G, we may write g = pk with p ∈ P and k ∈ K. Then
f1 (g) = f1 (pk) = χ(p)f1 (k) = χ(p)f2 (k) = f2 (pk) = f2 (g).
Lemma 2.6. If χ1 and χ2 are unramified, then
G
∼
(IndG
P χ1 )|K = (IndP χ2 )|K
as K-representations.
MOD p REPRESENTATIONS OF SL2 (Qp )
7
Proof. By the above lemma, we have
K
K
G
∼
∼
∼
(IndG
P χ1 )|K = IndK∩P (χ1 |K∩P ) = IndK∩P (χ2 |K∩P ) = (IndP χ2 )|K .
Lemma 2.7. For χ unramified, we have
K ∼
(IndG
P χ) = E,
I ∼ 2
(IndG
P χ) = E .
Proof. The first isomorphism follows from Lemma 2.5 and the fact χ is unramified. The
second isomorphism follows from the decomposition K = P (O)ωI ∪ I.
Remark 2.8. One has the function f0 ∈ IndG
P χ,
f0 (g) = 1, ∀g ∈ K
K
as a basis for (IndG
P χ) , and the functions f1 , f2 :
(
(
1 if g ∈ I
0
f1 (g) =
f2 (g) =
0 if g ∈ ωI
1
if g ∈ I
if g ∈ ωI
I
as a basis for (IndG
P χ) .
Now, let f0 be as in the remark, φ0 = j(f0 ) ∈ G, then
(
1
if val(x) ≥ 0
.
φ0 (g) =
val(x)
λ
if val(x) ≤ 0
Write 1R for the characteristic function for a subset R ⊂ F .
P
−1
Lemma 2.9.
a∈O/$O n(a/$)φ0 = (1 − λ )1$−1 O .
P
P
Proof.
a∈O/$O n(a/$)φ0 (x) =
a∈O/$O φ0 (x + a/$).
If val(x) < −1, then val(x + a/$) = val(x) < 0,
X
φ0 (x + a/$) = qλval(x) = 0.
a∈O/$O
If val(x) ≥ 0, then val(x + a/$) ≥ 0 if a = 0, val(x + a/$) = −1 if a 6= 0,
X
φ0 (x + a/$) = (q − 1)λ−1 + 1 = 1 − λ−1 .
a∈O/$O
If val(x) = −1, then there exists a unique a0 such that a0 + $x ≡ 0(mod $),
X
φ0 (x + a/$) = (q − 1)λ−1 + 1 = 1 − λ−1 .
a∈O/$O
Lemma 2.10.
P
a∈O/$2 O
n(a/$2 )φ0 = (1 − λ−1 )1$−2 O .
8
CHUANGXUN CHENG
P
P
2
2
Proof.
a∈O/$2 O n(a/$ )φ0 (x) =
a∈O/$2 O φ(x + a/$ ).
If val(x) < −2, then val(x + a/$2 ) = val(x) < 0,
X
φ0 (x + a/$2 ) = q 2 λval(x) = 0.
a∈O/$2 O
If val(x) ≥ 0, then val(x+a/$2 ) ≥ 0 if a = 0, val(x+a/$2 ) = −1 if $||a, val(x+a/$2 ) =
−2 if $ - a,
X
φ0 (x + a/$2 ) = 1 + (q − 1)λ−1 + (q 2 − q)λ−2 = 1 − λ−1 .
a∈O/$2 O
If val(x) = −2, then x = $−2 x0 for some x0 ∈ O\$O,
X
X
φ0 ($−2 (x0 +a)) = 1+(q −1)λ−1 +(q 2 −q)λ−2 = 1−λ−1 .
φ0 (x+a/$2 ) =
a∈O/$2 O
a∈O/$2 O
If val(x) = −1, write x = $−1 x0 for some x0 ∈ O\$O,
X
X
φ0 (x+a/$2 ) =
φ0 ($−2 (x0 $+a)) = 1+(q−1)λ−1 +(q 2 −q)λ−2 = 1−λ−1 .
a∈O/$2 O
a∈O/$2 O
Remark 2.11. (1) Let C(F, E) be the Schwartz space of locally constant functions on F
with compact support taking values in E, then C(F, E) ⊂ G and the quotient is a free
E-module of rank 1 generated by φ0 .
(2) By the above two lemmas and Lemma 2.4, we see that if λ 6= 1, φ0 generates G as
G-module. Therefore, f0 generates S(G, χ) as G-module.
I
G I
Recall that (IndG
P χ) has dimension 2. Let {f1 , f2 } be a basis of (IndP χ) characterized
by
f1 (Id) = 1, f1 (β) = 0,
f2 (Id) = 0, f2 (β) = 1.
has dimension 1 with basis f0 characterized by f0 (Id) = 1
G χ)K
We also know that
P (Ind
−1
$
0
and f0 (β) = f0 (
ω) = χ($−1 ) = λ, therefore f0 = f1 + λf2 . It is easy to see
0
$
the following lemma.
Lemma 2.12. (1) βf1 = f2 , βf2 = f1 .
(2) αf1 = λ−1 f1 , αf2 = λf2 .
Now we can prove the irreducibility of IndG
P χ when χ 6= 1 is unramified.
Proposition 2.13. If χ 6= 1 is unramified, then IndG
P χ is irreducible as a G-representation.
Proof. Let W ⊂ IndG
P χ be a non-zero subrepresentation. Since I1 is a pro-p group,
W I1 6= 0. Also, I/I1 acts trivially on W I1 , so W I 6= 0. Let f be a non-zero element of
I
W I ⊂ (IndG
P χ) = hf1 , f2 i. Write f = a1 f1 + a2 f2 . By the above lemma, we have
βf = a2 f1 + a1 f2 ∈ W,
αf = a1 λ−1 f1 + a2 λf2 ∈ W,
so f1 , f2 ∈ W . Therefore f0 ∈ W , W = IndG
P χ by Remark 2.11(2).
MOD p REPRESENTATIONS OF SL2 (Qp )
9
We still have to investigate the case χ = 1. It is obvious that IndG
P 1 has a 1dimensional subrepresentation consists constant functions. Define the special series Sp
∼ 1
as IndG
P 1/hconsti. Use the isomorphism P \G = P (F ), we give another model for Sp. Let
∼
C(P) be the space of locally constant functions on P1 (F ). Then IndG
P 1 = C(P), we may
realize Sp on C(P)/hφ0 i. Now we use the idea in [2] to prove the irreducibility of Sp.
Lemma 2.14. The following sequence of I1 -modules is exact:
0 → Eφ0 → (C(P))I1 → (C(P)/Eφ0 )I1 → 0,
where the action on Eφ0 is trivial.
Proof. From
0 → Eφ0 → C(P) → C(P)/Eφ0 → 0,
we have exact sequence
0 → Eφ0 → (C(P))I1 → (C(P)/Eφ0 )I1 .
It suffices to prove the last map is surjective. Let ψ ∈ (C(P)/Eφ0 )I1 and let φ be a pullback
of ψ in C(P). We need to show that φ is also I1 -invariant. For any i ∈ I1 , i · φ − φ gose to
zero in (C(P)/Eφ0 )I1 , there exists a constant ci ∈ E such that i · φ − φ = ci . Let (1 : x)
be any point in O0 , by Lemma 2.2, there exists an element i in the stabilizer of ∞ in I1
such that 0 · i = (1 : x), then we have
i · φ(∞) − φ(∞) = φ(∞ · i) − φ(∞) = 0
so that ci = 0. Thus
φ((1 : x)) − φ(0) = iφ(0) − φ(0) = 0,
φ is constant on O0 . Similarly, φ is constant on O∞ . By Lemma 2.1, φ is I1 -invariant. Proposition 2.15. (1) The representation Sp is irreducible.
(2) Sp is the only quotient of IndG
P 1.
Proof. The proof is exactly the same as proof of Theorem 28, 29 of [2].
2.3. Ramified principal series. We sketch the proof of the irreducibility of IndG
P χ for
ramified χ. The proof is almost the same as in the case χ is unramified. The difference is
that we do not have f0 and φ0 in this case.
I1
Lemma 2.16. dim(IndG
P χ) = 2.
Proof. This follows readily from the decomposition G = P I1 ∪ P βI1 . This decomposition
follows from Lemma 2.1.
I1 characterized by
Remark 2.17. We can choose a basis {f1 , f2 } of (IndG
P χ)
f1 (Id) = 1, f1 (β) = 0,
f2 (Id) = 0, f2 (β) = 1.
As before, we have βf1 = f2 , βf2 = f1 , αf1 = λ−1 f1 , αf2 = λf2 .
10
CHUANGXUN CHENG
Lemma 2.18. The images of f1 and f2 in the space G are given by
(
0
if val(x) ≥ 0
φ1 (x) = j(f1 )(x) =
−1
χ(x ) if val(x) < 0,
φ2 (x) = j(f2 )(x) = 1O .
Proof. It is an easy computation from the definition.
P
$
a
Lemma 2.19.
1O = λ1$O .
a∈O/$O
0 $−1
Proof.
(
X
a∈O/$O
$
a
1O )(x) =
0 $−1
X
λ1O ($−2 x + a/$)
a∈O/$O
=λ
X
1$2 O (x + a$).
a∈O/$O
P
If val(x) ≤ 0, then val(x + a$) ≤ 0, a∈O/$O 1$2 O (x + a$) = 0;
(
1
a 6= 0 P
1$2 O (x + a$) = 1;
If val(x) ≥ 2, then val(x + a$) =
val(x) a = 0, a∈O/$O
If val(x) = 1, then there existsPa unique a0 ∈ O/$O, such that val(x + a0 $) = 2,
val(x + a$) = 1 for other a, so a∈O/$O 1$2 O (x + a$) = 1
Proposition 2.20. If χ is ramified, IndG
P χ is irreducible.
Proof. First we observe that f1 and f2 generates S(G, χ) by the above computation. Let
I1 6= 0. Also, the
W ∈ IndG
P χ be a non-zero subrepresentation. Since I1 is a pro-p group, W
I1
I
I
I-action can be diagonalized on W 1 . Let f be a non-zero element of W 1 ⊂ (IndG
P χ) =
hf1 , f2 i. Write f = a1 f1 + a2 f2 . Since I acts by different characters on f1 and f2 , at least
one of f1 and f2 is contained in W . Furthermore, β normalizes I1 and interchanges f1 and
f2 . Therefore f1 , f2 ∈ W and W = IndG
P χ.
3. Supersingular representations
We first recall the classifications of irreducible representations for GL2 (κ) and SL2 (κ).
Fix an embedding κ → E.
For any r ∈ Z≥0 , consider the (r + 1)-dimensional representation of the finite group
GL2 (κ): Symr E 2 where GL2 (κ) acts through its natural action on the canonical basis of
2
E
we use the fixed embedding κ → E. This representation can be identified with
Lr. Here,r−i
y i with the action of GL2 (κ) given by
i=0 Ex
a b
xr−i y i = (ax + cy)r−i (bx + dy)i
c d
a b
where
∈ GL2 (κ), a, b, c, d are seen in E via the fixed embedding.
c d
MOD p REPRESENTATIONS OF SL2 (Qp )
11
j
If 0 ≤ j ≤ f − 1, we denote by (Symr E 2 )F r the (r + 1)-dimensional representation
with the action of GL2 (κ) given by
j
j
j
j
a b
xr−i y i = (ap x + cp y)r−i (bp x + dp y)i .
c d
f
Note that (Symr E 2 )F r = Symr E 2 .
Proposition 3.1. Let r0 , r1 , ..., rf −1 and m be integers such that 0 ≤ ri ≤ p − 1 and
0 ≤ m < q − 1, then the representations
(Symr0 E 2 ) ⊗E (Symr1 E 2 )F r ⊗E ... ⊗E (Symrf −1 E 2 )F r
f −1
⊗E detm
are irreducible and non-equivalent. These are all the irreducible representations of GL2 (κ).
Proof. See for example Section 2.2 of [5].
By restricting to SL2 (κ), we get the following proposition.
Proposition 3.2. Let r0 , r1 , ..., rf −1 be integers such that 0 ≤ ri ≤ p − 1, then the representations
f −1
(Symr0 E 2 ) ⊗E (Symr1 E 2 )F r ⊗E ... ⊗E (Symrf −1 E 2 )F r
are irreducible and non-equivalent. These are all the irreducible representations of SL2 (κ).
Proof. The proof is almost the same as the proof of the above proposition. The only thing
we need to check is that SL2 (κ)reg has q conjugacy classes, where SL2 (κ)reg is the subset
of SL2 (κ) of elements of order prime to p.
3.1. mod p representations of GL2 (Qp ). All results in this subsection are copied from
Section 4 of [4]. From now on in this section, we assume that F = Qp , although some of
the results in this section are not specific to this case. Let Z denote the center of GL2 (Qp ),
let r be an integer with 0 ≤ r ≤ p − 1 and Symr E 2 be the representation of GL2 (Zp ) (via
the natural projection GL2 (Zp ) → GL2 (Fp )) and extend it to ZGL2 (Zp ) by letting p act
trivially. For convenience, we write σr = Symr E 2 . Denote
GL (Q )
c-IndGL22 (Zpp)Z σr
the E-vector space of functions f : GL2 (Qp ) → Symr E 2 with compact support modulo
Z and f (kg) = σr (k)f (g) (for k ∈ GL2 (Zp )Z and g ∈ GL2 (Qp )). It is a GL2 (Qp )
representation with right regular action.
GL (Q )
Lemma 3.3. EndGL2 (Qp ) (c-IndGL22 (Zpp)Z σr ) ' E[T ].
This lemma is a special case of Proposition 8 of [3]. See Section 3.1 of [3] for the
definition of the operator T and more details.
Now we can state the following theorem from [2], [3] and [4].
Theorem 3.4. The smooth irreducible admissible representations of GL2 (Qp ) over E are
the following:
(1) the one dimensional representations χ ◦ det
(2) the representations
GL (Q )
(c-IndGL22 (Zpp)Z σr /(T − λ)) ⊗ (χ ◦ det)
12
CHUANGXUN CHENG
for 0 ≤ r ≤ p − 1, λ ∈ E × and (r, λ) 6∈ {(0, ±1), (p − 1, ±1)}
(3) the representations
GL (Q )
Ker(c-IndGL22 (Zpp)Z 1/(T − 1) 1) ⊗ (χ ◦ det)
(4) the representations
GL (Q )
(c-IndGL22 (Zpp)Z σr /T ) ⊗ (χ ◦ det).
We denote
GL (Q )
π(r, λ, χ) := (c-IndGL22 (Zpp)Z σr /(T − λ)) ⊗ (χ ◦ det)
×
for any r, λ, χ. If x ∈ E × , we denote unr(x) : Q×
p → E the unramified character sending
p to x.
Remark 3.5. (a) The representations in (2) are actually isomorphic to principal series.
More precisely, if (r, λ) is as in (2), then
GL (Qp )
unr(λ)
p)
π(r, λ, 1) ' IndP̃ (Q2
⊗ ω1r unr(λ−1 ).
This is also true for finite extensions of Qp .
(b) The representation Ker(π(0, 1, 1) → 1) in (3) is called the Steinberg representation.
GL (Q )
It is isomorphic to (IndP̃ (Q2 ) p 1)/hconsti. This representation with its twists constitute
p
the so called special series.
(c) The representations in (4) are the supersigular ones.
(d) There are intertwinings between the above representations:
π(r, λ, χ) ' π(r, −λ, χunr(−1))
π(0, λ, χ) ' π(p − 1, , λ, χ) (λ 6= ±1)
π(r, 0, χ) ' π(p − 1 − r, 0, χω1r ).
×
Denote by ωn the fundamental character I(Q̄p /Qp ) F×
pn ,→ E , where I(Q̄p /Qp ) ⊂
r+1
Gal(Q̄p /Qp ) is the inertia group. Denote by ind(ω2 ) the unique irreducible representation of Gal(Q̄p /Qp ) over E such that its restriction to the inertia group I(Q̄p /Qp ) is
p(r+1)
and its determinant is ω1r+1 . Also let unr(x) be the unramified characω2r+1 ⊕ ω2
ter Gal(Q̄p /Qp ) Gal(F̄p /Fp ) → E × sending F rob−1 ∈ Gal(F̄p /Fp ) to x ∈ E × . Here
F rob is given by (x 7→ xp ). Now we can give Breuil’s semi-simple modulo p Langlands
correspondence for GL2 (Qp ).
×
Definition 3.6. Let r ∈ {0, ..., p − 1}, λ ∈ E, χ : Q×
p → E and [p − 3 − r] the unique
integer in {0, ..., p − 2} congruences to p − 3 − r modulo p − 1. With the above notation,
we give the following semi-simple modulo p correspondence:
(1) if λ = 0,
(ind(ω2r+1 )) ⊗ χ ↔ π(r, 0, χ)
(2) if λ 6= 0,
r+1
ω1 unr(λ)
0
⊗ χ ↔ π(r, λ, 1)ss ⊗ χ ⊕ (π([p − 3 − r], λ−1 , 1) ⊗ ω r+1 )ss ⊗ χ
0
unr(λ−1 )
where ss means semi-simplification.
MOD p REPRESENTATIONS OF SL2 (Qp )
13
Remark 3.7. This correspondence can be refined into a non-semi-simple correspondence.
See [4] for more details.
3.2. Supersingular representations. In this subsection, we prove Proposition 1.7.
First, recall that
GL (Q )
c-IndGL22 (Zpp)Z σr
is the E-vector space of functions f : GL2 (Qp ) → Symr E 2 with compact support modulo
Z and f (kg) = σr (k)f (g) (for k ∈ GL2 (Zp )Z and g ∈ GL2 (Qp )). G acts on this space
as (gf )(g 0 ) = f (g 0 g). As in [3], for g ∈ GL2 (Qp ) and v ∈ Symr E 2 , we denote [g, v] the
GL (Q )
element in c-IndGL22 (Zpp)Z σr defined by:
[g, v](g 0 ) = σr (g 0 g)v if g 0 ∈ GL2 (Zp )Zg −1
[g, v](g 0 ) = 0 if g 0 6∈ GL2 (Zp )Zg −1 .
GL (Q )
Then we have g[g 0 , v] = [gg 0 , v], and [gk, v] = [g, σr (k)v]. Every element in c-IndGL22 (Zpp)Z σr
1 0
is a finite sum Σi [gi , vi ]. In the following, write ϑ =
.
0 p
Lemma 3.8. Suppose 0 ≤ r ≤ p − 1. Denote [Id, xr ]− and [ϑ, y r ]− the images of [Id, xr ]
GL (Q )
and [ϑ, y r ] in (c-IndGL22 (Zpp)Z σr )/T . Then
˜
GL (Q )
((c-IndGL22 (Zpp)Z σr )/T )I1 = E[Id, xr ]− ⊕ E[ϑ, y r ]− ,
GL (Q )
((c-IndGL22 (Zpp)Z σr )/T )I1 = E[Id, xr ]− ⊕ E[ϑ, y r ]− .
Proof. The first equality is from Theorem 3.2.4 and Corollary 4.1.4 of [3]. We prove the
GL (Q )
second equality. For simplicity, we write V = (c-IndGL22 (Zpp)Z σr )/T . Note that we have a
short exact sequence
1 → I1 → I˜1 → 1 + pZp → 1,
I˜1 /I1 ' 1 + pZp is abelian and pro-p. Consider the action of I˜1 /I1 on the space V I1 . First,
it is an extension of 1 dimensional representations since I˜1 /I1 is abelian. Second, I˜1 /I1
has an invariant vector in each 1 dimensional representation since I˜1 /I1 is a pro-p group.
Therefore, I˜1 /I1 acts trivially on V I1 ,
˜
V I1 = V I1 = E[Id, xr ]− ⊕ E[ϑ, y r ]− .
GL (Q )
Lemma 3.9. c-IndGL22 (Zpp)Z σr = E[SL2 (Qp )][Id, xr ] ⊕ E[SL2 (Qp )][ϑ, y r ].
GL (Q )
Proof. Let [g, v] ∈ c-IndGL22 (Zpp)Z σr . If ordp (det(g)) ≡ 1 (mod 2), then g ∈ ϑSL2 (Qp )Z,
[g, v] ∈ E[SL2 (Qp )][ϑ, y r ] since σr is irreducible as SL2 (Fp ) representation. Similarly, if
ordp (det(g)) ≡ 0 (mod 2), then [g, v] ∈ E[SL2 (Qp )][Id, xr ].
It is easy to see that E[SL2 (Qp )][Id, xr ] = {[g, v]|ordp (det(g)) is even} and E[SL2 (Qp )][ϑ, y r ] =
{[g, v]|ordp (det(g)) is odd}. So E[SL2 (Qp )][Id, xr ] ∩ E[SL2 (Qp )][ϑ, y r ] = 0, the lemma follows.
14
CHUANGXUN CHENG
We write Vr0 = E[SL2 (Qp )][Id, xr ] and Vr1 = E[SL2 (Qp )][ϑ, y r ], then from the definition
of Hecke action, (see for example [4] Section 2.5), T (Vr0 ) ⊂ Vr1 and T (Vr1 ) ⊂ Vr0 .
Lemma 3.10. [ϑ, y r ]− 6∈ V̄r0 = E[SL2 (Qp )][Id, xr ]− , [Id, xr ]− 6∈ V̄r1 = E[SL2 (Qp )][ϑ, y r ]− .
Proof. We prove the first statement. Assume that [ϑ, y r ]− ∈ V̄0 = E[SL2 (Qp )][Id, xr ]− ,
then [ϑ, y r ] ∈ Vr0 +T (Vr0 +Vr1 ). We may write [ϑ, y r ] = ζ0 +T (η0 +η1 ) with ζ0 , η0 ∈ Vr0 and
η1 ∈ Vr1 , then we get [ϑ, y r ] = (ζ0 + T (η1 )) + T (η0 ). By the above lemma, [ϑ, y r ] = T (η0 ),
GL (Q )
[ϑ, y r ]− = 0 in (c-IndGL22 (Zpp)Z σr )/T . This is a contradiction.
Now we can prove Proposition 1.7.
Proof. By Corollary 4.1.3 of [4],
GL (Q )
GL (Q )
(c-IndGL22 (Zpp)Z σr )/T ' (c-IndGL22 (Zpp)Z σp−1−r )/T ⊗ (ω r ◦ det),
we may assume that r ≤ p − 2. By Lemma 3.10, π̃r |SL2 (Qp ) is reducible.
Assume that (ρ, W ) is a non-zero and proper subrepresentation of π̃r |SL2 (Qp ) . Since I1
is a pro-p group, W I1 6= 0. By the above computation, there exists (a, b) ∈ E 2 \(0, 0) such
that a[Id, xr ]− + b[ϑ, y r ]− ∈ W I1 ⊂ W . If a = 0 (resp. b = 0), then W = V̄r1 (resp.
W = V̄r0 ) since [ϑ, y r ]− (resp. [Id, xr ]− ) generates W = V̄r1 (resp. W = V̄r0 ) (this is from
the proof of Lemma 3.10). Next we show that the case ab 6= 0 cannot happen. Assume
that a 6= 0 and b 6= 0. Without loss of generality, we may assume that a = 1 and b 6= 0.
µr 6= 1. Let µ̃ be a lift of µ in Qp . We have
If r 6= 0, then there exists µ ∈ F×
p such that −1
µ̃
0
µ̃−1 0
r
−
r
−
([Id, x ] + b[ϑ, y ] ) ∈ W . Since
(b[ϑ, y r ]− ) = µr b[ϑ, y r ]− , we see
0 µ̃
0 µ̃
that [ϑ, y r ]− ∈ W . Therefore W = π̃r , itis a contradiction.
p p−1 λ
p λ
If r = 0, then xr = y r = 1. Let gλ =
∈
SL
(Q
),
then
g
ϑ
=
. We
2
p
λ
0 1
0 p−1
have
X
[Id, 1]− + b[ϑ, 1]− +
gλ ([Id, 1]− + b[ϑ, 1]− ) ∈ W.
λ∈Zp /pZp
On the other hand,
[Id, 1]− + b[ϑ, 1]− +
X
gλ ([Id, 1]− + b[ϑ, 1]− )
λ∈Zp /pZp
−
−
= (p + 1)[Id, 1] + b([ϑ, 1] +
X
λ∈Zp /pZp
−
p λ
[
, 1]− )
0 1
−
= [Id, 1] + bT [Id, 1] = [Id, 1]− ∈ W,
where the second equality follows from the computation in section 2 of [4]. Then W = π̃r ,
which is a contradiction.
Assume that we have an isomorphism ξr : V̄r0 ' V̄r1 . Since (V̄r0 )I1 = E[Id, xr ]− and
(V̄r1 )I1 = E[ϑ, y r ]− , we may assume
that ξr ([Id, xr ]− ) = [ϑ, y r ]− . If r 6= 0, (p − 1)/2, we
a 0
consider the action of
for a ∈ O× . It acts as ar on [Id, xr ]− and as a−r on
0 a−1
[ϑ, y r ]− . So V̄r0 and V̄r1 are not isomorphic. If r = 0, then V̄r0 has an element [Id, 1]−
MOD p REPRESENTATIONS OF SL2 (Qp )
15
fixed by K, (V̄r1 )K = {0}, so V̄r0 and V̄r1 are not isomorphic. If r = (p − 1)/2, then the
isomorphism in Corollary 4.1.3 of [4] gives an isomorphism between V̄r0 and V̄r1 .
Remark 3.11. (1) Note that we have isomorphism
GL (Q )
GL (Q )
p
2
r
˜
ψr : (c-IndGL22 (Zpp)Z σr )/T →(c-Ind
GL2 (Zp )Z σp−1−r )/T ⊗ (ω ◦ det)
1
with ψr ([Id, xr ]− ) = [ϑ, y p−1−r ]− . Restricting to SL2 (Qp ), we have Vr0 ' Vp−1−r
.
(2) In [8], the author gives another method to prove the irreducibility of the GL2 (Qp )
GL (Q )
representation c-IndGL22 (Zpp)Z σp−1−r )/T . We can also deduce Proposition 1.7 using the
method in [8].
3.3. Mod p correspondence for SL2 (Qp ). We have proved Theorem 1.1. We can define
a mod p correspondence for SL2 (Qp ).
Let i : SL2 → GL2 be the inclusion map, pr : GL2 → P GL2 be the projection map. A
mod p Galois representation
ρ : GQp → GL2 (E)
induces by composition the projective Galois representation
pr ◦ ρ : GQp → P GL2 (E).
From Definition 3.6, the above map gives us a correspondence from representations of
GL2 (Qp ) to representations of SL2 (Qp ). Examining matters, we see that we associate to
each representation of GL2 (Qp ) all of its constituents on restriction to SL2 (Qp ). Nothing
very deep is happening here. As we have seen, every non-supersingular irreducible representation of GL2 (Qp ) stays irreducible after restricting to SL2 (Qp ). For every supersingular representation of GL2 (Qp ), we get a packet which has two irreducible representations
of SL2 (Qp ).
0
1
1
, Vp−1−r
}
, then two packets {Vr0 , Vr1 } and {Vp−1−r
Remark 3.12. By Remark 3.11, Vr0 ' Vp−1−r
p−1−r
r
) give the same
are the same. This corresponds to the fact that ind(ω2 ) and ind(ω2
projective Galois representation. (See [4] Lemma 4.2.2).
If r = (p − 1)/2, then Vr0 ' Vr1 , we have a packet contains a representation with
multiplicity two.
4. mod p conductors and newforms
In this section, we develop a theory of mod p newforms for GL2 (F ) and SL2 (F ). For
complex representations, this has been done by Casselman for GL2 (F ) and by Lansky and
Raghuram for SL2 (F ).
In [6], Casselman proved the following theorem.
Theorem 4.1 (Casselman). Let (π, V ) be an irreducible admissible infinite-dimensional
representation of GL2 (F ) over C. Let be the central character of π. Let
a b
Γm = {
∈ GL2 (O)| c ≡ 0( mod pm )},
c d
a b
∈ Γm }.
Vm = {v ∈ V | π(g)v = (a)v, ∀g =
c d
16
CHUANGXUN CHENG
Then we have the following.
(1) There exists a nonnegative integer m such that Vm 6= (0).
(2) Let c(π) be the least nonnegative integer m with the property that Vm 6= (0), (we call
c(π) the conductor of π.) then for any m ≥ c(π), dim Vm = m + 1 − c(π).
The assertion dim Vc(π) = 1 is sometimes referred to as multiplicity one for newforms,
and the unique vector in Vc(π) is called the newform of π. The extra dimensions of Vm
when m > c(π) are coming from old forms.
In [11], Lansky and Raghuram developed a theory dealing with representations of
SL2 (F ) over C. Basically, they examined the representations of SL2 (F ) case by case
and computed dim Vm . One interesting phenomenon is that the naive multiplicity one
results for newforms are not true. See [11] for details.
Now we understand mod p principal series of GL2 (F ) and SL2 (F ). When F = Qp ,
we also understand the supersingular representations of GL2 (Qp ) and SL2 (Qp ). We can
compute the conductors and newforms for these representations.
4.1. Case GL2 (F ). Let
a b
Γm = {
∈ GL2 (O)| c ≡ 0( mod pm )}.
c d
Let (π̃, V ) be an irreducible representation of GL2 (F ) over E with central character .
Define
a b
Vm = {v ∈ V | π̃(g)v = (a)v, ∀g =
∈ Γm }.
c d
If there exists a nonnegative integer m such that Vm 6= (0), we define c(π̃) to be the least
nonnegative integer m with the property that Vm 6= (0). Note that if (π̃, V ) is admissible,
then by definition, every vector is GL2 (OF )-finite and the set of vectors fixed by any open
subgroup of GL2 (OF ) has finite dimension. Thus c(π̃) exists.
We have the following theorem for mod p representations of GL2 (F ).
Theorem 4.2. (1) If π̃ is a principal series (resp. a special representation), then c(π̃) ∈
{0, 1, 2} (resp. c(π̃) ∈ {0, 1}), and for any m ≥ c(π̃), dim Vm = m + 1 − c(π̃).
(2) If F = Qp , and π̃ = π(r, 0, 1) is supersingular, then c(π̃) ∈ {0, 1}, and dim Vc(π̃) = 1.
Proof. If π̃ = IndG̃
χ ⊗χ2 (χ1 6= χ2 ) is a principal series (resp. π̃ = IndG̃
χ⊗χ/hconsti is a
P̃ 1
P̃
special representation), the computation is exactly the same as Casselman’s computation
in Section 1 of [6]. In particular, we have c(π̃) = (conductor of χ1 ) + (conductor of χ2 )
(resp. c(π̃) = (conductor of χ)). Note that the mod p characters of F × are tamely
ramified. Therefore c(π̃) ∈ {0, 1, 2} (resp. c(π̃) ∈ {0, 1}).
If F = Qp , π̃ = π(0, 0, 1) is supersingular, then it has trivial central character. We see
that [Id, 1]− is the unique vector fixed by K and c(π̃) = 0.
˜
If F = Qp , π̃ = π(r, 0, 1) and r 6= 0, we see that I˜1 acts trivially on V1 and V1 ⊂ V I1 . By
˜
Lemma 3.8, V I1 is two dimensional with basis [Id, xr ]− and [ϑ, y r ]− . An easy computation
shows that V1 = h[Id, xr ]− i and c(π̃) = 1.
MOD p REPRESENTATIONS OF SL2 (Qp )
17
4.2. Case SL2 (F ). Let G = SL2 (F ), we use the definition from Section 1. The following
easy but useful lemma is proved in Section 3.2 of [11].
Lemma 4.3. Let m ≥ 1. A complete set of representatives
the double
for cosetspace
1 0
1 0
Km \K/P (O) is given by {1, w, xi , yi }1≤i≤m−1 , where xi =
, yi =
and
$i 1
ι$i 1
ι is a fixed element in O× \(O× )2 .
Let χ be a character of F × . Let π(χ) = IndG
P χ be the principal series. Let η be a
×
character of O with η(−1) = χ(−1). Let m ≥ c(η). Then we have
π(χ)η,m =HomKm (η, π(χ))
(4.1)
=HomP (O) (η, χ) ⊕ Homw−1 Km w∩P (O) (η w , χ)⊕
m−1
M
Homx−1 Km xi ∩P (O) (η xi , χ) ⊕
m−1
M
i
i=1
Homy−1 Km yi ∩P (O) (η yi , χ)
i
i=1
Here η g is defined by η g (g −1 xg) = η(x). If m = 0, only the first term appears. If m = 1,
only the first two terms appear. By analyzing this equation, we can prove Theorem 1.9.
Proof of Theorem 1.9. (1) Since G = P K, we have
(
1
|Km \G/P | = |Km \K/P (O)| =
2m
m=0
m > 1.
The statement follows from equation (4.1).
(2) We use equation (4.1). Let η be a character of O× with η(−1) = χ(−1). Let
m ≥ c(η). The space HomP (O) (η, χ) is nonzero if and only if η = χ as characters of O× .
The space Homw−1 Km w∩P (O) (η w , χ) is nonzero if and only if η = χ−1 as characters of O× .
The space Homx−1 Km xi ∩P (O) (η xi , χ) is nonzero if and only if η = χ on 1 + $min{1,m−i} O
i
and m − i ≥ c(η). The space Homy−1 Km yi ∩P (O) (η yi , χ) is nonzero if and only if η = χ on
i
1 + $min{1,m−i} O and m − i ≥ c(η). Then counting the nonzero Hom0 s in (4.1) finishes
the computation.
0
(3) If π = E[SL2 (Qp )][Id, 1]− , then π K = h[Id, 1]− i and π K = ∅. Similarly, if π =
0
E[SL2 (Qp )][ϑ, 1]− , then π K = h[ϑ, 1]− i and π K = ∅.
(4) If π = E[SL2 (Qp )][Id, xr ]− , then π1 = h[Id, xr ]− i and π10 = ∅. Similarly, if π =
E[SL2 (Qp )][ϑ, y r ]− , then π10 = h[ϑ, y r ]− i and π1 = ∅.
References
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arxiv.org/abs/1010.1654
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Duke Math. J. 75 (1994), 261-292
[3] Barthel, L, Livne, R. Irreducible modular representations of GL2 of a local field. Duke Math. J. 75
(1994), 261-292
[4] Breuil, Christophe. Sur quelques représentations modulaires et p-adiques de GL2 (Qp ) I. Compositio
Math. 138, 2003, 165-188.
[5] Breuil, Christophe. Representations of Galois and of GL2 in characteristic p. Columbia Notes, available at http://www.ihes.fr/~breuil/publications.html
[6] Casselman, W On some results of Atkin and Lehner. Math. Ann. 201, 301-314, 1973
18
CHUANGXUN CHENG
[7] Casselman, W Introduction to the theory of admissible representations of p-adic reductive groups.
Unpublished notes distributed by P. Sally.
[8] Emerton, Matthew On a class of coherent rings, with applications to the smooth representations theory
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