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What is the Langlands Programme?
Shaun Stevens
University of East Anglia
6th March 2012
Number Theory
Finding non-trivial solutions of polynomial equations over Z
or Q, or even their existence, is hard:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
• x5 − 2x + 3 = 0;
• 3x2 + 4y 2 − 5z 2 = 0;
• 3x3 + 4y 3 + 5z 3 = 0;
• y 2 = x3 − 4x + 4.
Number Theory
Finding non-trivial solutions of polynomial equations over Z
or Q, or even their existence, is hard:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
• x5 − 2x + 3 = 0;
• 3x2 + 4y 2 − 5z 2 = 0;
• 3x3 + 4y 3 + 5z 3 = 0;
• y 2 = x3 − 4x + 4.
Other groups
There are simple necessary conditions for the existence of
solutions in Z:
• the existence of a real solution;
Number Theory
Finding non-trivial solutions of polynomial equations over Z
or Q, or even their existence, is hard:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
• x5 − 2x + 3 = 0;
• 3x2 + 4y 2 − 5z 2 = 0;
• 3x3 + 4y 3 + 5z 3 = 0;
• y 2 = x3 − 4x + 4.
Other groups
There are simple necessary conditions for the existence of
solutions in Z:
• the existence of a real solution;
• the existence of a solution modulo n, for all n ∈ N
⇐⇒ the existence of a solution modulo pr , for all p, r
Number Theory
Finding non-trivial solutions of polynomial equations over Z
or Q, or even their existence, is hard:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
• x5 − 2x + 3 = 0;
• 3x2 + 4y 2 − 5z 2 = 0;
• 3x3 + 4y 3 + 5z 3 = 0;
• y 2 = x3 − 4x + 4.
Other groups
There are simple necessary conditions for the existence of
solutions in Z:
• the existence of a real solution;
• the existence of a solution modulo n, for all n ∈ N
⇐⇒ the existence of a solution modulo pr , for all p, r
⇐⇒ the existence of a solution in Zp := lim Z/pr Z.
←−
Local fields: the p-adic numbers Qp
Motivation
Local Class
Field Theory
As well as the usual absolute value, Q has a p-adic absolute
value for each prime p:
a
n for ab coprime to p.
p = p−n ,
b p
Representations
Local
Langlands
Other groups
Thus pn → 0 as n → ∞.
Local fields: the p-adic numbers Qp
Motivation
Local Class
Field Theory
As well as the usual absolute value, Q has a p-adic absolute
value for each prime p:
a
n for ab coprime to p.
p = p−n ,
b p
Representations
Local
Langlands
Other groups
Thus pn → 0 as n → ∞. This valuation is non-archimedean:
|x + y|p ≤ max {|x|p , |y|p } .
Local fields: the p-adic numbers Qp
Motivation
Local Class
Field Theory
As well as the usual absolute value, Q has a p-adic absolute
value for each prime p:
a
n for ab coprime to p.
p = p−n ,
b p
Representations
Local
Langlands
Other groups
Thus pn → 0 as n → ∞. This valuation is non-archimedean:
|x + y|p ≤ max {|x|p , |y|p } .
Qp is the completion of Q with respect to | · |p .
∪
Zp = {x ∈ Qp : |x|p ≤ 1}, the ring of p-adic integers.
∪
pZp = {x ∈ Qp : |x|p < 1}, the unique maximal ideal.
There is only one prime in Zp .
Local fields: the p-adic numbers Qp
Some properties of Qp :
• Q is dense in Qp and Z is dense in Zp .
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
Local fields: the p-adic numbers Qp
Some properties of Qp :
• Q is dense in Qp and Z is dense in Zp .
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
• Every non-zero α ∈ Qp can be uniquely written
α =
X
an pn ,
n≥n0
with an ∈ {0, . . . , p − 1} and an0 6= 0; moreover
|α|p = p−n0 .
Local fields: the p-adic numbers Qp
Some properties of Qp :
• Q is dense in Qp and Z is dense in Zp .
Motivation
Local Class
Field Theory
• Every non-zero α ∈ Qp can be uniquely written
α =
Representations
Local
Langlands
Other groups
X
an pn ,
n≥n0
with an ∈ {0, . . . , p − 1} and an0 6= 0; moreover
|α|p = p−n0 .
• Every non-zero α ∈ Qp can be uniquely written
α = pn0 u,
with u ∈ Zp a unit.
Local fields: the p-adic numbers Qp
Some properties of Qp :
• Q is dense in Qp and Z is dense in Zp .
Motivation
Local Class
Field Theory
• Every non-zero α ∈ Qp can be uniquely written
α =
Representations
Local
Langlands
Other groups
X
an pn ,
n≥n0
with an ∈ {0, . . . , p − 1} and an0 6= 0; moreover
|α|p = p−n0 .
• Every non-zero α ∈ Qp can be uniquely written
α = pn0 u,
with u ∈ Zp a unit.
• Every ball B in Qp is both open and closed; every point of
B is the centre of the ball!
Local-Global (Hasse) Principle
The Qp and R = Q∞ are the only completions of Q so:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
The existence of solutions in Qp , for all p ≤ ∞, to a
rational polynomial equation should say something
about the existence of solutions in Q.
Local-Global (Hasse) Principle
The Qp and R = Q∞ are the only completions of Q so:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
The existence of solutions in Qp , for all p ≤ ∞, to a
rational polynomial equation should say something
about the existence of solutions in Q.
• The existence of local (p-adic or real) solutions is much
easier to determine because we can use analytic
techniques.
Local-Global (Hasse) Principle
The Qp and R = Q∞ are the only completions of Q so:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
The existence of solutions in Qp , for all p ≤ ∞, to a
rational polynomial equation should say something
about the existence of solutions in Q.
• The existence of local (p-adic or real) solutions is much
easier to determine because we can use analytic
techniques.
• The existence of p-adic solutions, for all p, to a rational
quadratic form does imply the existence of a rational
solution (Hasse–Minkowski).
Local-Global (Hasse) Principle
The Qp and R = Q∞ are the only completions of Q so:
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
The existence of solutions in Qp , for all p ≤ ∞, to a
rational polynomial equation should say something
about the existence of solutions in Q.
• The existence of local (p-adic or real) solutions is much
easier to determine because we can use analytic
techniques.
• The existence of p-adic solutions, for all p, to a rational
quadratic form does imply the existence of a rational
solution (Hasse–Minkowski).
• 3x3 + 4y 3 + 5z 3 = 0 has p-adic solutions, for all p, but no
rational solution (Selmer).
Galois Theory
If there are no solutions, where do solutions exist?
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
Galois Theory
If there are no solutions, where do solutions exist?
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
Example: x2 + 1 = 0 has no solutions in R, but two solutions
x = ±i in C.
Nothing distinguishes i from −i.
Galois Theory
If there are no solutions, where do solutions exist?
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Example: x2 + 1 = 0 has no solutions in R, but two solutions
x = ±i in C.
Nothing distinguishes i from −i.
Other groups


 field isomorphisms 
f : C → C such that
Gal(C/R) =
f (x) = x for all x ∈ R 
= {1, c}.
Galois Theory
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
x5 − 2x + 3 = 0 has no solutions in Q, but five solutions
α1 , . . . , α5 in C, but
• we cannot write them down;
• they are indistinguishable if starting from Q.
Galois Theory
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
x5 − 2x + 3 = 0 has no solutions in Q, but five solutions
α1 , . . . , α5 in C, but
• we cannot write them down;
• they are indistinguishable if starting from Q.
Put L = Q(α1 , . . . , α5 ), a subfield of C.
• L is a Q-vector space, of finite dimension over Q.


 field isomorphisms 
f : L → L such that
Gal(L/Q) =
f (x) = x for all x ∈ Q 
' S5 .
Galois Theory
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
x5 − 2x + 3 = 0 has no solutions in Q, but five solutions
α1 , . . . , α5 in C, but
• we cannot write them down;
• they are indistinguishable if starting from Q.
Put L = Q(α1 , . . . , α5 ), a subfield of C.
• L is a Q-vector space, of finite dimension over Q.


 field isomorphisms 
f : L → L such that
Gal(L/Q) =
f (x) = x for all x ∈ Q 
' S5 .
Moreover, Gal(L/Q) acts as Q-linear maps on L; we get a
(linear) representation of Gal(L/Q) which is the regular
representation of S5 .
Absolute Galois Theory
Put all such fields together:
Q = algebraic closure of Q (in C)
α ∈ C such that α is a root
=
of some fα (X) ∈ Q[X]
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
Then


 field isomorphisms 
GQ := Gal(Q/Q) = f : Q → Q such that
f (x) = x for all x ∈ Q 
is a big group! It acts linearly on L via its quotient
GQ /GL ' Gal(L/Q) ' S5 .
Elements (really conjugacy classes) are hard to write down!
Absolute Galois Theory
Motivation
Local Class
Field Theory
Representations
Local
Langlands
• What finite groups are quotients of GQ ?
Given K a field K, V a K-vector space, and a representation
ρ : GQ → AutK (V ) with finite image, put
Lρ = {x ∈ Q : σ(x) = x for all σ ∈ ker ρ}.
Other groups
Then Gal(Lρ /Q) ' GQ /ker ρ ' im ρ.
Absolute Galois Theory
Motivation
Local Class
Field Theory
Representations
Local
Langlands
• What finite groups are quotients of GQ ?
Given K a field K, V a K-vector space, and a representation
ρ : GQ → AutK (V ) with finite image, put
Lρ = {x ∈ Q : σ(x) = x for all σ ∈ ker ρ}.
Other groups
Then Gal(Lρ /Q) ' GQ /ker ρ ' im ρ.
Note that, if ρ is 1-dimensional, then Gal(Lρ /Q) is abelian.
1-dimensional representations of GQ correspond to
abelian extensions of Q.
Absolute Galois Theory
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
• Representations of GQ occur via action on solutions of
diophantine equations, for example via (´etale) cohomology.
GQ acts on the points of order dividing `n on the elliptic curve
E : y 2 = x3 − 4x + 4,
E [`n ] ' (Z/`n Z)2 ;
taking the inverse limit we get a representation of GQ on
E [`∞ ] = lim E [`n ] ' Z2` ,
←−
i.e. a two-dimensional representation.
Local-Global Again
Motivation
Local Class
Field Theory
Since Q is dense in Qp , we get an injective map
GQp := Gal(Qp /Qp ) ,→ GQ
Representations
Local
Langlands
Other groups
so we can try first to understand GQp , for each p.
Local-Global Again
Motivation
Local Class
Field Theory
Since Q is dense in Qp , we get an injective map
GQp := Gal(Qp /Qp ) ,→ GQ
Representations
Local
Langlands
Other groups
so we can try first to understand GQp , for each p.
The idea of the (local) Langlands programme is to understand
the representations of GQp in terms of representations of
certain matrix groups over Qp .
Start with 1-dimensional representations; that is, understand
the abelian extensions of Qp .
Basic structure of Q×
p
We have a group homomorphism
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
×
| · |p : Q×
p → R+ ,
with image pZ and kernel the group of units Up = Z×
p : we have
a split exact sequence
1 −→ Up −→ Q×
p −→ Z −→ 0,
p 7−→ 1.
Up is compact open in Q×
p , with filtration
Upn = 1 + pn Zp ,
n ≥ 1.
Algebraic extensions of Qp
Qp
F/Qp algebraic
| · |p extends uniquely to F
Qp
Zp
Fp
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Zp
∪
pZp
ring of integers
maximal ideal
oF
∪
pF
Fp
residue field
kF = oF /pF
p
prime element
If F/Qp is finite, pF = $F oF
$F
Other groups
pZp
Algebraic extensions of Qp
Qp
F/Qp algebraic
| · |p extends uniquely to F
Qp
Zp
Fp
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Zp
∪
pZp
ring of integers
maximal ideal
oF
∪
pF
Fp
residue field
kF = oF /pF
p
prime element
If F/Qp is finite, pF = $F oF
$F
Other groups
pZp
Unramified F/Qp : only extend the residue field kF /Fp .
Totally ramified F/Qp : only extend the image |F × |p of | · |p .
The Weil group
For F an algebraic extension of Qp , there is a natural map
Motivation
Gal(F/Qp )
/ / Gal(kF /Fp ).
Local Class
Field Theory
Representations
Local
Langlands
Other groups
If kF /Fp is finite then Gal(kF /Fp ) is generated by Frobenius
Frob−1 : x 7→ xp .
The Weil group
For F an algebraic extension of Qp , there is a natural map
Motivation
Gal(F/Qp )
/ / Gal(kF /Fp ).
Local Class
Field Theory
Representations
If kF /Fp is finite then Gal(kF /Fp ) is generated by Frobenius
Local
Langlands
Frob−1 : x 7→ xp .
Other groups
1
/ Ip
/ GQp
/ Gal(Fp /Fp )
/0
The Weil group
For F an algebraic extension of Qp , there is a natural map
Motivation
Gal(F/Qp )
/ / Gal(kF /Fp ).
Local Class
Field Theory
Representations
If kF /Fp is finite then Gal(kF /Fp ) is generated by Frobenius
Local
Langlands
Frob−1 : x 7→ xp .
Other groups
1
/ Ip
/ GQp
O
/ Gal(Fp /Fp )
O
/0
1
/I
p
?
/W
p
?
/ hFrobi
/0
Wp is the Weil group, the inertia group Ip is open in Wp .
Local Class Field Theory
Motivation
There is a natural isomorphism of topological groups
ab
ap : Q×
p −→ Wp ,
Local Class
Field Theory
Representations
Local
Langlands
Other groups
in which p 7−→ Frob; we have
1
/ Up
1
ap
/ I ab
p
/ Q×
p
/Z
/0
/Z
/ 0.
ap
/ W ab
p
Local Class Field Theory
Motivation
Local Class
Field Theory
Dualizing, we get a natural bijection
irreducible representations
1-dimensional
←→ representations of W
of GL1 (Qp )
p
Representations
Local
Langlands
Other groups
We will identify these sets.
Local Class Field Theory
Motivation
Local Class
Field Theory
Dualizing, we get a natural bijection
irreducible representations
1-dimensional
←→ representations of W
of GL1 (Qp )
p
Representations
Local
Langlands
We will identify these sets.
Other groups
The local Langlands correspondence (Harris–Taylor, Henniart)
generalizes this to n-dimensional representations of Wp .
Local Class Field Theory
Motivation
Local Class
Field Theory
Dualizing, we get a natural bijection
irreducible representations
1-dimensional
←→ representations of W
of GL1 (Qp )
p
Representations
Local
Langlands
We will identify these sets.
Other groups
The local Langlands correspondence (Harris–Taylor, Henniart)
generalizes this to n-dimensional representations of Wp .
• Do we just change 1 to n?
Local Class Field Theory
Motivation
Local Class
Field Theory
Dualizing, we get a natural bijection
irreducible representations
1-dimensional
←→ representations of W
of GL1 (Qp )
p
Representations
Local
Langlands
We will identify these sets.
Other groups
The local Langlands correspondence (Harris–Taylor, Henniart)
generalizes this to n-dimensional representations of Wp .
• Do we just change 1 to n?
• What sorts of representations?
Representations of p-adic groups
Motivation
Local Class
Field Theory
A smooth (complex) representation of G = GLn (Qp ) is a
homomorphism
π : G −→ AutC (V),
for V a complex vector space, such that
Representations
Local
Langlands
Other groups
StabG (v) is open, for all v ∈ V.
Representations of p-adic groups
Motivation
Local Class
Field Theory
A smooth (complex) representation of G = GLn (Qp ) is a
homomorphism
π : G −→ AutC (V),
for V a complex vector space, such that
Representations
Local
Langlands
StabG (v) is open, for all v ∈ V.
Other groups
The only finite-dimensional irreducible smooth representations
of G are 1-dimensional, of the form
g 7→ χ(det(g)),
×
for χ : Q×
p → C a (smooth) character.
Representations of p-adic groups
Motivation
Local Class
Field Theory
A smooth (complex) representation of G = GLn (Qp ) is a
homomorphism
π : G −→ AutC (V),
for V a complex vector space, such that
Representations
Local
Langlands
StabG (v) is open, for all v ∈ V.
Other groups
The only finite-dimensional irreducible smooth representations
of G are 1-dimensional, of the form
g 7→ χ(det(g)),
×
for χ : Q×
p → C a (smooth) character.
Schur’s Lemma holds so every irreducible smooth
representation π of G has a central character ωπ : Z(G) → C× .
Langlands Parameters
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
A Langlands parameter for G is a smooth semisimple
n-dimensional representation
ϕ : Wp −→ GLn (C).
Note that we do not require irreducibility.
Langlands Parameters
Motivation
Local Class
Field Theory
Representations
A Langlands parameter for G is a smooth semisimple
n-dimensional representation
ϕ : Wp −→ GLn (C).
Local
Langlands
Note that we do not require irreducibility.
Other groups
Given nowPa number of representations ϕi : Wp → GLni (C)
with n = i ni , we can form their direct sum
M
ϕ=
ϕi : Wp −→ GLn (C),
i
with image in a Levi subgroup.
Parabolic (Harish-Chandra) induction
Given P = M n N a parabolic subgroup of G, we have
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
P/N ' M ' GLn1 (Qp ) × · · · × GLnk (Qp ),
P
with i ni = n. Any irreducible representation of M
decomposes as a tensor product
ρ1 ⊗ · · · ⊗ ρk ,
for ρi an irreducible representation of GLni (Qp ).
Parabolic (Harish-Chandra) induction
Given P = M n N a parabolic subgroup of G, we have
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
P/N ' M ' GLn1 (Qp ) × · · · × GLnk (Qp ),
P
with i ni = n. Any irreducible representation of M
decomposes as a tensor product
ρ1 ⊗ · · · ⊗ ρk ,
for ρi an irreducible representation of GLni (Qp ).
We can form the (normalized) parabolically induced
representation
ρ1 × · · · × ρk := Ind G
P ρ1 ⊗ · · · ⊗ ρk .
The semisimplification of this representation is independent of
the order of the representations ρi .
Parabolic (Harish-Chandra) induction
Motivation
Local Class
Field Theory
An irreducible representation of G which does not appear as a
submodule of any properly parabolically induced representation
is called cuspidal:
Representations
Local
Langlands
Theorem (Harish-Chandra, Jacquet)
Other groups
For any irreducible representation π of G, there is a cuspidal
representation ρ of a Levi subgroup M such that
π is a submodule of Ind G
P ρ,
for some P = M N a parabolic; moreover (M, ρ) is unique up
to conjugacy.
Example: GL2 (Qp )
M = GL1 (Qp ) × GL1 (Qp ) has representations χ1 ⊗ χ2 .
Motivation
Local Class
Field Theory
Representations
Local
Langlands
±1
• If χ1 χ−1
2 6= | · |p then χ1 × χ2 is irreducible;
• If χ1 χ−1
2 = | · |p then χ1 × χ2 has length 2, with
1-dimensional submodule; the quotient is called a
Steinberg representation:
Other groups
−1/2
0 → 1G → | · |1/2
→ StG → 0.
p × | · |p
−1
• If χ1 χ−1
2 = | · |p then we get the same composition
factors, reversed.
All other irreducible representations of G are cuspidal.
Example: GL2 (Qp )
M = GL1 (Qp ) × GL1 (Qp ) has representations χ1 ⊗ χ2 .
Motivation
Local Class
Field Theory
Representations
Local
Langlands
±1
• If χ1 χ−1
2 6= | · |p then χ1 × χ2 is irreducible;
• If χ1 χ−1
2 = | · |p then χ1 × χ2 has length 2, with
1-dimensional submodule; the quotient is called a
Steinberg representation:
Other groups
−1/2
0 → 1G → | · |1/2
→ StG → 0.
p × | · |p
−1
• If χ1 χ−1
2 = | · |p then we get the same composition
factors, reversed.
All other irreducible representations of G are cuspidal.
Following Deligne, we use the representations of SL2 (C) to
distinguish StG from 1G .
Local Langlands Correspondence for GL2
Motivation
Local Class
Field Theory
Representations
Local
Langlands
There is a canonical bijection
(smooth irreducible )
(msmooth W -semisimple )
p
←→
representations
representations
of GL2 (Qp )
Wp × SL2 (C) → GL2 (C)
[Kutzko, 1980]
Other groups
irreducible χ1 × χ2 ←→ χ1 ⊕ χ2 ,
1G ←→ 1 ⊕ 1,
StG ←→ 1 ⊗ St2 ,
cuspidal ←→ irreducible as Wp -representation.
Local Langlands Correspondence for GLn
Representations
There is a canonical bijection
(smooth irreducible )
(msmooth W -semisimple )
p
←→
representations
representations
of GLn (Qp )
Wp × SL2 (C) → GLn (C)
Local
Langlands
[Harris–Taylor, Henniart 1998]
Motivation
Local Class
Field Theory
Other groups
Local Langlands Correspondence for GLn
Representations
There is a canonical bijection
(smooth irreducible )
(msmooth W -semisimple )
p
←→
representations
representations
of GLn (Qp )
Wp × SL2 (C) → GLn (C)
Local
Langlands
[Harris–Taylor, Henniart 1998]
Motivation
Local Class
Field Theory
Other groups
The local Langlands correspondence for GLn reduces to
(irreducible cuspidal )
( irreducible smooth
representations
of GLn (Qp )
←→
semisimple representations
Wp → GLn (C)
)
Local Langlands Correspondence for GLn
Representations
There is a canonical bijection
(smooth irreducible )
(msmooth W -semisimple )
p
←→
representations
representations
of GLn (Qp )
Wp × SL2 (C) → GLn (C)
Local
Langlands
[Harris–Taylor, Henniart 1998]
Motivation
Local Class
Field Theory
Other groups
The local Langlands correspondence for GLn reduces to
(irreducible cuspidal )
( irreducible smooth
representations
of GLn (Qp )
←→
semisimple representations
Wp → GLn (C)
• What does canonical mean here?
)
Local Langlands Correspondence for GLn
Motivation
Local Class
Field Theory
Representations
Local
Langlands
There is a unique system of bijections
(smooth irreducible )
(continuous W -semisimple )
p
rn
−−→
representations
representations
of GLn (Qp )
Wp × SL2 (C) → GLn (C)
such that
Other groups
• r1 is given by local class field theory;
• rn (π ⊗ χ ◦ det) = rn (π) ⊗ r1 (χ);
• r1 (ωπ ) = det rn (π);
• rn respects L-functions L(π1 × π2 , s) and -factors of
pairs of representations.
Cuspidal representations of GLn (Qp )
In order to make use of the Langlands correspondence, it would
be helpful to have an explicit correspondence.
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
Cuspidal representations of GLn (Qp )
In order to make use of the Langlands correspondence, it would
be helpful to have an explicit correspondence.
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Theorem (Bushnell–Kutzko, 1993)
There is an explicit list of pairs (J, λ), consisting of a
compact-mod-centre open subgroup of G = GLn (Qp ) and an
irreducible representation λ of J, such that:
Other groups
• every irreducible cuspidal representation of G is equivalent
to some Ind G
J λ;
G 0
0 0
• Ind G
J λ ' Ind J 0 λ iff (J, λ) is conjugate to (J , λ ).
[Howe–Moy for p > n.]
Cuspidal representations of GLn (Qp )
In order to make use of the Langlands correspondence, it would
be helpful to have an explicit correspondence.
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Theorem (Bushnell–Kutzko, 1993)
There is an explicit list of pairs (J, λ), consisting of a
compact-mod-centre open subgroup of G = GLn (Qp ) and an
irreducible representation λ of J, such that:
Other groups
• every irreducible cuspidal representation of G is equivalent
to some Ind G
J λ;
G 0
0 0
• Ind G
J λ ' Ind J 0 λ iff (J, λ) is conjugate to (J , λ ).
[Howe–Moy for p > n.]
Using this, for p - n, Bushnell–Henniart have given an effective
description of the Langlands correspondence.
Local Langlands Conjecture for Sp2n
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
Local Langlands Conjecture for Sp2n
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
There is a canonical surjective map
(smooth irreducible )
( smooth W -semisimple )
p
−→
representations
representations
of Sp2n (Qp )
Wp × SL2 (C) → SO2n+1 (C)
with finite fibres, called L-packets.
Local Langlands Conjecture for Sp2n
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
There is a canonical surjective map
(smooth irreducible )
( smooth W -semisimple )
p
−→
representations
representations
of Sp2n (Qp )
Wp × SL2 (C) → SO2n+1 (C)
with finite fibres, called L-packets.
• The fibre over ϕ should be in bijection with the set of
irreducible representations of
Aϕ = π0 ZSO2n+1 (C) (ϕ)/Z(SO2n+1 (C)) .
Local Langlands Conjecture for Sp2n
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
There is a canonical surjective map
(smooth irreducible )
( smooth W -semisimple )
p
−→
representations
representations
of Sp2n (Qp )
Wp × SL2 (C) → SO2n+1 (C)
with finite fibres, called L-packets.
• The fibre over ϕ should be in bijection with the set of
irreducible representations of
Aϕ = π0 ZSO2n+1 (C) (ϕ)/Z(SO2n+1 (C)) .
• One would like to reduce to cuspidal representations, but
unfortunately things are not so easy.
Local Langlands Correspondence for Sp4
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
×
Let ω : Q×
p → C be the unramified quadratic character
(trivial on Up and with ω(p) = −1). For G = Sp4 (Qp ), we
have the Langlands parameter
ϕ = ω ⊗ St3 ⊕ ω ⊕ 1.
Here Aϕ is the Klein 4-group so the corresponding L-packet
has cardinality 4.
Local Langlands Correspondence for Sp4
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
×
Let ω : Q×
p → C be the unramified quadratic character
(trivial on Up and with ω(p) = −1). For G = Sp4 (Qp ), we
have the Langlands parameter
ϕ = ω ⊗ St3 ⊕ ω ⊕ 1.
Here Aϕ is the Klein 4-group so the corresponding L-packet
has cardinality 4.
Two of the representations come from
Ind G
P ω| · | ⊗ ω,
where P = M N with M ' GL1 (QP ) × GL1 (Qp ), but the other
two are cuspidal!
Local Langlands Correspondence for Sp4
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
×
Let ω : Q×
p → C be the unramified quadratic character
(trivial on Up and with ω(p) = −1). For G = Sp4 (Qp ), we
have the Langlands parameter
ϕ = ω ⊗ St3 ⊕ ω ⊕ 1.
Here Aϕ is the Klein 4-group so the corresponding L-packet
has cardinality 4.
Two of the representations come from
Ind G
P ω| · | ⊗ ω,
where P = M N with M ' GL1 (QP ) × GL1 (Qp ), but the other
two are cuspidal!
Note: the image of ϕ is not contained in any proper Levi
subgroup of SO5 (C).
Discrete series representations
Motivation
Local Class
Field Theory
Representations
Local
Langlands
The Local Langlands correspondence for GLn (Qp ) reduces to
( irreducible discrete )
(
)
semisimple representations
series representations ←→
Wp × SL2 (C) → GLn (C)
of GLn (Qp )
with image in no Levi sbgp
Other groups
• There is a classification of discrete series representations in
terms of cuspidal representations (Zelevinsky):
they are generalizations of Steinberg representations.
Discrete series representations
The Local Langlands correspondence for Sp2n (Qp ) reduces to
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
( irreducible discrete )
(
)
semisimple representations
series representations −→ Wp × SL2 (C) → SO2n+1 (C)
of Sp2n (Qp )
with image in no Levi sbgp
• There is a classification of discrete series representations in
terms of cuspidal representations (Sally–Tadi´c for n = 2;
Mœglin–Tadi´c in general).
For Sp4 , the two irreducible subquotients of
Ind G
P ω| · | ⊗ ω
are discrete series representations.
Cuspidal representations of Sp2n (Qp ), p 6= 2
Theorem (S. 2008)
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
There is an explicit list of pairs (J, λ), consisting of a compact
open subgroup of G = Sp2n (Qp ) and an irreducible
representation λ of J, such that:
• every irreducible cuspidal representation of G is equivalent
to some Ind G
J λ.
[For sufficiently large p, this is also due to Kim–Yu (for a
general connected reductive group).]
Cuspidal representations of Sp2n (Qp ), p 6= 2
Theorem (S. 2008)
Motivation
Local Class
Field Theory
Representations
Local
Langlands
Other groups
There is an explicit list of pairs (J, λ), consisting of a compact
open subgroup of G = Sp2n (Qp ) and an irreducible
representation λ of J, such that:
• every irreducible cuspidal representation of G is equivalent
to some Ind G
J λ.
[For sufficiently large p, this is also due to Kim–Yu (for a
general connected reductive group).]
The hope is to use this to make the local Langlands
correspondence for Sp2n explicit, at least when p - n.
Motivation
Local Class
Field Theory
Representations
What is the Langlands Programme?
Local
Langlands
Other groups
Shaun Stevens
University of East Anglia
6th March 2012