Representations of p-adic groups

Theorems and notes

Martin Hillel Weissman

Contents

Chapter 1. The additive group 11. Representations of cyclic groups 12. Characters of Zp 23. Representations of Qp 34. Integration, functions, and distributions 45. Invariants and coinvariants 56. Fourier transform, modules, and sheaves 6

Chapter 2. The multiplicative group 91. Representations of F×p and Z×p 9

2. Representations of Q×p 10

3. Local class field theory for Q×p 10

4. The category of representations of Q×p 13

Chapter 3. The structure of GL2(Qp) 151. Borel subgroups, tori 152. Bruhat-Tits building 183. Topological properties 204. Parahoric subgroups, more decompositions 20

Chapter 4. Smooth representations 231. Fundamentals 232. Parabolic induction and corestriction 243. Principal series 244. Unitarity 245. Intertwining operators 24

Chapter 5. Hecke algebras and Langlands perspective 251. Spherical representations 252. Iwahori-spherical representations 253. Langlands parameters 25

Bibliography 27

Index 29

3

CHAPTER 1

The additive group

1. Representations of cyclic groups

Definition 1.1. Let G be a group. A representation of G is a pair (π, V )where V is a complex vector space, and π : G→ GL(V ) is a group homomorphism.

Definition 1.2. A character of G is a representation (π, V ) in which V = C.The set of characters of G forms a group: Hom(G,C×).

Proposition 1.3. If G is a finite cyclic group, then Hom(G,C×) is a cyclicgroup of the same cardinality as G. If G is an infinite cyclic group, then Hom(G,C×)is isomorphic to C×, and there is one such isomorphism for each of the two gener-ators of G.

Definition 1.4. Let (π, V ) be a representation of a group G. Let χ : G→ C×be a character. The (G,χ)-isotypic subspace of V is defined by

V χ = {v ∈ V : π(g)v = χ(g) · v for all g ∈ G}.

When χ is the trivial character, we write V G for the (G,χ)-isotypic subspace; thisis called the space of G-invariants of V :

V G = {v ∈ V : π(g)v = v for all g ∈ G}.

Proposition 1.5. Let G be a finite cyclic group. Then for all g ∈ G,∑χ∈Hom(G,C×)

χ(g) =

{0 if g 6= 1

#G if g = 1.

Similarly, for all χ ∈ Hom(G,C×),∑g∈G

χ(g) =

{0 if χ 6= 1

#G if χ = 1.

Theorem 1.6. If G is a finite cyclic group, and (π, V ) is a representation ofG, then

V =⊕

χ∈Hom(G,C×)

V χ.

Note 1.7. If G is an infinite cyclic group, a representation of G is the same asa module over the commutative group ring C[G].

Definition 1.8. Let (π, V ) be a representation of a group G. Let χ : G→ C×be a character. The (G,χ)-coisotypic quotient of V is defined by

Vχ = V/V (χ), where V (χ) = SpanC{π(g)v − χ(g) · v : v ∈ V, g ∈ G}.

1

2 1. THE ADDITIVE GROUP

When χ is the trivial character, we write VG for the (G,χ)-coisotypic quotient; thisis called the space of G-coinvariants of V :

VG = V/ SpanC{π(g)v − v : v ∈ V, g ∈ G}.

Theorem 1.9. If (π, V ) is a representation of a finite cyclic group G, andχ : G→ C× and ψ : G→ C× are two characters, then the composition of inclusionand projection gives a linear map

V χ ↪→ V � Vψ.

This linear map is an isomorphism if χ = ψ and is the zero map otherwise.

Example 1.10. The following is a representation of Z. Let V = C[Z] be thevector space with basis Z. Define π : Z → GL(V ) to be the shift operator: if∑an · [n] ∈ V , and s ∈ Z, then

[π(s)](∑

an · [n])

=∑

an[n+ s] =∑

an−s[n].

Then V χ = {0} for all characters χ ∈ Hom(Z,C×). However, Vχ is one-dimensionalfor all characters χ.

2. Characters of ZpRefer to Serre [Ser73] for the structure theory of Zp and Qp.

Definition 2.1. A p-adic integer is a sequence a = (a0, a1, . . .) for which

• ai ∈ Z/piZ for all i ≥ 0.• If 0 ≤ i ≤ j then ai = aj modulo piZ.

The set of p-adic integers is denoted Zp.

Proposition 2.2. The p-adic integers form a ring of characteristic zero undertermwise addition and multiplication.

Proposition 2.3. The p-adic integers form a closed subset of∏i(Z/piZ),

where each factor is given the discrete topology.

Proposition 2.4. Addition and multiplication are continuous, as functionsfrom Zp × Zp to Zp.

Proposition 2.5. If x ∈ Zp, then the set {x + pjZp : j ≥ 0} forms a basis ofopen neighborhoods of x in Zp.

Proposition 2.6. The closed nontrivial subgroups of (Zp,+) are precisely thesubgroups pjZp for j ≥ 0. Each of these subgroups is open, and Zp/pjZp is isomor-phic to Z/pjZ.

Proposition 2.7. The set of continuous characters χ : Zp → C× is a groupunder pointwise multiplication. It will be denoted Homc(Zp,C×).

Definition 2.8. A smooth representation of Zp is a representation (π, V ) of(Zp,+) such that for all v ∈ V , there exists an open subgroup H ⊂ Zp for whichv ∈ V H = {v ∈ V : π(h)v = v for all h ∈ H}.

Proposition 2.9. If χ : Zp → C× is a continuous character, then Ker(χ) isa nontrivial closed (and hence open) subgroup. Hence the smooth one-dimensionalrepresentations (χ,C) of Zp are precisely the continuous characters.

3. REPRESENTATIONS OF Qp 3

Theorem 2.10. Let (π, V ) be a smooth representation of Zp. Then

V =⊕

χ∈Homc(Zp,C×)

V χ.

3. Representations of QpDefinition 3.1. A p-adic number is a sequence a = (. . . , a−1, a0, a1, . . .) for

which

• There exists an integer N such that ai = 0 for i < N . The largest (or ∞if there is not a largest) such N is called the valuation of a and is writtenval(a).

• With N as above, ai ∈ pNZ/piZ for all i ≥ N .• If i ≤ j then ai = aj modulo piZ.

The set of p-adic numbers is denoted Qp.

Definition 3.2. For every j ∈ Z, define Dj = {a ∈ Qp : val(a) ≥ j}. Notethat these form a chain under inclusion,

· · · ⊃ D−2 ⊃ D−1 ⊃ D0 = Zp ⊃ D1 ⊃ D2 ⊃ · · · ,

and Qp =⋃j Dj and {0} =

⋂j Dj .

Proposition 3.3. For every j ∈ Z, Dj is a group under termwise addition.Thus Qp itself is a group under termwise addition.

Proposition 3.4. Suppose that N,M ∈ Z, and k ≥ 0. Suppose also thati ≥ M , i ≥ N + k, j ≥ N , and j ≥ M + k. Then multiplication descends to awell-defined Z-bilinear map:

pNZpiZ

× pMZpjZ

→ pM+NZpM+N+kZ

.

Moreover, this multiplication map commutes with the natural projections when a ≥0, b ≥ 0:

pNZpi+aZ ×

pMZpj+bZ

//

��

pM+NZpM+N+kZ

=

��pNZpiZ ×

pMZpjZ

// pM+NZpM+N+kZ

Definition 3.5. Let a and b be p-adic numbers, with val(a) = N and val(b) =M . For each k ≥ (N + M), define ck ∈ pM+NZ/pM+N+kZ to be the result ofmultiplying ai and bj for any sufficiently large i, j, as in the previous proposition.The resulting sequence c = (ck) is called the product of a and b.

Proposition 3.6. With this product, the p-adic numbers form a field of charac-teristic zero under termwise addition and multiplication, containing Zp as a subring.

Proposition 3.7. The valuation satisfies the following properties.

(1) For any integer j, val(pj) = j.(2) For all a, b ∈ Qp, val(a + b) ≥ min(val(a), val(b)). This is an equality if

val(a) 6= val(b).

4 1. THE ADDITIVE GROUP

(3) Restricting to the nonzero elements of Qp gives a group homomorphismval : Q×p � Z.

Definition 3.8. Define a subset U of Qp to be open if pjU ∩Zp is open in Zpfor all integers j. This defines a topology on Qp.

Proposition 3.9. Each subgroup Dj is open and closed in Qp. Every closednontrivial subgroup of Qp is one of the subgroups Dj.

Proposition 3.10. Addition and multiplication are continuous, as functionsfrom Qp ×Qp to Qp.

Proposition 3.11. If x ∈ Qp, then the cosets {x + Dj : j ≥ 0} forms a basisof open neighborhoods of x in Qp.

Proposition 3.12. If χ : Qp → C× is a continuous character, then Ker(χ) is anontrivial closed subgroup. The group of such characters is denoted Homc(Qp,C×).

Definition 3.13. Define a character ψ◦ : Qp → C× by

ψ◦(a) = e2πia0 ,

where we recall that a ∈ Qp stands for a sequence (aj : j ∈ Z) in which (inparticular) a0 ∈ p−NZ/Z.

Theorem 3.14. If ψ : Qp → C× is a continuous character, then there exists aunique element y ∈ Qp such that

ψ(x) = ψ◦(xy) for all x ∈ Qp.

This gives a group isomorphism from Homc(Qp,C×) to Qp.

Definition 3.15. Define, for any y ∈ Qp, the character ψy : Qp → C× by

ψ(x) = ψ◦(xy) for all x ∈ Qp.

Definition 3.16. The group Homc(Qp,C×) is viewed as a topological groupvia the isomorphism from Homc(Qp,C×) to the additive group Qp.

4. Integration, functions, and distributions

See Weil [Wei95] for integration and Fourier analysis on p-adic fields.

Definition 4.1. Let Cc(Qp) be the vector space of C-valued compactly sup-ported functions on Qp. Let C∞(Qp) be the vector space of locally constant C-valued functions on Qp. Let C∞c (Qp) = Cc(Qp) ∩ C∞(Qp). The vector spaceC∞c (Qp) is called the Schwartz space on Qp.

Proposition 4.2. The vector space C∞(Qp) is a commutative ring, underpointwise multiplication. The vector space C∞c (Qp) is a commutative ring-without-identity (sometimes called a rng), under pointwise multiplication.

Definition 4.3. Let A be a compact open subset of Qp. The characteristicfunction of A is the element chA ∈ C∞c (Qp) defined by chA(a) = 1 if a ∈ A andchA(a) = 0 otherwise.

Proposition 4.4. The complex vector space C∞c (Qp) is spanned by the set{chA : A is compact open in Qp}.

5. INVARIANTS AND COINVARIANTS 5

Theorem 4.5. There exists a unique linear map∫·dx : C∞c (Qp) → C which

satisfies the following axioms:

(1)∫

chZp(x)dx = 1.

(2) If A is a compact open subset of Qp and a ∈ Qp, then∫

chA(x)dx =∫cha+A(x)dx.

Definition 4.6. More generally, suppose that V is any complex vector space.Let Cc(Qp, V ) and C∞(Qp, V ) and C∞c (Qp, V ) be defined as before, but usingV -valued functions instead of C-valued functions.

If A is a compact open subset of Qp and v ∈ V , let chA,v ∈ C∞c (Qp, V ) denotethe function defined by chA,v(a) = v if a ∈ A and chA(a) = 0 otherwise.

Theorem 4.7. There exists a unique linear map∫·dx : C∞c (Qp, V )→ V which

satisfies the following axioms:

(1)∫

chZp,v(x)dx = v.(2) If A is a compact open subset of Qp and a ∈ Qp and v ∈ V , then∫

chA,v(x)dx =∫

cha+A,v(x)dx.

Definition 4.8. A distribution on Qp is a C-linear map δ : C∞c (Qp) → C.The distributions form a complex vector space denoted C−∞(Qp).

Example 4.9. If a ∈ Qp, then evaluation at a gives a distribution eva ∈C−∞(Qp).

Example 4.10. If η ∈ C∞(Qp), then the map f 7→∫f(x)η(x)dx defines a dis-

tribution. This defines a series of inclusions of vector spaces C∞c (Qp) ⊂ C∞(Qp) ⊂C−∞(Qp).

5. Invariants and coinvariants

Definition 5.1. A smooth representation of Qp is a representation (π, V ) of(Qp,+) such that for all v ∈ V , there exists an open subgroup H ⊂ Qp for whichv ∈ V H = {v ∈ V : π(h)v = v for all h ∈ H}.

Proposition 5.2. If (π, V ) is a smooth representation of Qp, then for allv ∈ V , the function (x 7→ π(x)v) is an element of C∞(Qp, V ).

Definition 5.3. Let (π, V ) be a smooth representation of Qp. Let χ ∈Homc(Qp,C×) be a continuous character. Define the subspace

V (χ) = SpanC{π(a)v − χ(a)v : a ∈ Qp, v ∈ V }.

Proposition 5.4. The subspace V (χ) ⊂ V is stable under the action of Qp,i.e., if v ∈ V (χ) and b ∈ Qp, then π(b)v ∈ V (χ).

Definition 5.5. The χ-coisoytpic quotient of V , also called the twisted Jacquetmodule is the quotient V/Vχ. When χ is the trivial character, the quotient is calledthe space of Qp-coinvariants, or simply the Jacquet module of V .

Proposition 5.6. Suppose that (π, V ) is a smooth representation of Qp andχ ∈ Homc(Qp,C×). Then V (χ)χ = {0} and Vχ(χ) = {0}, and thus (Vχ)χ = Vχ.

Theorem 5.7. With (π, V ) and χ as above, we have v ∈ V (χ) if and only ifthere exists an integer N such that∫

χ(x) · chDN (x) · [π(x)](v)dx = 0.

6 1. THE ADDITIVE GROUP

In this case, the integral vanishes with N replaced by any smaller integer.

Definition 5.8. Suppose that (π, V ) and (σ,W ) are smooth representations ofQp. An intertwining operator from (π, V ) to (σ,W ) is a C-linear map φ : V →Wsuch that for all a ∈ Qp,

φ ◦ π(a) = σ(a) ◦ φ.The set of intertwining operators forms a complex vector space, denoted HomQp(V,W )(when π and σ are implicitly understood).

Proposition 5.9. The category of smooth representations of Qp and intertwin-ing operators forms a cocomplete abelian category. In particular, arbitrary directsums of smooth representations are naturally smooth representations, kernels andcokernels of intertwining operators are smooth representations.

Proposition 5.10. Let (π, V ) and (σ,W ) be two smooth representations ofQp, and φ : V → W a Qp-intertwining operator. Then for any character χ ∈Homc(Qp,C×), we have φ(V (χ)) ⊂ W (χ). Thus φ descends uniquely to a linearmap φχ from Vχ to Wχ.

Theorem 5.11. For any χ ∈ Homc(Qp,C×), the rules (V 7→ Vχ) and (φ 7→ φχ)define an exact functor from the category of smooth representations of Qp to thecategory of complex vector spaces.

More explicitly, given (π, V ) and (σ,W ) and (ρ, U) smooth representations ofQp and Qp-intertwining operators U → V and V →W , suppose that

0→ U → V →W → 0

is exact. Then the sequences

0→ U(χ)→ V (χ)→W (χ)→ 0,

0→ Uχ → Vχ →Wχ → 0

are also exact.

Theorem 5.12. Suppose that (π, V ) is a smooth representation of Qp. If Vχ =0 for all χ ∈ Homc(Qp,C×), then V = 0.

6. Fourier transform, modules, and sheaves

Definition 6.1. Suppose f ∈ C∞c (Qp). The Fourier transform of f is thefunction Ff : Qp → C given by

Ff(y) =

∫Qpf(x)ψy(x)dx =

∫Qpf(x)ψ◦(xy)dx.

Theorem 6.2. The Fourier transform gives a complex linear isomorphism fromC∞c (Qp) to itself. Its inverse is given by the transform:

F−1g(x) =

∫Qpg(y)ψ◦(xy)dy.

Definition 6.3. Suppose that f1, f2 ∈ C∞c (Qp). The convolution of f1 andf2 is the function f1 ∗ f2 ∈ C∞c (Qp) given by

[f1 ∗ f2](x) =

∫Qpf1(y)f2(xy−1)dy.

6. FOURIER TRANSFORM, MODULES, AND SHEAVES 7

Proposition 6.4. Convolution is C-bilinear, and gives C∞c (Qp) the structureof a commutative ring-without-identity.

Definition 6.5. We write (C∞c (Qp), ·) for the commutative rng given by point-wise multiplication, and (C∞c (Qp), ∗) for the commutative rng given by convolution.

Theorem 6.6. If f1, f2 ∈ C∞c (Qp), then

F(f1 ∗ f2) = Ff1 · Ff2.

In this way, the Fourier transform is an isomorphism of rings-without-identity from(C∞c (Qp), ∗) to (C∞c (Qp), ·).

Proposition 6.7. Suppose that (π, V ) is a smooth representation of Qp. ThenV is a module over the rng (C∞c (Qp), ∗) via the map C∞c (Qp)× V → V :

(f, v) 7→ π(f) · v :=

∫Qpf(x) · (π(x)v)dx.

Proposition 6.8. The previous definition identifies the category of smooth rep-resentations of Qp with the category of (C∞c (Qp), ∗)-modules V such that C∞c (Qp) ·V = V . (Note that this condition is not automatic since C∞c (Qp) is an ring-without-identity!). These are called smooth modules over C∞c (Qp).

Definition 6.9. Suppose that (π, V ) is a smooth representation of Qp, viewednaturally as a module over (C∞c (Qp), ∗). The Fourier transform of (π, V ) is the(C∞c (Qp), ·)-module structure given by the map C∞c (Qp)× V → V :

(f, v) 7→ Fπ(f) · v :=

∫QpFf(x) · (π(x)v)dx.

Example 6.10. Suppose that ψ = ψy ∈ Homc(V,C×), viewed as a one-dimensionalsmooth representation of Qp. The Fourier transform is a (C∞c (Qp), ·)-module struc-ture on C given by

(f, z) 7→∫QpFf(x)ψ◦(yx) · vdx = f(y) · v.

Definition 6.11. Suppose that U is a compact open subset of Qp. DefineC∞U (Qp) to be the subring of C∞c (Qp) consisting of functions vanishing outside ofU , with identity element given by chU .

Definition 6.12. Let U be a compact open subset of Qp. When (π, V ) is asmooth representations of Qp, let VU = Fπ(C∞U (Qp))V = Fπ(chU ) · V .

Proposition 6.13. Suppose that U ⊃ T are compact open subsets of Qp, and(π, V ) is a smooth representation of Qp. Then chU · chT = chT and the operatorFπ(chT ) sends VU to VT .

Theorem 6.14 (cf. §III, [Rod77]). Suppose that (π, V ) is a smooth represen-tation of Qp. There is a unique sheaf of complex vector spaces on Qp, given byU 7→ VU for all compact open subsets U ⊂ Qp, with restriction maps VU → VTgiven as above whenever U ⊃ T . This gives an equivalence of categories from thecategory of smooth representations of Qp to the category of sheaves of complex vectorspaces on Qp.

Proposition 6.15. Keep (π, V ) and the associated sheaf as in the previoustheorem. Then for any y ∈ Qp, the fiber of the sheaf at y is the vector space Vψy .

CHAPTER 2

The multiplicative group

1. Representations of F×p and Z×pRecall that the group F×p is cyclic. However, there is no canonical generator of

F×p (except if p = 2 or p = 3).

Proposition 1.1. The group Hom(F×p ,C×) is cyclic of order p− 1.

Definition 1.2. If j ≥ 0, the congruence subgroup Up,j is the subgroup ofZ×p given by

Up,j = {u ∈ Z×p : val(u− 1) > j}.

Proposition 1.3. For any j ≥ 0, we have Up,j = 1 + pj+1Zp.

Proposition 1.4. The ring homomorphism Zp → Fp given by reduction modp (or equivalently sending a sequence (ai) to a0 ) gives a surjective homomorphismfrom Z×p to F×p , yielding a short exact sequence:

1→ Up,0 → Z×p → F×p → 1.

Proposition 1.5. There exists a unique homomorphism θ : F×p → Z×p splittingthe above short exact sequence.

Theorem 1.6. When p 6= 2, define r = 1, and when p = 2 define r = 2. Definethe exponential map exp : prZp → 1 + prZp by the usual power series

exp(x) =∑ xn

n!.

Then exp converges and gives a continuous group isomorphism from prZp (a groupunder addition) to 1 + prZp = Up,r−1 (a group under multiplication).

Corollary 1.7. If p 6= 2 then the group Z×p is isomorphic to F×p × (Zp,+). If

p = 2 then the group Z×2 is isomorphic to (Z/4Z)× × (Z2,+).

Endow the group Z×p with the subspace topology from Zp.

Proposition 1.8. The collection of subgroups Up,j are compact and open forall j ≥ 0. They form a basis of neighborhoods of 1 ∈ Z×p .

Definition 1.9. Let χ : Z×p → C× be a smooth character. Then for sufficientlylarge integers j, χ is trivial when restricted Up,j . The depth of χ is the smallestpositive integer j, for which χ is trivial when restricted to Up,j .

Proposition 1.10. If p 6= 2, then the characters of Z×p of depth bounded by

j are identified with the characters of F×p × Z/pjZ. If p = 2 and j ≥ 1, then

the characters of Z×p of depth bounded by j are identified with the characters of

(Z/4Z)× × Z/2j−1Z. The only depth-zero character of Z×2 is the trivial character.

9

10 2. THE MULTIPLICATIVE GROUP

2. Representations of Q×pThe valuation val : Q×p → Z gives a short exact sequence of groups:

1→ Z×p → Q×p → Z→ 1.

This sequence splits, by sending 1 ∈ Z to p ∈ Q×p .

Proposition 2.1. The map (u, n) 7→ u · pn defines a topological group isomor-phism from Z×p × Z to Q×p (where Q×p is given the subspace topology from Qp).

Corollary 2.2. The set of continuous characters Homc(Q×p ,C×) is identified

with Homc(Z×p ,C×)×Homc(Z,C×).

Definition 2.3. Suppose that χ ∈ Homc(Z×p ,C×) is a continuous character.

The inertial class of χ is the set of characters of Q×p which restrict to χ on Z×p .Namely, the inertial class consists of all characters of the form:

χs(u · pn) = χ(u) · sn,

as s ranges over all nonzero complex numbers.

Proposition 2.4. The continuous characters of Q×p are parameterized by triplesof characters.

p 6= 2: by triples (χ, χ◦, s) with χ ∈ Hom(F×p ,C×) and χ ∈ Homc(Zp,C×)

and s ∈ C×;p = 2: by triples (χ, χ◦, s) with χ ∈ Hom((Z/4Z)×,C×) and χ ∈ Homc(Zp,C×)

and s ∈ C×.

3. Local class field theory for Q×p

Theorem 3.1. Suppose that q = pf for some positive integer f . Then thepolynomial Φq = (Xq−1 − 1)/(Xp−1 − 1) is irreducible over Fp and

Fq = Fp[X]/(Φq(X))

is a field of order q. Moreover Fq is a Galois extension of Fp and Gal(Fq/Fp) isisomorphic to Z/fZ. An isomorphism is characterized by the following: for everyelement γ ∈ Gal(Fq/Fp), there exists a unique r(γ) ∈ Z/fZ such that

γ(x) = xp−r(γ)

for all x ∈ Fq.

Note: The sign in the exponent −r(γ) is the geometric normalization forlocal class field theory.

Proposition 3.2. If K is a finite Galois extension of Fp, then K is isomorphicto Fq for some q = pf as above.

Proposition 3.3. If n is a positive integer and p does not divide n, then ndivides pf − 1 for some positive integer f .

Theorem 3.4. Suppose that q = pf for some positive integer f . Then thepolynomial Φq = (Xq−1 − 1)/(Xp−1 − 1) is irreducible over Qp and define

Qq = Qp[X]/(Φq(X)).

Let ζ denote the image of X in Qq, i.e., a primitive (q − 1)th root of unity.

3. LOCAL CLASS FIELD THEORY FOR Q×p 11

Then val : Q×p → Z extends uniquely to a homomorphism val : Q×q → Z in such

a way that val−1(N) is a subring Zq of Qq. Moreover Qq is a Galois extension ofQp and Gal(Qq/Qp) is isomorphic to Z/fZ.

Definition 3.5. The maximal unramified extension of Qp is the composi-tum Qunrp of all fields Qq as q ranges over powers of p.

Proposition 3.6. The maximal unramified extension Qunrp /Qp has the follow-ing properties:

(1) val : Q×p → Z extends to a homomorphism val : Qunrp → Z, and Zunrp =

val−1(N) is a subring of Qunrp .(2) The ideal pZunrp is the unique maximal ideal in Zunrp , and the quotient

field Zunrp /pZunrp is an algebraic closure of Fp, which we call Fp.

(3) If n is a positive integer, then Qunrp contains a primitive nth root of unityif and only if p does not divide n.

Proposition 3.7. Every automorphism α ∈ Gal(Qunrp /Qp) preserves the valu-ation, and thus the subring pZunrp and its maximal ideal. This identifies Gal(Qunrp /Qp)with Gal(Fp/Fp).

Definition 3.8. The profinite completion of Z is the abelian group Z whoseelements are families (an : 1 ≤ n ∈ Z) such that an ∈ Z/nZ for all n ≥ 1 andwhenever m divides n, am = an modulo m. This forms a group under addition,and is given the subspace topology from

∏n(Z/nZ); thus Z is a compact abelian

topological group.

Proposition 3.9. For any integer a, the constant family (a) is an element of

Z. In this way, Z is a dense subgroup of Z.

Proposition 3.10. Define a map from Z to∏p Zp, by sending each family

(an) to its restriction to prime-power indices. This defines a topological group

isomorphism from Z to∏p Zp.

Proposition 3.11. There is a unique isomorphism r : Gal(Fp/Fp)→ Z, whichsatisfies the following properties:

(1) If α ∈ Gal(Fp/Fp) and α(x) = xp for all x ∈ Fp, then r(α) = −1.

(2) For any q = pf , r restricts to an isomorphism from Gal(Fp/Fp) to f · Z.

Thus r also lifts to an isomorphism r : Gal(Qunrp /Qp)→ Z.

The maximal unramified extension of Qp is obtained by adjoining nth roots ofunity for n not divisible by p. Now we study the effect of adjoining p-power rootsof unity.

Definition 3.12. Let e be a positive integer. Consider the cyclotomic poly-nomial with integer coefficients:

Ψe =Xpe − 1

Xpe−1 − 1.

Then Ψe is irreducible over Qp. Define Qp(µpe) = Qp[X]/Ψe(X). It is called ap-cyclotomic extension of Qp, and the image of X in Qp(µpe) is a primitive(pe)th root of unity.

12 2. THE MULTIPLICATIVE GROUP

Proposition 3.13. For every positive integer e, Qp(µpe) is a cyclic Galoisextension of Qp, and there is a unique isomorphism of groups

re : Gal(Qp(µpe)/Qp)→ (Z/peZ)×

such that for all γ ∈ Gal(Qp(µpe)/Qp) and all primitive (pe)th roots of unity ζ,

γ(ζ) = ζre(γ).

Definition 3.14. The full p-cyclotomic extension of Qp is the compositumof the fields Qp(µpe) for all e ≥ 1. It is denoted by Qp(µp∞).

Theorem 3.15. The full p-cyclotomic extension of Qp is an abelian Galoisextension. There is a unique isomorphism r : Gal(Qp(µp∞)/Qp) → Z×p such that

for all γ ∈ Gal(Qp(µp∞)/Qp), all e ≥ 1, and all primitive (pe)th roots of unity ζ,

γ(ζ) = ζr(γ) mod pe .

Definition 3.16. Let Qabp be the compositum of the algebraic extensions Qunrp

and Qp(µp∞).

Proposition 3.17. The field Qabp is a Galois extension of Qp, and its Galoisgroup is a direct product of its subgroups:

Gal(Qabp /Qp) = Gal(Qabp /Qunrp )×Gal(Qabp /Qp(µp∞).

In this way, there is a unique isomorphism

r : Gal(Qabp /Qp)→ Z× Z×pcompatible with the previously defined isomorphisms.

Theorem 3.18 (Local Kronecker-Weber). If K/Qp is an abelian Galois exten-sion, then there exists an embedding of Qp-algebras from K into Qabp .

Definition 3.19. Fix an algebraic closure Qp of Qp, containing Qunrp . The

absolute Galois group of Qp is the Galois group Γ = Gal(Qp/Qp). The inertiasubgroup of Γ is the subgroup I = Gal(Qp/Qunrp ). These fit into a short exactsequence:

1→ I → Γ→ Z→ 1.

Definition 3.20. The Weil group W is the preimage of Z in Γ, in the aboveexact sequence. We give the Weil group the weakest Hausdorff topology for whichthe following statements hold:

(1) If K is a finite extension of Qunrp , then Gal(Qp/K) is open in I and inW.(2) The group structure Γ× Γ→ Γ is separately continuous.

Theorem 3.21. Let Wc be the closure of the commutator subgroup of W. ThenW/Wc is isomorphic to Q×p .

Corollary 3.22. There is a group isomorphism from Homc(W,C×) (the groupof continuous homomorphisms from W to C×) to Homc(Q×p ,C×).

This is “local Langlands for GL1”, stating in other words that the irreduciblesmooth representations of GL1(Qp) are naturally parameterized by continuous ho-momorphisms from W to GL1(C). For “local Langlands for GLn”, one replacesGL1 by GLn and refines the parameterization by equivalence relations.

4. THE CATEGORY OF REPRESENTATIONS OF Q×p 13

Corollary 3.23. There is a group isomorphism from Homc(Γ,C×) (the groupof continous homomorphisms from the absolute Galois group of Qp to C×) to thesubgroup of Homc(Q×p ,C×) consisting of characters of finite order.

4. The category of representations of Q×pSince Q×p ∼= Z× Z×p , we may decompose representations of Q×p in two steps.

Proposition 4.1. Suppose that (π, V ) is a smooth representation of Q×p . Then

for any character χ ∈ Homc(Z×p ,C×), the (Z×p , χ)-isotypic subrepresentation is

again a smooth representation of Q×p and

V =⊕

χ∈Homc(Z×p ,C×)

V χ.

is a decomposition of (π, V ) into a direct sum of smooth subrepresentations.

Within each Z×p -isotypic component, we have:

Definition 4.2. Suppose that (π, V ) is a smooth representation of Q×p which

is (Z×p , χ)-isotypic. We view (π, V ) as a module over C[T, T−1] by defining T · v =π(p)v.

Proposition 4.3. This defines an isomorphism of categories between (Z×p , χ)-

isotypic smooth representations of Q×p and C[T, T−1]-modules; the latter categoryis (by definition) the category of quasicoherent sheaves on the punctured affine lineover C.

One can think of the set Homc(Q×p ,C×) as an infinite discrete collection (in-

dexed by χ ∈ Homc(Z×p ,C×) of “punctured pancakes” (each a copy of Hom(Z,C×) =

C×).

Definition 4.4 (See Debacker’s notes, §9.2). Let C be an abelian categorywith full subcategories Ci indexed by a set I. We say that C =

∏Ci is a splitting

of the category if

(1) Every object V ∈ C has a unique direct sum decomposition V =⊕Vi for

objects Vi of Ci.(2) If Vi ∈ Ci and Vj ∈ Cj and i 6= j, then Hom(Vi, Vj) = 0.

Proposition 4.5. The category Rep∞(Q×p ) of smooth representations of Q×psplits,

Rep∞(Q×p ) =∏

χ∈Homc(Z×p ,C×)

Rep∞χ (Q×p ),

where the factors are the categories of smooth (Z×p , χ) isotypic representations. Each

of those categories is isomorphic to the category of C[T, T−1]-modules.

CHAPTER 3

The structure of GL2(Qp)

1. Borel subgroups, tori

Definition 1.1. Let F be a field, and V a finite-dimensional F -vector space.A flag in V is a sequence:

0 = W0 ⊂W1 ⊂W2 ⊂ · · · ⊂Wr ⊂Wr+1 = V,

where all inclusions are strict inclusions of vector spaces. The length of a flag W•is the number of proper nonzero spaces in the sequence, the number r in the flagabove. A maximal flag is a flag of maximal length (which equals dim(V )− 1).

Definition 1.2. Let W• = (W0,W1, . . . ,Wr+1) be a flag in a finite-dimensionalF -vector space V . A subflag of W• is an ordered subset of W• containing W0 = 0and Wr+1 = V .

When a vector space comes with an ordered basis, there are some standardflags.

Definition 1.3. Let V = Fn = F⊕· · ·⊕F . Define, for 0 ≤ d ≤ n the subspace

Wd =

(d⊕i=1

F

)⊕

(n⊕

i=d+1

0

).

Then the sequence

0 = W0 ⊂W1 ⊂W2 ⊂ · · · ⊂Wn = V,

is called the standard maximal flag in V . The subflags of the standard maximalflag are called the standard flags in V .

Definition 1.4. Let W• be a flag in V of length r. The type of W• is thevector of dimensions:

typ(W•) = (dim(W1), dim(W2), . . . , dim(Wr)).

Often, instead of the numerical invariant “type,” one uses a Dynkin diagramto encode the same information. The Dynkin diagram of GL(V ) is a collection ofdim(V )−1 dots, with edges connecting them in linear sequence. Here is the Dynkindiagram of GL7(F ).

The types of flags correspond to subdiagrams, i.e., diagrams obtained by delet-ing a subset of dots and all the edges they touch. The rule is the following: for aflag W•, delete the dth vertex if there exists an i such that dim(Wi) = d. Thusmaximal flags correspond to the deletion of all vertices.

15

16 3. THE STRUCTURE OF GL2(Qp)

[010]

[001]

[011][101]

[110]

(100)

(010)

(001)

(011)

(110)

(101)

Figure 1. The Fano plane. Lines in F32 correspond to points in

the Fano plane, and are labelled by their nonzero vector, i.e. (101)stands for the onzero vector (1, 0, 1) in F3

2. Planes in F2 correspondto lines in the Fano plane, and are labelled by the unique nonzerovector orthogonal to the plane. For example, [101] stands for theplane of vectors orthogonal to (1, 0, 1) with respect to the usual dotproduct over F2. Incidence of points and lines in the Fano planecorresponds to containment of lines in planes in F3

2.

Definition 1.5. The spherical building of GL(V ) is the simplicial complex,whose dimension-r simplices are the flags of length r in V , where the notion of “xis a face of y” means that “x is a subflag of y”.

Example 1.6. The Fano plane and its relation to the spherical building ofGL3(F2). There are seven one-dimensional subspaces and seven two-dimensionalsubspaces of F3

2. Their incidence is described by the Fano (finite projective) plane:

In the spherical building, the vertices and edges of the Fano plane – the flagsof length 1 and types 1 and 2 – become the vertices of the building. Edges in thebuilding correspond to incidence in the Fano plane.

Lemma 1.7. Let V ′1 and V ′2 be subspaces of a vector space V . Let V ′′1 = V/V ′1and let V ′′2 = V/V ′′2 . Suppose that dim(V ′1) = dim(V ′2). Then, for any linearisomorphisms α : V ′1 → V ′2 and β : V ′′1 → V ′′2 , there exists a linear automorphismg ∈ GL(V ) such that g restricts to α on V ′1 , and induces β on the quotient V/V ′1 .

Proposition 1.8. The group GL(V ) acts on its spherical building, preservingtypes of flags. It acts transitively on the set of flags of any given type.

Definition 1.9. Let F be a field, and V a finite-dimensional F -vector space.Let W• be a flag in V . The stabilizer of W• is the subgroup of GL(V ) consisitingof linear maps φ such that φ(Wi) ⊂Wi for all i.

A parabolic subgroup of GL(V ) is a subgroup which is the stabilizer of aflag. A Borel subgroup of GL(V ) is a parabolic subgroup which is the stabilizerof a maximal flag.

1. BOREL SUBGROUPS, TORI 17

Proposition 1.10. The group GL(V ) acts transitively on the set of Borelsubgroups, by conjugation.

Corollary 1.11. When V = F 2, GL(V ) = GL2(F ), every parabolic subgroupof GL(V ) is conjugate to the standard Borel subgroup B:

B =

{(a b0 d

): a, d ∈ F×, b ∈ F

}.

B is the parabolic subgroup of GL2(F ) stabilizing the standard maximal flag.

Definition 1.12. Let W• be a flag in V of length r. The associated gradationis the sequence of vector spaces:

Gr0(W•) = W1 = W1/W0, Gr1(W•) = W2/W1, . . . , Grr+1(W•) = V/Wr.

Definition 1.13. Let P be the parabolic subgroup of GL(V ) stabilizing aflag W•. Then P naturally acts on each step in the gradation Wi/Wi−1, giving ahomomorphism

P → GL(Gr0(W•))× · · · ×GL(Grr+1(W•).

The kernel of this homomorphism is called the unipotent radical of P .

Proposition 1.14. Let B be the standard Borel subgroup from before. Itsunipotent radical is the subgroup

U =

{(1 b0 1

): b ∈ F

}.

Definition 1.15. Let W• be a flag in V of length r. A splitting of W• is asequence of subspaces of V : S1, S2, . . . , Sr+1, such that for all i ≥ 1,

Wi = S1 ⊕ · · · ⊕ Si.

Proposition 1.16. Every flag has a splitting.

Definition 1.17. Let W• be a flag in V , and let P be the parabolic subgroupof GL(V ) stabilizing this flag. Let S1, . . . , Sr+1 be a splitting of the flag. ThenGL(S1)× · · · ×GL(Sr+1) is naturally a subgroup of P . Such subgroups are calledLevi subgroups of P .

Note that, if W• is a maximal flag in V , then any splitting consists of one-dimensional spaces, and so any Levi subgroup is isomorphic to (F×)r+1. Such Levisubgroups are the maximal (split) tori in GL(V ).

Proposition 1.18. If W• is a flag in V , then the Dynkin diagram of any Levisubgroup of the stabilizer of W• is precisely the subdiagram of the Dynkin diagramof GL(V ) matching the type of W•.

Proposition 1.19. Let W• be a flag of length r in V , and S1, . . . , Sr+1 asplitting of the flag. Let P be the parabolic subgroup of GL(V ) stabilizing W•. LetU be the unipotent radical of P . Let L = GL(S1) × · · · × GL(Sr+1) be the Levisubgroup of P coming from the splitting. Then P = L n U , i.e., L ∩ U = {1} andU is a normal subgroup of P and P = LU .

Using the standard splitting of F 2 as F ⊕ F , we find the standard torus inB:

T =

{(a 00 d

): a, d ∈ F×

}.

18 3. THE STRUCTURE OF GL2(Qp)

Proposition 1.20. Let P be a parabolic subgroup of GL(V ), and L a Levisubgroup of P . Then there exists a parabolic subgroup P− ⊂ GL(V ) such thatP ∩ P− = L. This is called the opposite parabolic.

Proposition 1.21. Let B be the standard Borel subgroup of GL2(F ), and Tthe standard torus in B. The opposite Borel subgroup is given by

B− =

{(a 0c d

): a, d ∈ F×, c ∈ F

}.

Definition 1.22. Let T be a maximal torus inG = GL(V ). DefineN = NG(T )be its normalizer. The group W = N/T is called the Weyl group of G with respectto the torus T .

Proposition 1.23. If dim(V ) = d, then the Weyl group is isomorphic to Sd,the symmetric group on d letters.

Theorem 1.24 (Bruhat decomposition). Fix a maximal torus T contained ina Borel subgroup B in G = GL(V ). Let W be the Weyl group of G with respect toT . Then

G =⊔w∈W

BwB.

(Note that T ⊂ B, so the coset wB is well defined for w ∈W = N/T .)

We have come close to proving that the set of subgroups B,N, T ⊂ G satisfyTits’ axioms for a BN-pair:

(1) G is generated by B and N .(2) B ∩N = T and T is a normal subgroup of N .(3) The group W = N/T is generated by a set S of elements of order 2.(4) If s ∈ S and w ∈ W , then sBw is contained in the union of BswB and

BwB.(5) No element s ∈ S normalizes B.

2. Bruhat-Tits building

Now we restrict attention to the field Qp. We have already flags and parabolicsubgroups coming from linear algebra over Qp. Now we discuss affine flags andparahoric subgroups.

Definition 2.1. Let V be a Qp-vector space of dimension n. A lattice in Vis a Zp-submodule of V which is free of rank n as a Zp-module. The standardlattice in Qnp is the lattice Znp .

Proposition 2.2. If V is a finite-dimensional Qp-vector space, then the groupGL(V ) acts transitively on the set of lattices in V . The stabilizer of the standardlattice Znp ⊂ Qnp is the subgroup GLn(Zp) ⊂ GLn(Qp). Note that if g ∈ GLn(Qp),

then g ∈ GLn(Zp) if and only if the matrix entries of g lie in Zp and det(g) ∈ Z×p .

Proposition 2.3. Let Λ and Λ′ be lattices in V . Then there exists a positiveinteger N such that pNΛ ⊂ Λ′.

Definition 2.4. If Λ and Λ′ are lattices, and Λ′ ⊂ Λ, then we say that Λ′ is asublattice of Λ. The index [Λ : Λ′] is the index of Λ′ as a subgroup of Λ.

2. BRUHAT-TITS BUILDING 19

Proposition 2.5. If Λ is a lattice in Qnp , then pΛ is a sublattice of Λ and[Λ : pΛ] = pn.

Proposition 2.6. If Λ′ is a sublattice of Λ then the index [Λ : Λ′] is a powerof p.

Proposition 2.7. Let V be a Qp vector space of dimension n. If Λ is a latticein V , then Λ/pΛ is naturally an Fp-vector space of dimension n. The lattices Λ′

satisfying Λ ⊃ Λ′ ⊃ pΛ are in natural inclusion-preserving bijection with the Fp-linear subspaces of Λ/pΛ.

Definition 2.8. Two lattices Λ and Λ′ in V are called homothetic if thereexists an integer N such that Λ = pN · Λ′, equivalently if there exists a nonzerox ∈ Q×p such that Λ = x · Λ′. A lattice class is an equivalence class of lattices inV under the relation of homothety, and we write [Λ] for the homothety class of Λ.

Definition 2.9. An affine flag in V of length r is a labeling of the verticesof the r-gon by lattice classes (i.e., a sequence of lattice classes up to circularpermutation):

[Λ1], [Λ2], . . . , [Λr]

such that for some representatives of each class, we have strict containments:

Λ1 ⊃ Λ2 ⊃ · · · ⊃ Λr ⊃ pΛ1.

The type is the labeling of the edges of the r-gon by the indices:

[Λ1 : Λ2], [Λ2 : Λ3], . . . , [Λr : pΛ1].

(Note that the type is well-defined, up to circular permutation)

Definition 2.10. The affine Dynkin diagram of GL(V ), where V is a Qp-vector space of dimension n, is the n-gon. Given an affine flag in V , its typeis encoded in the subdiagram whose connected components consist of e1, . . . , ervertices where pe1 , . . . , per is the type discussed above.

Proposition 2.11. The reduction-mod-p map gives a surjective homomor-phism from GLn(Zp) onto GLn(Z/pnZ).

Proposition 2.12. The group G = GL(V ) acts transitively on the set of lat-tices, preserving the relation of homothety. Thus G acts on the set of lattice classes,and on the set of affine flags as well. In fact, G acts transitively on the set of affineflags of any given type. This action factors through PGL(V ) = GL(V )/Z(GL(V )).

Definition 2.13. The (reduced) Bruhat-Tits building of GL(V ) is the simpli-cial complex whose r-simplicies are the affine flags in V of length r, where one affineflag is a face of another affine flag if its sequence of lattice classes is a subsequenceof the other.

Theorem 2.14. Fix a lattice class [Λ] in V , and let Λ = Λ/pΛ be the resultingvector space over Fp. Let X[Λ] be the subset of the Bruhat-Tits building consisting ofaffine flags containing [Λ]. Then the geometry of X[Λ] coincides with the spherical

building of Λ over Fp.

Corollary 2.15. The (reduced) Bruhat-Tits building of GL2(Qp) is a (p+1)-regular tree.

20 3. THE STRUCTURE OF GL2(Qp)

3. Topological properties

We endow GLn(Qp) with the subspace topology from Mn(Qp) ∼= Qn2

p .

Definition 3.1. For each positive integer n, letG(n) denote the kernel of reduc-tion from GLn(Zp) to GLn(Z/pnZ). These are called the principal congruencesubgroups of GLn(Zp).

Proposition 3.2. For all n > 0, the principal congruence subgroup G(n) is acompact open subgroup of GLn(Qp).

Lemma 3.3. For sufficiently large integers e, the exponential power series expdefines a homeomorphism from peMn(Zp) onto G(n), whose inverse is given by thelogarithm power series.

Proposition 3.4. The principal congruence subgroups form a neighborhood

basis of 1 in GLn(Qp), where the latter is given the subspace topology from Qn2

p .

Theorem 3.5. Let U+ be the subgroup of GLn(Qp) consisting of upper-triangularmatrices with ones along the diagonal. Let U− be the analogous group of lower-triangular matrices. Let T be the maximal torus of GLn(Qp) consisting of diagonalmatrices. Then for all n > 0, multiplication provides a bijection

(G(n) ∩ U−)× (G(n) ∩ T )× (G(n) ∩ U+)→ J.

Proposition 3.6. The group GLn(Zp) is maximal among compact subgroups ofGLn(Qp). All maximal compact subgroups of GLn(Qp) are conjugate to GLn(Zp).

Theorem 3.7. If B is any Borel subgroup of GLn(Qp), then GLn(Qp) = B ·GLn(Zp). Just remember: G = BK.

Corollary 3.8. The quotient space G/B is compact.

Proposition 3.9. If U is a compact open subgroup of G = GLn(Qp), thenG/U is countable.

Theorem 3.10. Any measure on GLn(Qp) which is right-translation-invariantis also left-translation-invariant, and will be called a Haar measure. There is aunique Haar measure for which GLn(Zp) has measure one.

4. Parahoric subgroups, more decompositions

Definition 4.1. Let V be a finite-dimensional Qp vector space. Let [Λ•] bean affine flag in V . The parahoric subgroup associated to [Λ•] is the stabilizerin GL(V ) of the sequence of lattices Λ1 ⊃ Λ2 ⊃ · · · ⊃ Λr after choosing some set ofrepresentatives for each lattice class. An Iwahori subgroup subgroup of GL(V )is a parahoric subgroup stabilizing a maximal affine flag.

Iwahori subgroups are minimal parahorics. On the other hand, subgroups likeGLn(Zp) ⊂ GLn(Qp) are maximal parahorics.

In what follows, fixG = GLn(Qp), and the standard Borel, torus, and unipotentgroups B = T nU . Let K = GLn(Zp) be the maximal parahoric associated to thelattice Znp .

Proposition 4.2. There is a unique maximal compact subgroup T◦ of T , givenby those diagonal matrices with entries in Z×p . The valuation induces an isomor-phism:

valT : T/T◦ → Zn.

4. PARAHORIC SUBGROUPS, MORE DECOMPOSITIONS 21

For ~t = (t1, . . . , tn) ∈ Q×p , view ~t as a diagonal matrix, i.e., an element of T .The valuation is given by

valT (~t) = (val(t1), . . . , val(tn)).

Definition 4.3. A sequence (a1, . . . , an) of integers is called dominant ifa1 ≥ a2 ≥ · · · ≥ an. Let Tdom denote the set of ~t ∈ T such that val(~t) is dominant.

Proposition 4.4. For every ~t ∈ T , there exists w ∈ W = N/T such thatval(wtw−1) is dominant.

Theorem 4.5 (Polar decomposition). G = KTK. In fact, G = KTdomK

Now we specialize to standard lattices in GL2(Qp). The standard lattice isΛ◦ = Z2

p. Let the standard sublattice be the lattice Λ′◦ = Zp ⊕ pZp of index pin Λ◦. The standard lattice corresponds to the maximal compact subgroup K =GL2(Zp) and the pair Λ◦ ⊃ Λ′◦ corresponds to the Iwahori subgroup:

J =

{(a bc d

)∈ GL2(Zp) : c ∈ pZp

}.

Define

T◦ = T ∩K =

{(a 00 d

): a, d ∈ Z×p

}.

The valuation map identifies T/T◦ with Z2.The center of G is the subgroup Z of scalar matrices, and define

Z◦ = Z ∩K =

{(a 00 a

): a ∈ Z×p

}.

Definition 4.6. The affine Weyl group is the group N/ZT◦. It fits into ashort exact sequence

1→ T/ZT◦ → N/ZT◦ → N/ZT,

leading to an exact sequence

1→ Z→ W →W → 1.

Proposition 4.7. The group G decomposes as a disjoint union:

G =⊔w∈W

JwJ.

CHAPTER 4

Smooth representations

In this chapter, G = GL2(Qp) and B = TU are the standard Borel subgroup,torus, unipotent subgroup of G, respectively. Write N = NG(T ) and W = N/T forthe Weyl group.

Define K = GL2(Zp), and J for the standard Iwahori subgroup.

1. Fundamentals

Definition 1.1. A smooth representation is a pair (π, V ) where V is acomplex vector space, π : G→ GL(V ) is a group homomorphism, and for all v ∈ Vthere exists an open subgroup H ⊂ G such that v ∈ V H .

Definition 1.2. Let (π, V ) and (σ,W ) be smooth representaitons of G. Anintertwining operator from V to W , or simply a morphism of smooth represen-tations from V to W , is a C-linear map φ : V →W such that

φ ◦ π(g) = σ(g) ◦ φ

for all g ∈ G. We write HomG(V,W ) for the complex vector space of intertwiningoperators from V to W .

Proposition 1.3. The category of smooth representations of G and intertwin-ing operators is a cocomplete abelian category. In particular, kernels and cokernelsof intertwining operators among smooth representations are also smooth represen-tations, and arbitrary direct sums of smooth representations are smooth represen-tations.

Definition 1.4. Let (π, V ) be a smooth representation of G. A subrepre-sentation of (π, V ) is a subspace W ⊂ V such that π(g)w ∈ W for all g ∈ G,w ∈W .

Definition 1.5. Let (π, V ) be a smooth representation of G. We say that(π, V ) is decomposable if there exist subrepresentations W1,W2 ⊂ V such thatW1 ⊕W2 = V .

Definition 1.6. Let (π, V ) be a smooth representation of G. We say that(π, V ) is reducible if there exists a subrepresentation W ⊂ V such that W 6= 0and W 6= V .

Theorem 1.7 (Schur’s Lemma). Let (π, V ) be an irreducible representation ofG. Then HomG(V, V ) = C.

(See Cartier, Representations of p-adic groups, in Corvallis, p.118, for a proofbased on work of Jacquet. Assume that we know that G has a countable basis forits topology)

23

24 4. SMOOTH REPRESENTATIONS

2. Parabolic induction and corestriction

Recall the torus T ∼= F× × F×, contained in the Borel subgroup B, containedin G = GL2(Qp).

Proposition 2.1. The map B → T , sending

(a b0 d

)to

(a 00 d

)is a

surjective homomorphism with kernel U . Thus we identify T with B/U .

Definition 2.2. Let χ = (χ1, χ2) be a smooth character of T , i.e., a represen-tation of T of the form

χ

(a 00 d

)= χ1(a) · χ2(d),

for some χ1, χ2 ∈ Homc(Q×p ,C×). The pullback of χ to B is the character of Bdefined by

χ

(a b0 d

)= χ

(a 00 d

)= χ1(a) · χ2(d).

Definition 2.3. Let (σ,W ) be a smooth representation of B. The smoothlyinduced representation of G is the pair (ρ, IndGBW ) where IndGB(W ) is the spaceof functions f : G→W satisfying the following two properties:

(1) f(bg) = σ(b)f(g) for all b ∈ B, g ∈ G.(2) There exists an open subgroup H ⊂ G (depending on f) such that f(gh) =

f(g) for all g ∈ G, h ∈ H.

This is a smooth representation of G, by the map ρ : G→ GL(IndGBW ) given by:

[ρ(g)f ](x) = f(xg).

Definition 2.4. Let χ be a smooth character of T . The parabolically in-duced representation of G is the result of pulling χ back to a smooth character ofB and smoothly inducing to G.

In the other direction, we can begin with a smooth representation of G andobtain a smooth representation of T in two steps.

Proposition 2.5. Let (π, V ) be a smooth representation of G. Let VU =V/V (U) be the resulting Jacquet module. Then V (U) is a B-stable subspace of Vand thus VU acquires a natural action of B.

Definition 2.6. Let (π, V ) be a smooth representation of G. The paraboliccorestriction (often just called the Jacquet module) is the result of viewing VU asa representation of B, then restricting it to a representation of T .

Proposition 2.7. If (π, V ) is a smooth representation of G, then VU is asmooth representation of T .

Theorem 2.8 (Frobenius reciprocity). Let (π, V ) be a smooth representationof G. Let (σ,W ) be a smooth representation of T . Then there is a natural C-linearisomorphism:

HomG(IndGBW,V ) ∼= HomT (W,VU ).

3. Principal series

4. Unitarity

5. Intertwining operators

CHAPTER 5

Hecke algebras and Langlands perspective

1. Spherical representations

2. Iwahori-spherical representations

3. Langlands parameters

25

Bibliography

[Rod77] Francois Rodier, Decomposition spectrale des representations lisses, Non-commutative

harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Springer, Berlin, 1977,pp. 177–195. Lecture Notes in Math., Vol. 587. MR 0460549 (57 #542)

[Ser73] J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973, Translated from

the French, Graduate Texts in Mathematics, No. 7. MR 0344216 (49 #8956)[Wei95] Andre Weil, Basic number theory, Classics in Mathematics, Springer-Verlag, Berlin, 1995,

Reprint of the second (1973) edition. MR 1344916 (96c:11002)

27

Index

character, 1

characteristic function, 4

coinvariants, 1coisotypic, 1

congruence subgroup, 7

distribution

on Qp, 5

intertwining operator, 6invariants, 1

isoytpic, 1

Jacquet module, 5

twisted, 5

p-adic integer, 2

p-adic number, 3

representation, 1

Schwartz space, 4

valuation, 3

29