Representation theory

Chapter IV: The endomorphism algebras, GL h and representation theory

4.2 Representation theory

We start this section with a useful result that expresses the cohomology of certain rigid analytic spaces in terms of the cohomology of a subspace.

Proposition 4.2.1. Let X be a rigid analytic space with an action of the group G, and letY be a subspace of X on which the subgroup H ofGacts. Suppose

(i) H is normal inG,

(ii) ^G_H is a finite abelian group, (iii) X is the disjoint union

X = a

g∈^G_H

g(Y).

Then

H_cⁱ(X,Ql) Ind^G_H H_cⁱ(Y,Ql) asG-representations.

Proof. We first prove the result in the case that ^G_H is cyclic of orderm, generated by g0 ∈G. In this case, (iii) becomes

X =Y tg₀(Y)t · · · tg₀^m−1(Y).

So we have

H_cⁱ(X,Ql) Mm−1

j=0

H_cⁱ(Y,Ql)

asQl-vector spaces. Letx_k be the element ofL_m−1

j=0 H_cⁱ(Y,Ql)withk-th coordinate equal to xand which is 0 everywhere else. We pick the above isomorphism so that

g₀· x_k = x_k+1, 0 ≤ k < m−1.

Define

f : Ind^G_H H_cⁱ(Y,Ql)→ H_cⁱ(X,Ql) Mm−1

j=0

H_cⁱ(Y,Ql)

g_o^kx7→ x_k.

We will show that this is an isomorphism of representations.

Letg ∈G. Then we can write

g= hg₀^l withh∈ H, 0 ≤ l ≤ m−1.

So it suffices to show that f is compatible with both the actions ofh ∈H and ofg₀. For 0 ≤ k < m−1,

g₀· (g₀^kx) =g₀^k+1x7−→^f x_k+1 =g₀· x_k =g₀· f(g₀^kx)

and sinceg₀^m ∈ H,

g₀·(g₀^m−1x)= g₀⁰(g₀^mx) 7−→^f (g₀^m · x)₀= g₀^m · x₀=g₀·x_m−1 =g₀· f(g₀^m−1x).

And forh∈ H, we havehg₀^k =g₀^kh⁰for someh⁰∈ H sinceH is normal inG, so

h·(g₀^kx)= g₀^k(h⁰·x) 7−→^f (h⁰·x)_k =g₀^k·(h⁰·x)₀ = (g₀^kh⁰)·x0= (hg₀^k)·x0= h·xk = h·f(g₀^kx).

So f is an isomorphism ofG-representations.

Now, for the general case where ^G_H is finite abelian, we can write ^G_H as a direct product ^G_H = ^G_H¹×^G_H²× · · · ×^G_Hⁿ, where eachG_jis a subgroup ofGcontainingHand

each^G_H^j is finite cyclic. LetH_j =G1· · ·G_j. Since ^G_H is abelian,g_j1G_j2 =G_j2g_j1 for allg_j1 ∈ G_j1, 1 ≤ j1,j2 ≤ n, so H_j is a subgroup ofG and H_j−1 is normal in H_j. Furthermore, we haveHn= Gand _H^Hj

j−1 = _G^G₁¹_···G^···G_j−1^j _(G1···G^G_j−1^j _)∩Gj = ^G_H^j. DefineXj

recursively by X₀ =Y and

X_j = a

gj∈ ^Hj

Hj−1

g_j(X_j−1).

(iii) implies that the above is indeed a disjoint union for all 1 ≤ j ≤ n, and that X = X_n. Applying the previous case repeatedly shows that

H_cⁱ(X,Ql) Ind^H_Hⁿ

n−1Ind^H_Hⁿ⁻¹

n−2· · ·Ind^H_H¹

0 H_cⁱ(Y,Ql) Ind^G_H H_cⁱ(Y,Ql).

Next, we include a result that will be needed later in Section 6.3 to show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower realizes the local Langlands and the Jacquet-Langlands correspondences. This result is useful as local Langlands and Jacquet-Langlands both behave well with respect to twists.

Lemma 4.2.2. (a) Let (ρ,V) be an irreducible representation of E^×, and let ι : D^× → E^× be the isomorphism

D^{× ι}

//

 _

E^×_{ _}

GLh(L_h)

φ7→(φ^T)⁻¹

//GLh(L_h).

Thenθ^∗

Res_θ(D^E× ×) ρ

ι^∗ρ⊗(χι^∗ρ◦Nrd)^∨, where χι^∗ρis the central character ofι^∗ρ, andNrd :D → Lis the reduced norm map of the division algebraD.

(b) Letπbe an irreducible representation ofGL_h(L), and let jbe the isomorphism j :GL_h(L) −→ GL_h(L)

g 7→ (g^T)⁻¹. Then ψ^∗

Res^GL_ψ(GL^h(L)

h(L))π

j^∗π ⊗ (χ_j^∗_π ◦ det)^∨, where χ_j^∗_π is the central character of j^∗π.

Proof. We will prove (a). The proof of (b) is almost identical.

First, note that, by Lemma 4.1.1, for our embedding of D^× into GL_h(L_h), the determinant map onGL_h(L_h), when restricted to D^×, has image inL, and is in fact the reduced norm map Nrd :D^× → L. In other words, the diagram

D^×_{ _} ^{Nrd //}

L_{ _}

GL_h(L_h)

det // L_h commutes.

Forφ ∈ D^×,v ∈V,

((θ^∗Resρ)(φ))(v)= (ρ((detφ)(φ^T)⁻¹))(v) = (ι^∗ρ((detφ)⁻¹φ))(v).

Letι^∗ρ⊗(χι^∗ρ◦Nrd)^∨act on the vector spaceV ⊗

Ql Hom

Ql(Ql,Ql). Consider the isomorphism

V →V ⊗

Ql Hom

Ql(Ql,Ql)

αv 7→ v⊗ fα (where fα : c7→αc).

Forφ ∈ D^×,

(((χι^∗ρ◦Nrd)^∨(φ))(fα))(c)= fα((χι^∗ρ◦Nrd)(φ⁻¹)(c))

= fα(ι^∗ρ(detφ⁻¹)(c))

= α(ι^∗ρ(detφ⁻¹))(c),

so((χι^∗ρ◦Nrd)^∨(φ))(fα)= f_α(ι^∗_ρ(detφ⁻¹₎₎, and

(ι^∗ρ(φ))(v)⊗ ((χι^∗ρ◦Nrd)^∨(φ))(f1) = (ι^∗ρ(φ))(v) ⊗ f_ι^∗_ρ(detφ⁻¹₎ 7→ (ι^∗ρ((detφ)⁻¹φ))(v),

as required.

C h a p t e r 5

THE DUALITY MAP AND THE EXTERIOR POWER MAP ON MODULI SPACES

5.1 Definitions of the duality map and the exterior power map

In this section, we will use the Serre dual and the exterior power of a π_L-divisible O_L-module to define maps from LT_h^∞ toM_h^∞.

We start with the Serre dual. Note that G^∨_m,n Gm,n, so G^∨_1,h−1 Gh−1,1 has dimensionh−1 and heighth. Let us fix such an isomorphism. The dual of aπ_L- divisibleO_L-moduleX has Tate module(Tπ_LX)^∨(1). In particular, the Tate module of the dual of (O_L/πⁿ_LO_L)^h is equal toO^h

L(1). Over ˘L(ζp^∞), we haveO^h

L(1) O^h

L

since ˘L(ζ_p^∞)contains thep^∞ roots of unity.

Definition 5.1.1. Define∨: LT_h^∞×_L˘ L(ζ˘ _p^∞)→ M_h^∞×_L˘ L˘(ζ_p^∞)by (X, β, α) 7→ (X^∨,(β^∨)⁻¹,(α^∨_ζ

pn)⁻¹) whereα^∨_ζ

pn is given by the composition

TπLX^∨−−−−−−→ (T_πLX)^∨(1)−−−−−−→ O _L^h(1) −−−−−−→ O _L^h.

Let us now look at the exterior power. Recall thatVh−1G1,h−1 G_h−1,1. We will fix such an isomorphism.

Definition 5.1.2. DefineVh−1 :LT_h^∞ → M_h^∞by (X, β, α) 7→*

,

h−1

^X,

h−1

^β,

h−1

^α+ - .

In the above, by a slight abuse of notation,Vh−1αis the level structure onVh−1X given by

O_L^h −−−−−−→^φ

^h−1

O_L^h

Vh−1α

−−−−−−→

^h−1

T_pX −−−−−−→

T_p*

,

^h−1

X+ - ,

whereφis the isomorphism

v_i 7→ (−1)ⁱv₁∧v₂∧ · · · ∧v_i−1∧v_i+1∧ · · · ∧v_h, v₁, . . . ,v_hbeing the standard basis vectors forO^h

L.

5.2 Properties of the exterior power map and group actions

In order to study the cohomology of the dual Lubin-Tate tower using the exterior power mapVh−1, it is necessary to first understand how the group actions behave with respect toVh−1.

Recall from Section 4.1 that we have embeddings of D^× and E^× in GL_h(L_h). We shall view D^× andE^× as subgroups ofGL_h(L_h) using these embeddings. We also recall the maps

θ :D^× → E^× ψ :GL_h(L)→ GL_h(L)

φ7→ (detφ)(φ⁻¹)^T, g7→ (detg)(g⁻¹)^T. Proposition 5.2.1. The mapV_h−1

:LT_h^∞ → M_h^∞is (a) D^×-equivariant if we letD^× act on M_h^∞ viaθ,

(b) GL_h(L)-equivariant if we letGL_h(L)act on M_h^∞ viaψ, (c) W_L-equivariant.

Proof.

(a) Supposeφ ∈ D^× is given by

φe_i =

h−1

X

j=0

α_jie_j.

Then

* ,

h−1^ φ+

-

e˜_i =(−1)ⁱφe0∧φe1∧ · · · ∧φe_i−1∧φe_i+1∧ · · · ∧φe_h−1

=(−1)ⁱ

h−1

X

j=0

α_j0e_j∧ · · · ∧

h−1

X

j=0

α_j,i−1e_j ∧

h−1

X

j=0

α_j,i+1e_j∧ · · · ∧

h−1

X

j=0

α_j,h−1e_j

= Xh−1

k=0

β_k(e₀∧e1∧ · · · ∧ek−1∧ek+1∧ · · · ∧eh−1)

for some β_k ∈ L_h. To write down an expression for β_k, it is helpful to first introduce the order-preserving bijections

b_l : {0,1, . . . ,l−1,l+1, . . . ,h−1} → {1,2, . . . ,h−1}

j 7→











j +1 if j < l, j if j > l.

Forτ ∈S_h−1, letτ_ik =b⁻¹_k ◦τ◦b_i. Then β_k = (−1)ⁱ X

τ∈Sh−1

sgn(τ)ατ_ik(0),0· · ·ατ_ik(i−1),i−1ατ_ik(i+1),i+1· · ·ατ_ik(h−1),h−1.

So the coefficient of ˜e_k in V_h−1φ

˜

e_i is given by

(−1)^i+kdet(φwith row and column containing αk,ideleted)

=C_k,i, the cofactor of the elementa_k,iofφ.

So

h−1

^φ =

* . . . . . . . ,

C0,0 C0,1 · · · C0,h−1

C1,0 C1,1 · · · C1,h−1

... ... . . . ...

C_h−1,0 C_h−1,1 · · · C_h−1,h−1 + / / / / / / / -

= (detφ)(φ⁻¹)^T.

(b) It suffices to prove the proposition forg ∈GL_h(L)∩Mat_h(O_L). For suchg, it acts onLT_h^∞ by

(X, β, α) 7→ (Y,q◦ β, α)

whereY,q, αare as defined in Section 3.1. Recall thatY^rig,q^rigandα^rig(which corresponds toα^rig :O_L^h −→ Tπ_LY^rig)satisfy the commutative diagram

(L/O_L)^{h αrig}

//

g

X^rig

q^rig

(L/O_L)^{h αrig}

//Y^rig = ^Xrig_αrig(ker(g)).

Applying the functorVh−1to the above diagram gives Vh−1(L/O_L)^h

Vh−1α^rig

//

Vh−1g

Vh−1X^rig

Vh−1q^rig

V_h−1(L/O_L)^h

Vh−1α^rig

//V_h−1

Y^rig.

By an argument similar to that in (a), the following diagram commutes:

O_L^h

ψ(g)=(detg)(g⁻¹)^T

//Vh−1O_L^h

Vh−1g

O_L^h

//Vh−1O_L^h.

SinceV_h−1α^rigcorresponds to(V_h−1α)^rig, using the above 2 diagrams, we get the commutative diagram

O^h

L

Vh−1α

//

ψ(g)

Tp

V_h−1

X

Vh−1q

O^h

L

Vh−1α

//T_pVh−1Y .

Henceψ(g)acts on M_h^∞ by

* ,

^h−1

X,

h−1^ β,

^h−1

α+ -

7→*

,

^h−1

Y,

h−1^ q◦

h−1^ β,

^h−1

α+ - ,

as required.

(c) Letw ∈WL. By the universal property of the exterior power, we have F₍Vh−1G1,h−1)/Fp =

h−1^

F_G1,h−1/

Fp, and

T_p* ,

^h−1

X+ -

=

h−1^ T_pX.

The first equality shows that (Vh−1β)^w = Vh−1β^w, and the second gives (V_h−1α)^w =V_h−1α^w, so the mapV_h−1

: LT_h^∞ → M_h^∞isW_L-equivariant.

Proposition 5.2.1 tells us that the exterior power mapVh−1isW_L-equivariant. Unlike Vh−1however, the duality map∨is notW_L-equivariant, one of the reasons being that the Tate module of the dual of(L/O_L)^hisO^h

L(1). As such, it is difficult to understand theWL-action on M_h^∞ by considering the duality map. However, the duality map∨ can still be used to study the cohomology of M_h^∞as aE^××GL_h(L)-representation, and hence can be used to show that the supercuspidal part of the cohomology ofM_h^∞ in the middle degree realizes the Jacquet-Langlands correspondence up to certain twists.

C h a p t e r 6

THE GEOMETRY AND COHOMOLOGY OF THE DUAL LUBIN-TATE TOWER

6.1 Geometry of the dual Lubin-Tate tower

Before looking at the tower, it is helpful to first understand the level 0 situation. Let us writeLT_hfor the formal schemeM¹

h, andM_hforM^h−1

h .

Proposition 6.1.1. The mapVh−1 :LT_h → M_hinduces an isomorphism

h−1

^: (LT_h)₀→ (M_h)₀.

Proof. By Theorem 3.1.1, we have a non-canonical isomorphism (LT_h)₀≈SpfO_L˘[[t₁, . . . ,t_h−1]],

which shows, by considering the duality map∨, that (M_h)₀≈SpfO_L˘[[T₁, . . . ,T_h−1]].

Let P = (P,Q,F,V⁻¹) be the O_L-display corresponding to the π_L-divisible O_L- module G1,h−1, so that P = D(G1,h−1) and Q = V D(G1,h−1) with F and V⁻¹ as given in the definition of D(Gm,n). Let S ∈ C, and let SpecA ⊆ S be an open affine subset of S. By Grothendieck-Messing theory (cf. [Mes72]), π_L-divisible O_L-modules over AliftingG1,h−1correspond to s.e.s.

0−→ M −→ P⊗_O˘

L A−→ N −→0 of A-modules lifting the Hodge filtration

0−→ Q

πLP −→ P

πLP −→ P

Q −→0.

Since Q

π_LP = V D(G1,h−1)

π_LD(G₁,h−1) = hV e0, . . . ,V e_h−1i

hπ_Le₀, . . . , π_Le_h−1i = hπ_Le0,e1, . . . ,e_h−1i hπ_Le₀, . . . , π_Le_h−1i, the above condition is equivalent to

M

m_AM = Q

π_LP = he1, . . . ,e_h−1i hπ_Le₁, . . . , π_Le_h−1i.

Applying Nakayama’s lemma, we see thatM must be of the form he1+t1π_Le0, . . . ,e_h−1+t_h−1π_Le0i for somet1, . . . ,th−1∈ A.

ApplyingVh−1to the s.e.s.

0−→ he₁+t₁π_Le₀, . . . ,e_h−1+t_h−1π_Le₀i −→ he₀, . . . ,e_h−1i −→ he₀i −→0, (6.1) we get the s.e.s.

0−→ h(e1+t1π_Le0)∧· · ·∧(e_h−1+t_h−1π_Le0)i −→ he˜0, . . . ,e˜_h−1i −→ he˜1, . . . ,e˜_h−1i −→0 where

(e₁+t₁π_Le₀)∧ · · · ∧(e_h−1+t_h−1π_Le₀)= e˜₀−t₁π_Le˜₁−t₂π_Le˜₂− · · · −t_h−1π_Le˜_h−1. Let us rewrite (6.1) as follows:

0−→ hf₁, . . . , f_h−1i −→^ρ he₀, . . . ,e_h−1i −→^τ hg₀i −→0 f_i 7→ e_i+t_iπ_Le₀

e₀ 7→ g₀

e_i 7→ −t_iπ_Lg_i, i ,0.

Applying∨, we get the s.e.s.

0−→ hg^∨₀i ^τ

∨

−−→ he₀^∨, . . . ,e^∨_h−1i ^ρ

∨

−−→ hf₁^∨, . . . , f_h−1^∨ i →0 where

(τ^∨g₀^∨)(e_i)= g^∨₀(τe_i)= 









g₀^∨(g₀)= 1, ifi= 0, g₀^∨(−tiπLg₀) =−tiπL, ifi, 0, (ρ^∨e_i^∨)(f_j)= e^∨_i (ρf_j)= e^∨_i (e_j +t_jπ_Le₀) = 







1, ifi = j, 0, ifi , j. So the above s.e.s. is the same as

0−→ he^∨₀−t₁π_Le^∨₁− · · · −t_h−1π_Le^∨_h−1i −→ he^∨₀, . . . ,e^∨_h−1i −→ he^∨₁, . . . ,e^∨_h−1i −→0.

Therefore we have a commuting diagram

(M_h)₀≈ SpfO_L˘[[T₁, . . . ,T_h−1]]

(LT_h)₀≈SpfO_L˘[[t₁, . . . ,t_h−1]]

Vh−1 22

∨

,,

(M_h)₀ ≈SpfO_L˘[[S₁, . . . ,S_h−1]]

T7→i

Si

OO

which shows that V_h−1 induces an isomorphism from (LT_h)₀ to (Mh)₀ since the duality map∨: LT_h→ M_hclearly does.

Corollary 6.1.2. The mapV_h−1

: LT_h → M_hinduces an isomorphism

^h−1

: (LT_h)_m → (M_h)_(h−1)m for anym∈Z.

Proof. We recall the definition of the mapθ : D^× → E^×. Using the above embed- dings of D^× andE^×intoGL_h(L_h), we haveθ(φ) = (detφ)(φ⁻¹)^T.

Letm∈Z. Fix someφ ∈D^×with val(φ) = m, then val(θ(φ))= v_L(det(θ(φ))) = v_L

det

(detφ)(φ⁻¹)^T

= v_L((detφ)^h−1) = (h−1)v_L(detφ)= (h−1)m.

So the action of φinduces an isomorphism from(LT_h⁰)₀to (LT_h⁰)_m, and the action ofθ(φ) induces an isomorphism from(M_h⁰)0to(M_h⁰)_(h−1)m.

By Proposition 5.2.1, we have the following commutative diagram:

(LT_h⁰)₀

Vh−1

//

φ

(M_h⁰)₀

θ(φ)

(LT_h⁰)_m

Vh−1 //(M_h⁰)_(h−1)m

where the top map is an isomorphism by Proposition 6.1.1. HenceVh−1induces an isomorphism from (LT_h⁰)_m to (M_h⁰)_(h−1)m, as desired.

Proposition 6.1.3. For anyn ≥ 0,m,n∈Z, each connected component of (LT_hⁿ)_m is mapped isomorphically onto some connected component of(M_hⁿ)_(h−1)m under the mapVh−1 :LT_h^∞ → M_h^∞.

Proof. By Proposition 6.1.1,Vh−1induces an isomorphism on the formal schemes Vh−1: (LT_h)0→ (M_h)0, hence an isomorphism on their generic fibers.

By considering the action ofO_D^× on the connected components of (LT_hⁿ)₀, we see that each connected component of (LT_hⁿ)0 is finite étale over (LT_h⁰) of the same degree. Using duality, we see that each connected component of(M_hⁿ)0is also finite étale over(M_h⁰)₀of the same degree. Since

(LT_hⁿ)₀

Vh−1

//

finite étale

(M_hⁿ)₀

finite étale

(LT_h⁰)0

Vh−1 // (M_h⁰)0

commutes, this shows that V_h−1 induces an isomorphism from each connected component of(LT_hⁿ)₀to some connected component of(M_hⁿ)₀.

Now, by considering the action of the groupsD^×andE^×, we see that, for anym ∈Z, the mapVh−1induces an isomorphism from each connected component of(LT_hⁿ)_m to some connected component of(M_hⁿ)_(h−1)m.

Corollary 6.1.4. Suppose(p(q−1),h−1)= 1. ThenVh−1: LT_h^∞ → M_h^∞induces an isomorphism

(LT_hⁿ)_m → (M_hⁿ)_(h−1)m for anyn≥ 0,m,n∈Z.

Proof. We first consider the casem= 0. Since(p(q−1),h−1)= 1, by Proposition 4.1.4, the mapθ0,nis bijective. By Theorem 3.2.4, this means that the map induced by Vh−1 on the connected components is a bijection. Together with Proposition 6.1.3, this shows that Vh−1 induces an isomorphism from (LT_hⁿ)0 to (M_hⁿ)0. The result then follows for all m ∈ Z by considering the action of the groups D^× and E^×.

6.2 Cohomology of the dual Lubin-Tate tower

In this section, we will reinterpret the results of the previous section in terms of the cohomology of the dual Lubin-Tate tower. In order to avoid unnecessarily cumbersome notation, from here on, all cohomology is understood to meanl-adic cohomology withQlcoefficients, wherel , pis an odd prime. We will also slightly abuse notation, and whenever the Lubin-Tate tower or the dual Lubin-Tate tower appears, we actually mean its change base from ˘LtoC.

Lemma 6.2.1. The kernelker(θ) acts trivially onLT_h⁰, so the action ofD^× onLT_h⁰ induces an action ofθ(D^×) = Im(θ) ≤ E^× onLT_h⁰. This action can be extended to an action ofE^θ = hO^×

Lh,Im(θ)iby lettingO^×

Lh act trivially onLT_h⁰. Proof. Supposek ∈ker(θ). The action ofkon LT_h⁰is given by

(X, β) 7→ (X, β◦k⁻¹).

But k ∈ ker(θ) = µ_h−1(O_L) ⊂ O^×

L and X has multiplication by O^×

Lh, so (X, β) = (X, β◦ k⁻¹). So k acts trivially on LT_h⁰, and we get an action of Im(θ) on LT_h⁰ induced by the action ofD^×.

To see that we can extend the above action of Im(θ)to an action ofE^θ = hO^×

Lh,Im(θ)i whereO^×

Lhacts trivially, we just need to check thatO^×

Lh∩Im(θ) ⊆ Im(θ)acts trivially onLT_h⁰, but this is clear.

Proposition 6.2.2. LetE^θ ≤ E^×act on LT_h⁰as in Lemma 6.2.1. Then for alli ≥ 0, H_cⁱ(M_h⁰) Ind^E_E^×θ H_cⁱ(LT_h⁰)

asE^××W_L representations.

Proof. The proposition is clear fori > 0 since H_cⁱ(LT_h⁰) = 0 = H_cⁱ(M_h⁰) fori > 0.

Consideri =0. LetY₀be the image of

h−1

^: LT_h⁰→ M_h⁰. By Proposition 5.2.1, and the fact thatO^×_L

h acts trivially on M_h⁰, we have H_c⁰(Y₀) H_c⁰(LT_h⁰)

asE^θ×W_L-representations, where the action ofE^θonLT_h⁰is as described in Lemma 6.2.1.

By Proposition 4.1.3,E^θ = {ϕ∈ E^× : (h−1)|val(ϕ)}and a full set of representatives forE^×/E^θ is given by($⁰)^k for k ∈ {0,1, . . . ,h−2}. It is clear that

M_h⁰=Y0t$⁰(Y0)t · · · t$^0h−2(Y0).

So by Proposition 4.2.1,

H_cⁱ(M_h⁰) Ind^E_E^×θ H_cⁱ(LT_h⁰) asE^××W_L representations, as desired.

We now prove an analogous result forH_cⁱ(M_h^∞).

Theorem 6.2.3. Letd = (q−1,h−1),e=v_p(h−1)andcbe the maximum integer

≤ esuch thatO_Lcontains thep^croots of unity. Letθ∗H_cⁱ(LT_h^∞)be the pushforward of the representationH_cⁱ(LT_h^∞)under the map

θ :D^× θ(D^×).

Then for alli ≥ 0, ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_h^∞)

Ind^E_θ(D^× ×)θ∗H_cⁱ(LT_h^∞) Ind^E_θ(D^× ×)θ∗* ,

H_cⁱ(LT_h^∞) K +

- Ind^E_θ(D^× ×)θ∗H_cⁱ(LT_h^∞)^K

as E^× × GLh(L) × WL representations, where K = ker(θ₀) = µ_pcd(O_L) and H_cⁱ(LT_h^∞)^K is the subrepresentation ofH_cⁱ(LT_h^∞)fixed byK.

Proof. LetY_nbe the image of

^h−1

: LT_hⁿ→ M_hⁿ.

By Lemma 4.1.2 (a) and Proposition 4.1.4 (c), forn > 2ee_L, the kernel ofθ0,ncan be written as a direct product

ker(θ0,n)= K_n,1⁰ ×K_n,2⁰ , and the det mod(1+π_LⁿO_L) map gives an isomorphism

K = ker(θ₀)= µ_pcd(O_L) ^{det mod(1+π}

n LO_L)

−−−−−−−−−−−−−−→

K_n,1⁰ .

LetK_n = 1+π^n−ee_L ^LO_L ≤ D^×. Proposition 4.1.4 (c) tells us that we have an exact sequence

1−→1+πⁿ_LO_L −→ K_n ^{det mod(1+}

π_LⁿO_L)

−−−−−−−−−−−−−−→ K_n,2⁰ −→1.

Recall from Proposition 6.1.3 thatV_h−1

maps each connected component of(LT_hⁿ)_m isomorphically onto some connected component of(M_hⁿ)_(h−1)m. Since

K ×K_n

det mod(1+πⁿ_LO_L)

−−−−−−−−−−−−−−−− K_n,1⁰ ×K_n,2⁰ =ker(θ0,n),

by Theorem 3.2.4, two connected components ofLT_hⁿwill each be mapped isomor- phically to the same connected component of M_hⁿunder theVh−1 map if and only

if they are in the same orbit ofK×K_n. Furthermore, 1+π_LⁿO_L ≤ D^×acts trivially onLT_hⁿ, so

ψ^∗H_cⁱ(Y_n) θ∗* ,

H_cⁱ(LT_hⁿ) K×K_n +

- asθ(D^×)×GL_h(L)×W_L-representations.

Claim:

H_cⁱ(M_hⁿ) Ind^E_E^×θInd_θ(^Eθ_D×)H_cⁱ(Y_n) Ind^E_θ(D^× ×) H_cⁱ(Y_n).

Proof of claim: LetX_nbe given by X_n= a

m∈Z

(M_hⁿ)_(h−1)m.

Note thatX_ncontainsY_nas a subspace. By Propositions 4.1.3 (d) and 4.1.4 (b), E^θ

Imθ cokerθ₀= O^×

E

Imθ₀

det mod(1+πⁿ_LO_L)

−−−−−−−−−−−−−−→

(O_L/πⁿ_LO_L)^×

Imθ0,n = cokerθ0,n

is a finite abelian group.

D^× acts transitively on the connected components of LT_hⁿ, so for any ϕ < Im(θ), ϕ(Y_n)will be disjoint fromY_n. Furthermore, the orbit ofY_nunderO^×

EisX_nsinceO^×

E

acts transitively on the connected components of (M_hⁿ)_(h−1)m for anym ∈ Z. This shows that

X_n= a

ϕ∈_Im(θ)^Eθ

ϕ(Y_n).

By Proposition 4.2.1, we have

H_cⁱ(X_n) Ind^E_θ(D^θ ×)H_cⁱ(Y_n).

Recall from Proposition 4.1.3 (d) that the valuation map gives an isomorphism E^×

E^θ Z (h−1)Z. So a full set of representatives for ^E×

E^θ is{1, $⁰, $⁰², . . . , $^0h−2}, and it is clear that M_hⁿ = X_nt$⁰(X_n)t · · · t$^0h−2(X_n).

So, again, by Proposition 4.2.1,

H_cⁱ(M_hⁿ) Ind^E_E^×θ H_cⁱ(X_n).

This proves the claim.

The map

H_cⁱ(LT_h^n−eeL) → H_cⁱ(LT_hⁿ) factors through ^H

ic(LT_hⁿ)

Kn since K_n = 1+ π^n−ee_L ^LO_L acts trivially on LT_h^n−eeL. So lim−−→n

Hⁱ_c(LT_hⁿ)

Kn = H_cⁱ(LT_h^∞), and lim−−→

n

H_cⁱ(LT_hⁿ)

K ×K_n = H_cⁱ(LT_h^∞) K .

It remains to show that ^H

ci(LT_h^∞)

K H_cⁱ(LT_h^∞)^K. Formsufficiently large, the determi- nant of the elements ofK are distinct mod(1+π^m_LO_L), so the orbits of the induced action of K on the connected components of LT_h^m each have size d. We define a subspaceZ_mofLT_h^m by taking one connected component from each orbit. Let kbe a generator ofK. ThenLT_h^m is the disjoint union

LT_h^m = Z_mtk(Z_m)t · · · tk^d−1(Z_m), so we have an identification

H_cⁱ(LT_h^m)

d−1

M

j=0

H_cⁱ(Z_m)

of Ql-vector spaces such that the action of K on Ld−1

j=0H_cⁱ(Z_m) is given by k · (z₀,z₁, . . . ,z_d−1)= (z_d−1,z₀, . . . ,z_d−2). Define an isomorphism of vector spaces

H_cⁱ(LT_h^m) K

−−→ H_cⁱ(LT_h^m)^K [(z₀, . . . ,z_d−1)]7−→ (z, . . . ,z), where z = _d¹P_d−1

j=0 z_j. Using the fact that K is in the center of D^×, an easy computation shows that the above is also an isomorphism of representations.

Putting these together, we have ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_hⁿ)

Ind^E_θ(D^× ×)ψ^∗H_cⁱ(Y_n) Ind_θ(^E×_D×)θ∗* ,

H_cⁱ(LT_hⁿ) K ×K_n +

- ,

and ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_h^∞)

Ind^E_θ(D^× ×)θ∗* ,

lim−−→

n

H_cⁱ(LT_hⁿ) K×K_n +

-

Ind^E_θ(D^× ×)θ∗* ,

H_cⁱ(LT_h^∞) K +

- Ind^E_θ(D^× ×)θ∗H_cⁱ(LT_h^∞)^K,

as desired.

Corollary 6.2.4. Suppose(q−1,h−1)= 1. There is an action ofθ(D^×) ⊂ E^×on H_cⁱ(LT_h^∞)induced from the action ofD^×. Then for alli ≥ 0,

ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_h^∞)

Ind_θ(^E×_D×)H_cⁱ(LT_h^∞) asE^××GL_h(L)×W_L representations.

Just like for LT_h^∞, we will consider the functor

H_cⁱ(M_h^∞) : RepE^× →Rep(GL_h(L)×W_L) ρ7→Hom_E^×(H_cⁱ(M_h^∞), ρ).

The following is an easy corollary of Theorem 6.2.3.

Corollary 6.2.5. For alli ≥ 0, ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_h^∞)[ρ]

= H_cⁱ(LT_h^∞)f θ^∗

Res_θ(^E×_D×) ρg .

Proof. By Theorem 6.2.3 and Frobenius reciprocity, ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_h^∞)[ρ]

= Hom_E^× ψ^∗

Res^GL_ψ(GL^h(L)

h(L))H_cⁱ(M_h^∞) , ρ

= Hom_E^×* ,

Ind^E_θ(D^× ×)θ∗* ,

H_cⁱ(LT_h^∞) K +

- , ρ+

-

= Hom_θ(D^×₎* ,

θ∗* ,

H_cⁱ(LT_h^∞) K +

-

, Res_θ(D^E× ×) ρ+ -

= Hom_D^×* ,

H_cⁱ(LT_h^∞) K , θ^∗

Res^E_θ(D^× ×) ρ + - .

SinceK acts trivially onθ^∗

Res^E_θ(D^× ×) ρ

, the above is equal to Hom_D^×

H_cⁱ(LT_h^∞), θ^∗

Res^E_θ(D^× ×) ρ = H_cⁱ(LT_h^∞)f θ^∗

Res^E_θ(D^× ×) ρg .

6.3 Geometric realization of local correspondences and vanishing results In this section, we will use Corollary 6.2.5 and results on the cohomology of the Lubin-Tate tower to deduce results for the cohomology of the dual Lubin-Tate tower.

In particular, we will show that the supercuspidal part of the cohomology of the dual Lubin-Tate tower in the middle degree realizes the local Langlands and the Jacquet-Langlands correspondences (up to appropriate twists).

Theorem 6.3.1. Forπ ∈Cusp(GL_h(L)),

H_c^h−1(M_h^∞)_cusp(JL⁻¹(π)) =π ⊗rec(π^∨⊗ (χπ◦det)⊗ (| · | ◦det)^h−12 )(h−1), where χπis the central character ofπandH_c^h−1(M_h^∞)_cuspis the supercuspidal part ofH_c^h−1(M_h^∞).

Proof. By Corollary 6.2.5, we have the commutative diagram

Irr⁰E^×

ρ7→θ^∗ Res^E_θ(D^×_×

)ρ

//

H_cⁱ(M_h^∞)cusp

Irr⁰D^×

H_cⁱ(LT_h^∞)cusp

Cusp(GL_h(L))×Irr(W_L)

π⊗r7→ψ^∗

Res^{G Lh}_ψ( ^(L)

G Lh(L))π

⊗r

//Cusp(GL_h(L))×Irr(W_L)

where Irr⁰D^× ⊂ IrrD^× is the subset consisting of representations of the form JL⁻¹(π) for someπ ∈Cusp(GL_h(L))(similarly for Irr⁰E^×).

Note that

• θ^∗

Res^E_θ(D^× ×) ρ

∈Irr⁰D^× since

ρ= JL⁻¹(π)withπ ∈Cusp(GL_h(L))

⇒ ρ⊗(χπ◦Nrd)^∨ =JL⁻¹(π ⊗(χπ◦det)^∨),

• π ∈Cusp(GL_h(L)) since ψ^∗

Res^GL_ψ(GL^h(L)

h(L))π

= π ⊗(χ_π◦det)^∨ ∈Cusp(GL_h(L)).

The commutative diagram only gives us H_cⁱ(M_h^∞)_cusp[ρ] as a ψ(GL_h(L)) ×W_L- representation. But by considering the duality map ∨ : LT_h^∞ → M_h^∞, we know that H_cⁱ(M_h^∞)_cusp realizes the Jacquet-Langlands correspondence, so we already understand H_cⁱ(M_h^∞)_cusp[ρ] as aGL_h(L)representation.

Let us define

F : Irr⁰E^× →Cusp(GL_h(L))×Irr(W_L)

JL⁻¹(π) 7→π ⊗rec(π^∨⊗ (χπ◦det)⊗ (| · | ◦det)^h−12 )(h−1).

By the above, it will suffice to check thatFmakes the diagram commute. Applying Theorem 3.2.1, and using the fact that JL and rec are compatible with twists, we see that H_c^h−1(LT_h^∞)cusp

fρ⊗ (χρ◦Nrd)^∨g

andψ^∗

Res^GL_ψ(GL^h(L)

h(L))F(ρ)

are both equal to

(JL(ρ) ⊗(χ_JL(ρ) ◦det)^∨) ⊗rec(JL(ρ)^∨⊗ (χ_JL(ρ) ◦det)⊗ (| · | ◦det)^h−12 )(h−1), as desired.

We can also use Theorem 3.2.2 to deduce that Theorem 6.3.2. Fori , h−1, ρ ∈RepE^×,

H_cⁱ(M_h^∞)_cusp(ρ) =0.

Proof. This is immediate from Theorem 3.2.2 and the commutative diagram

RepE^×

ρ7→θ^∗ Res_θ^E×

(D×)ρ

//

H_cⁱ(M_h^∞)cusp

RepD^×

H_cⁱ(LT_h^∞)cusp

Rep(GL_h(L)×W_L)

π⊗r7→ψ^∗

Res^{G Lh}_ψ( ^(L)

G Lh(L))π

⊗r

//Rep(GL_h(L)×W_L).

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