ANÁLISIS DE ESFUERZOS EN SOLIDWORKS DE LAS PIEZAS

In this section we will explicitly compute the local parameters of the local lift from GSO(X, Fv) to GSp(2, Fv) for the unramified case as we did in Chapter 5. As

in Chapter 5, the groups GSO(X, Fv),GSp(n, Fv), etc are all denoted simply by

is finite. Also “Ind ” always means unnormalized induction, and whenever we use normalized induction, we use the notation “n-Ind ”. Thus, for example, if π is the principal series representation of G = GL(4) fully induced from the standard parabolic P by the four unramified characters χ1, χ2, χ3, χ4, then we have π ∼=

n-IndG_P(χ1χ2χ3χ4) = IndGP( ˜χ1χ˜2χ˜3χ˜4) , where ˜χ1 = | · |3/2χ1, ˜χ2 =| · |1/2χ1, ˜χ3 =

| · |−1/2_χ

2, and ˜χ4 =| · |−3/2χ4.

First we need the following, which seems to be well-known by now.

Lemma B.4.1. Letτ be the character on the standard parabolicP2 of GSp(2)which

is trivial on the unipotent radical and sends diag (α1, α2, γα−11, γα

−1

2 )7→τ0(γ)τ1(α1)τ2(α2)

for unramified characters τ0, τ1, and τ2. If Π is the unramified subquotient of the

representation of GSp(2) give by n-IndGSp_P (2)

2 τ, then the Langlands parameter of Π

is

diag (χ0($), χ0χ1($), χ0χ1χ2($), χ1χ2($))∈GSp(2,C) = LGSp(2)◦, where $ is the uniformizer of F.

By using this together with the techniques from Chapter 5, we prove

Proposition B.4.2. Assume π is an unramified representation of GL(4) with a unitary central character which is fully induced from four unramified characters

χ1, χ2, χ3, χ4, i.e. π ∼= n-Ind

GL(4)

P χ1χ2χ3χ4, and the central character ofπ isχ2, i.e.

χ1χ2χ3χ4 =χ2. Leteπ be the unramified representation of GSO(X)that corresponds

to πby Proposition B.2.3. If π_ecorresponds to an unramified Π∈Irr(GSp(2)) under theta correspondence, then the Langlands parameter of Π is

diag (χ1($), χ3($), χ2($), χ4($))∈GSp(2,C).

Proof. Let _eπ∼= Ind_QGSO(₃ X)µ whereµ acts onMQ3 by

diag (β1, β2, β3, λβ3−1, λβ

−1 2 , λβ

−1

1 )7→µ0(λ)µ1(β1)µ2(β2)µ3(β3),

withQ3the parabolic subgroup preserving the flaghx12i ⊂ hx12, x13i ⊂ hx12, x13, x14i,

and xij’s as in Section B.2. Then we have µ0 = χ | · |3/2_χ 1 , µ1 = | · |2_χ 1χ2 χ , µ2 = | · |χ1χ3 χ , µ3 = χ1χ4 χ .

There is a non-zero R-homomorphism

ωX,2 −→Π⊗Ind

GSO(X)

Q3 µ= Ind

GSp(2)×GSO(X)

Since there is a natural injection IndGSp(2)_GSp(2)×_×GSO(_Q X)

3 Π⊗µ ,→Ind

R

R∩(GSp(2)×Q3)Π⊗µ

(see Lemma 5.1.4), we have a non-zero R-homomorphism

ωX,2 −→IndRR∩(GSp(2)×Q3)Π⊗µ.

Let Qbe the parabolic subgroup preserving the flag hx12, x13, x14i. Then by induc-

tion in stages, we have a non-zero R-homomorphism

ωX,2 −→IndRSQInd

MQ

SQ₃Π⊗µ, where SQ and SQ3 are as in Proposition 5.1.1.

Now if we take the Jacquet module of ωX,2 with respect to NQ, the Frobenius

reciprocity and Proposition 5.1.1 give non-zero R-homomorphisms 0⊂J(2) ⊂J(1) ⊂J(0) −−−−→ϕ IndMQ

SQ3Π⊗µ.

First assume that kerϕ⊇J(1)_{. Then there is a non-zero M}

Q-homomorphism I(0) =J(0)/J(1) −→IndMQ

SQ3Π⊗µ.

Then if we take the Jacquet module of I(0) _{with respect to N}

Q3, the Frobenius

reciprocity and Proposition 5.1.1 give a non-zero MQ3-homomorphism

IndMQ3

SP₀,Q₃σ0,3 ∼

=σ0,3 −→Π⊗µ,

whereσ0,3is as in Proposition 5.1.1. Then on the element of the form ((1), β1, β2, β3)∈

MQ3 acts σ0,3 by the character

((1), β1, β2, β3)7→ |β1|2|β2|2|β3|2.

On the other hand, it acts on Π⊗µby the character

((1), β1, β2, β3)7→µ1(β1)µ2(β2)µ3(β3).

Therefore we must have µ3 = | · |2. But this is impossible, because µ3 = χ1χ4/χ

and so |µ3|<| · |. Thus kerϕ+J(1).

Then by restricting ϕto J(1)_{, we have non-zero M}

Q-homomorphisms

0⊂J(2) ⊂J(1) ϕ

0

−−−−→IndMQ

SQ3Π⊗µ.

Then if kerϕ0 ⊇J(2)_{, we have a non-zero M}

Q-homomorphism I(1) =J(1)/J(2) −→IndMQ

Then by exactly the same argument as above, this leads to a contradiction. There- fore kerϕ0 ₊J(2).

Then by restricting ϕ0 toJ(2)_{, we have a non-zero M}

Q-homomorphism J(2)(= I(2))−→IndMQ

SQ₃Π⊗µ. If we take the Jacquet module of I(2) _{with respect toN}

Q2, the Frobenius reciprocity

and Proposition 5.1.1 give a non-zero MQ3-homomorphism

IndMQ3

SP₂,Q₃σ2,3

ϕ00

−−−−−→Π⊗µ.

Then the element of the form ((1), β1, β2, β3) ∈ MQ3 acts on Ind

MQ3

SP2,Q3σ2,3 by the

character

((1), β1, β2, β3)7→ |β1|2τ2(β2)τ3(β3),

and on Π⊗µby the character

((1), β1, β2, β3)7→µ1(β1)µ2(β2)µ3(β3).

Thus we have

µ1 =| · |2, µ2 =τ2, µ3 =τ3.

Moreover, since µ1 =| · |2χ1χ2/χand χ1χ2χ3χ4 =χ2, we have

χ=χ1χ2 =χ3χ4.

Next notice that MQ3 ∼= GSp(2)×G

3 m. Then by restricting ϕ 00 _{to the elements} of the form ((g),1,1,1) ∈ GSp(2) × _G3 m ∼= MQ3, we have a non-zero GSp(2)- homomorphism IndGSp(2)_P 2 σ 0 2,3 −→Π⊗µ −1 0 ,

where σ₂0_,3 is the character on P2 given by

σ0_2,3(     α1 ∗ ∗ ∗ 0 α2 ∗ ∗ 0 0 λα−₁1 ∗ 0 0 0 λα₂−1     = (| · |2τ₃−1)(α1)(| · |2τ2−1)(α2)(| · |−1τ2τ3)(λ),

and Π⊗µ−₀1 is the twist of Π by µ0 through the multiplier character ν, i.e. (Π⊗

µ0)(g) =µ0(g)Π(g). Hence we have a non-zero GSp(2)-homomorphism

(IndGSp(2)_P

2 σ

0

2,3)⊗µ0 −→Π,

where the twisting is again via ν. It is easy to see that (IndGSp(2)_P 2 σ 0 2,3)⊗µ0 ∼= Ind GSp(2) P2 τ,

where τ is the character onP2 given by τ(     α1 ∗ ∗ ∗ 0 α2 ∗ ∗ 0 0 λα−₁1 ∗ 0 0 0 λα−₂1     ) = (| · |2_µ−1 3 )(α1)(| · |2µ−21)(α2)(| · |−1µ0µ2µ3)(λ) = (| · |2χχ−₁1χ₄−1)(α1)(| · |χχ−11χ −1 3 )(α2)(| · |−3/2χχ−21)(λ) = (| · |2χ3χ−11)(α1)(| · |χ4χ1−1)(α2)(| · |−3/2χ1)(λ).

Thus Π is an unramified quotient of n-IndGSp(2)_P

2 τ 0_{, where τ}0 _{is given by} τ0(     α1 ∗ ∗ ∗ 0 α2 ∗ ∗ 0 0 λα−₁1 ∗ 0 0 0 λα−₂1     ) = (χ3χ−11)(α1)(χ4χ−11)(α2)(χ1)(λ).

Thus by Lemma B.4.1, the Langlands parameter of Π is

diag (χ1($), χ3($), χ2($), χ4($))∈GSp(2,C).