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Base change and K -theory for GL(n, R) II

Roger Plymen; joint work with Sergio Mendes

September 2009

Roger Plymen; joint work with Sergio Mendes Base change and K -theory for GL(n, R) II

The Weil group

The idea is to parametrize the irreducible admissible representationsof G = GL(n,C) in a way which is well-adapted to L-functions

The Weil group (local Langlands group) of GL(n,C) is

WC := C×

The Langlands parameters are morphisms φ : WC → G∨ = GL(n,C);the image of φ comprises semisimple elements. The boundedL-parameters φ correspond to tempered representations of G

The irreducible representations of WC are given by

χ(z) = r2te i`θ

with ` ∈ Z and t ∈ C, bounded iff t ∈ iR

The Weil group

WC := C×

χ(z) = r2te i`θ

The Weil group

WC := C×

χ(z) = r2te i`θ

The Weil group

WC := C×

χ(z) = r2te i`θ

with ` ∈ Z and t ∈ C, bounded iff t ∈ iRRoger Plymen; joint work with Sergio Mendes Base change and K -theory for GL(n, R) II

Langlands correspondence

Theorem (Zelobenko and Naimark)

The local Langlands correspondence for the tempered dual of GL(n,C) is

χ1 ⊕ χ2 ⊕ · · · ⊕ χn 7→ χ1 � χ2 � · · ·� χn

The � notation refers to the representation induced from thestandard Borel subgroup of GL(n,C) to GL(n,C)

The L-factor attached to χ1 ⊕ · · · ⊕ χn is the product of theindividual L-factors:

L(s, χ) = 2(2π)−(s+t+|`|/2)Γ(s + t + |`|/2)

χ1 ⊕ χ2 ⊕ · · · ⊕ χn 7→ χ1 � χ2 � · · ·� χn

L(s, χ) = 2(2π)−(s+t+|`|/2)Γ(s + t + |`|/2)

χ1 ⊕ χ2 ⊕ · · · ⊕ χn 7→ χ1 � χ2 � · · ·� χn

L(s, χ) = 2(2π)−(s+t+|`|/2)Γ(s + t + |`|/2)

The Weil group WR

The Weil group WR is

WR :=< j > C×

withj2 = −1, jc = cj

Short exact sequence

1→WC →WR → G (C/R)→ 1

The Weil group WR

The Weil group WR is

WR :=< j > C×

withj2 = −1, jc = cj

Short exact sequence

1→WC →WR → G (C/R)→ 1

The Weil group

The idea is to parametrize the irreducible admissible representationsof G = GL(n,R) in a way which is well-adapted to L-functions

The Weil group (local Langlands group) of GL(n,R) is WR

The Langlands parameters are morphisms φ : WR → G∨ = GL(n,C);the image of φ comprises semisimple elements. The boundedL-parameters φ correspond to tempered representations of G

Base change is defined by restriction of an L-parameter from WR toWC

This determines the base change map

Irrt GL(n,R)→ Irrt GL(n,C)

The Weil group

The commutative diagram

E/F be a Galois extension of the local field F ; here, F = R, E = CThen we construct a commutative diagram

K top∗ (G (E ))

µE−−−−→ K ∗(Irrt(G (E ))

y ybcE/F

K top∗ (G (F )) −−−−→

K ∗(Irrt(G (F ))

The horizontal map is the Baum-Connes correspondence; the verticalmap is determined by the local Langlands correspondence

We have used the strong Morita equivalence [RJP] true for every localfield F :

C ∗r (G (F )) ∼ C0(Irrt G (F ))

K top∗ (G (E ))

µE−−−−→ K ∗(Irrt(G (E ))

y ybcE/F

K top∗ (G (F )) −−−−→

K ∗(Irrt(G (F ))

C ∗r (G (F )) ∼ C0(Irrt G (F ))

K top∗ (G (E ))

µE−−−−→ K ∗(Irrt(G (E ))

y ybcE/F

K top∗ (G (F )) −−−−→

K ∗(Irrt(G (F ))

C ∗r (G (F )) ∼ C0(Irrt G (F ))

Base change for characters

We haveC× = WC

R× ∼= W abR

Let χ be a character of WR. Then base change is

(BC )(χ) = χ ◦ NC/R, NC/R(z) = zz

We haveC× = WC

R× ∼= W abR

(BC )(χ) = χ ◦ NC/R, NC/R(z) = zz

We haveC× = WC

R× ∼= W abR

(BC )(χ) = χ ◦ NC/R, NC/R(z) = zz

The base change map

The base change map is proper and sends

ξ1 ⊕ · · · ⊕ ξr ⊕ IndWRWC

χ1 ⊕ · · · ⊕ IndWRWC

toξ1 ◦ N ⊕ · · · ⊕ ξr ◦ N ⊕ χ1 ⊕ χσ ⊕ · · · ⊕ χq ⊕ χσq

Base change creates a map Rr+q → Rr+2q sending

(t1, . . . , tr , t′1, . . . , t

′q) 7→ (2t1, . . . , 2tr , t

′1, t′1, . . . , t

′q, t′q)

The corresponding map S r+q → S r+2q is nullhomotopic and soK (Rr+q)→ K (Rr+2q) is the zero map.

The base change map

The base change map is proper and sends

ξ1 ⊕ · · · ⊕ ξr ⊕ IndWRWC

χ1 ⊕ · · · ⊕ IndWRWC

toξ1 ◦ N ⊕ · · · ⊕ ξr ◦ N ⊕ χ1 ⊕ χσ ⊕ · · · ⊕ χq ⊕ χσq

Base change creates a map Rr+q → Rr+2q sending

(t1, . . . , tr , t′1, . . . , t

′q) 7→ (2t1, . . . , 2tr , t

′1, t′1, . . . , t

′q, t′q)

The corresponding map S r+q → S r+2q is nullhomotopic and soK (Rr+q)→ K (Rr+2q) is the zero map.

The base change map

If q = 0 then the base change map is proper:

ξ1 ⊕ · · · ⊕ ξr 7→ ξ1 ◦ N ⊕ · · · ⊕ ξr ◦ N

BC : IndG(R)B(R)(ξ1 ⊗ ξ2 ⊗ · · · ⊗ ξr ) 7→ IndG(C)

B(C)(ξ1 ◦ N ⊗ · · · ⊗ ξr ◦ N)

The Weyl group Sr now enters the picture. Since K (Rr/Sr ) = 0when r > 1, we get the zero map when r > 1

The base change map

Theorem

If n > 1 then the base change map on the left-hand-side of thecommutative diagram is zero:

jC/R = 0

The case of GL(1)

For GL(1) we have a commutative diagram

R(U(1))µC−−−−→ K 1(Irrt(GL(1,C))

y ybcC/R

R(Z/2Z) −−−−→µR

K 1(Irrt(GL(1,R))

The map jC/R has an interpretation in terms of K -cycles. The K -cycle

(C0(R), L2(R),−id/dx)

is equivariant with respect to C× and R×, and therefore determines a class/∂C ∈ KC×

1 (EC×) and a class /∂R ∈ KR×1 (ER×).

The case of GL(1)

Theorem

For GL(1) the base change map is given by

jC/R : R(U(1))→ R(O(1)), (nj) 7→ (n0, n0)

In terms of K-cycles, base change in dimension 1 admits the followingdescription:

/∂C 7→ (/∂R, /∂R)

Weil group WF

Let F be a local nonarchimedean field, so F/Qp or F = Fq((x)). Thelocal Langlands group is LF := WF × SL(2,C) and the idea is toparametrize the smooth dual of GL(n,F ) by morphismsLF → G∨ = GL(n,C) which are ΦF -semisimple.

For the unramified unitary principal series, the L-parameters are givenby

φ = χ1 ⊗ 1⊕ · · · ⊕ χn ⊗ 1

where each χj is an unramified character of WF . Local classfieldtheory gives F× ∼= W ab

An invariant of a finite Galois extension E/F : the residue degreef = f (E/F ). For example, given F/Qp we have qF = pf .

Weil group WF

φ = χ1 ⊗ 1⊕ · · · ⊕ χn ⊗ 1

Weil group WF

φ = χ1 ⊗ 1⊕ · · · ⊕ χn ⊗ 1

Weil group WF

We have a canonical map

1→ IF →WFdF→ Z→ 0

and each unramified character is χ(w) = zdF (w). Under the basechange map bE/F : χ 7→ χ|WE we have

For all w ∈WE :(bE/Fχ)(w) = (z f )dE (w)

The unramified unitary principal series has L-parameters(z1, . . . , zn) ∈ Tn/Sn. Under base change, we have

(z1, . . . , zn) 7→ (z f1 , . . . , z

and Tn/Sn ∼ T via (z1, . . . , zn) 7→ z1 · · · zn

Weil group WF

We have a canonical map

1→ IF →WFdF→ Z→ 0

and each unramified character is χ(w) = zdF (w). Under the basechange map bE/F : χ 7→ χ|WE we have

For all w ∈WE :(bE/Fχ)(w) = (z f )dE (w)

The unramified unitary principal series has L-parameters(z1, . . . , zn) ∈ Tn/Sn. Under base change, we have

(z1, . . . , zn) 7→ (z f1 , . . . , z

and Tn/Sn ∼ T via (z1, . . . , zn) 7→ z1 · · · zn

Weil group WF

K -theory will see the unramified unitary principal series of GL(n,F )as a circle T and will see base change as follows:

K 1(T)→ K 1(T), Z→ Z, m 7→ f ·m

For the arithmetically unramified unitary principal series we havesomething similar

Tn//Sn → Tn//Sn

each coordinate zj 7→ z fj .

The L-parameter of the Steinberg representation St(n,F ) is 1⊗ R(n)with R(n) the n-dimensional rep of SL(2,C). Giving it an unramifiedtwist and performing base change we get T→ T, z 7→ z f and thisis seen by K 1 as multiplication by f .

Weil group WF

K 1(T)→ K 1(T), Z→ Z, m 7→ f ·m

Tn//Sn → Tn//Sn

Weil group WF

K 1(T)→ K 1(T), Z→ Z, m 7→ f ·m

Tn//Sn → Tn//Sn

Let π be a (unitary) cuspidal rep of GL(n,F ) with Langlandsparameter ρ. Suppose that E/F is cyclic of degree n and that ρ ismonomial, induced from a regular character η of E×. We have

BC (π) = η � ησ � · · ·� ησn−1

an irreducible rep of GL(n,E ). The rep π is not isolated: we cantwist by χ(z) = zvalF ◦det, z ∈ C, |z | = 1.

Now BC (π) is an element in the unitary principal series of GL(n,E )

This leads to the map T→ Tn, z 7→ (z f , . . . , z f ) which, afterapplying the Chern character, induces a map which is zero except on1-forms:

Hodd(Tn)→ H1(T), dθj 7→ f (dθ), 1 ≤ j ≤ n

This is how K 1 sees base change for many cuspidal reps of GL(n)

BC (π) = η � ησ � · · ·� ησn−1

Cycles in chamber homology; the tree of SL(2)

E/F be a Galois extension of the local field F . Then we construct acommutative diagram with chamber homology on the left

HG(E)∗ (β1G (E ))

µE−−−−→ K ∗(Irrt(G (E ))

y ybcE/F

HG(F )∗ (β1G (F )) −−−−→

K ∗(Irrt(G (F ))

β1GL(n) = R× βSL(n)

For example β1GL(2) = R× βSL(2). Upon Galois extension E/Qp,the valency of the tree will increase from p + 1 to pf + 1 so thatmultiplication by f in H1 must be related to this ......

HG(E)∗ (β1G (E ))

µE−−−−→ K ∗(Irrt(G (E ))

y ybcE/F

HG(F )∗ (β1G (F )) −−−−→

K ∗(Irrt(G (F ))

HG(E)∗ (β1G (E ))

µE−−−−→ K ∗(Irrt(G (E ))

y ybcE/F

HG(F )∗ (β1G (F )) −−−−→

K ∗(Irrt(G (F ))

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