EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 6, No. 3, 2013, 282-298
ISSN 1307-5543 – www.ejpam.com

K-theory, Chamber Homology and Base Change for GL(2)

Wemedh Aeal

School of Mathematics,The University Of Manchester, Greater Manchester, United Kingdom

Abstract. In this work on GL(2) we have found that it is hard to compute the chamber homology
groups from the quotient space β1GL(2)/GL(2) (Mobius band), so we introduced a new way to com-
pute the chamber homology groups by restricting to the original quotient space (edge) before taking
the real line R. We have not yet given a full description of what happening under base change when
we work on the cuspidal representation but, we somehow, gave a way to compute the base change
effect of some type of cuspidal representations which are the admissible pairs. The base change of a
principal series representations is always a principal series. Similarly, the base change of a twist of
Steinberg representation is again a twist of Steinberg. However, an irreducible Galois representation
can certainly restrict to a reducible one. Thus it is possible for the base change of a cuspidal to be
principal series. In fact, if π is any irreducible admissible representation of GL(2, F) then one can find
an extension E/F such that BC(π) is either unramified or Steinberg. .

2010 Mathematics Subject Classifications: 58B34, 11S70, 46L80, 11S31, 19K33, 11F85

Key Words and Phrases: Local Langlands, Base change, K-theory, Chamber Homology, Baum-Conns
map, representation theory

1. Introduction

Let G = GL(n, F) and let C∗r G denote the reduced C∗-algebra of G. According to the
Baum-Connes correspondence, we have a canonical isomorphism [2]

µF : K top
j β1G→ K j C∗r G,

where β1G denotes the enlarged building of G. In noncommutative geometry, isomorphisms
of C∗-algebras are too restrictive to provide a good notion of isomorphisms of noncommutative
spaces, and the correct notion is provided by strong Morita equivalence of C∗-algebras. The
noncommutative C∗-algebra C∗r G is strongly Morita equivalent to the commutative C∗-algebra
C0(I r r t G) where I r r t G denotes the tempered dual of G [18]. Consequently, we have

K j C∗r G ∼= K j I r r t G

Email address: wemedh.aeal@manchester.ac.uk, wemedh@hotmail.com

http://www.ejpam.com 282 c© 2013 EJPAM All rights reserved.



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 283

and this leads to the following formulation of the Baum-Connes correspondence:

K top
j β1G ∼= K j I r r t G.

This in turn leads to the following diagram

K top
j (β

1G(E))

��

µE // K j(I r r t G(E))

K j(BCE/F )
��

K top
j (β

1G(F)) µF
// K j(I r r t G(F))

where the left-hand vertical map is the unique map which makes the diagram commuta-
tive. This work will be concerned with presenting an explicit construction of the local Lang-
lands correspondence between so-called cuspidal representations of GL2(F) and certain 2-
dimensional representations of WF , where the residue characteristic char 6= 2. We will focus
on the classification and the construction of the cuspidal representations for GL2(F), it was
originally treated by [12] and [13].

2. Chamber Homology for GL(2)

Consider the pair (L,κ) where L is a Levi subgroup of a parabolic subgroup of G, and κ
is an irreducible cuspidal representation of L. Two pairs (L1,κ1), (L2,κ2) are called inertially
equivalent if there exist g ∈ G and an unramified character χ of L2 such that

L2 = Lg
1 , and κg

1 = κ2⊗χ,

where Lg
1 := g−1 L1 g and κg

1(x) = κ1(g x g−1) for all x ∈ Lg
1 . Let [L,κ]G be the inertial

equivalence class of the pair (L,κ) and let B(G) be the set of all inertial equivalence classes.
This set is called the Bernstein spectrum of G.

Definition 1. Let s ∈ B(G), an s-type is a pair (J ,σ) consisting of a compact open subgroup
J of G and an irreducible smooth representation σ of J such that for any irreducible smooth
representation π of G, the restriction of π to J contains σ if and only if π is an object of Rs(G),
[8]. It has been proved by [6] and [9] that there exists an s-types for each point s ∈B(G).

Now, let OF denote the ring of integers of F ,$ ∈ F be a uniformizer, and p be the maximal
ideal of OF . Also, let

Π = Πn =

�

0 In−1
$F 0

�

, and si =











Ii−1
0 1
1 0

In−i−1











,

for every i ∈ {1, . . . , n − 1}, and s0 = Πs1Π−1 denotes the standard involutions in G. The
finite Weyl group is W0 = 〈s1, s2, . . . , sn−1〉, and the affine Weyl group defined as follows



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 284

W = 〈s0, s1, . . . , sn−1〉. The extended affine Weyl group is denoted by W =Wo 〈Π〉. Its clear
thatW∩ GL(n,OF ) =W0. The standard Iwahori subgroup is

I =















O ×F OF . . . OF

p
F

. . . . . .
...

...
. . . . . . OF

p
F

. . . p
F
O ×F















.

Let Σ be the apartment attached to the diagonal torus and let∆ be the unique chamber in this
apartment which is stabilized by 〈Π〉I . Let Ji be the maximal standard parahoric subgroups of
G,

Ji = I〈s0, s1, . . . si−1, si+1, . . . , sn−1〉I

where J0 = GL(n,OF ). We see that Ji are the stabilizers of the vertices, the stabilizer of the
facets of dimension n− 1 of ∆ are K0, K1, . . . , Kn−1, where Ki = I〈si〉I . The enlarged building
β1G is labelled, this means there exists a simplicial map

ℑ : β1G→∆,

this map is dimensions preserver. This labelling is unique and it allows us to fix an orientation
of the simplices. The chamber homology groups are obtained by totalizing the bicomplex:

0 R(J0)⊕R(J1)⊕ . . .⊕R(Jn−1)oo

��

· · ·oo

��

R(K0)⊕R(K1)⊕ . . .⊕R(Kn−1)oo

��

R(I)oo

��

0 R(J0)⊕R(J1)⊕ . . .⊕R(Jn−1)oo · · ·oo R(K0)⊕R(K1)⊕ . . .⊕R(Kn−1)oo R(I)oo

the vertical maps are given by 1−iΠ.
We assume that C =R(J0)⊕R(J1)⊕ . . .⊕R(Jn−1), C

′
=R(K0)⊕R(K1)⊕ . . .⊕R(Kn−1)

and C
′′
=R(I). By totalizing the above bicomplex, we obtain this chain complex

0 Coo · · ·oo C
′

i−1⊕C
′

i
oo C

′

i ⊕C
′

i+1
oo · · ·oo C

′′
oo 0oo .

Definition 2 ([3]). The homology groups of this totalized complex are the chamber homology
groups.

Now, for each point s ∈B(G) let

C (s) =R(J
0
(s))⊕R(J

1
(s))⊕ . . .⊕R(J

n−1
(s)),

C
′
(s) =R(K

0
(s))⊕R(K

1
(s))⊕ . . .⊕R(K

n−1
(s)),

and
C
′′
(s) =R(I(s)).



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 285

We associate a sub-bicomplex

0 C (s)oo

��

· · ·oo

��

C
′
(s)oo

��

C
′′
(s)oo

��

0 C (s)oo · · ·oo C
′
(s)oo C

′′
(s)oo

in which each vertical map is 0. The homology groups of the chain complex

0 C (s)oo · · ·oo C
′′
(s)oo 0oo

is denoted by h j(s), we call this complex the little complex. When we totalize the associated
bicomplex, we get the chain complex

0 C (s)oo · · ·oo C
′

i−1(s)⊕C
′

i (s)
oo C

′

i (s)⊕C
′

i+1(s)
oo · · ·oo C

′′
(s)oo 0oo .

Theorem 1 ([1]). The homology groups H j(s) of this complex are given by

H
0
(s) = h

0
(s), H

n
(s) = h

n−1
(s)

H
i+1
(s) = h

i
(s)⊕ h

i+1
(s), 0≤ i ≤ n− 2

H
ev
(s) = h

0
(s)⊕ h

1
(s)⊕ . . .⊕ h

n−1
(s) = H

odd
(s)

The even (resp. odd) chamber homology is precisely the total homology of the little complex.

Now if we back to our case, let F be non-archimedean p-adic local field, G = GL(2, F) and
β1GL(2) be the enlarged building of G. The enlarged building of G can be defined as

β1G = βSL(2)×R

with an action

GL(2)× β1GL(2)−→ β1GL(2)

GL(2)× βSL(2)×R−→ β1GL(2)

(x , y, t) 7−→ (x y, t + valF (det x)).

The enlarged building β1GL(2) has the structure of polysimplicial complex, but we have
β1G = βSL(2)×R. The action of SL(2) on its tree could be extended to an action of GL(2).
We will investigate the chamber homology H j(β1GL(2)) of G acting on its enlarged build-
ing properly. The quotient β1GL(2)/GL(2) is a Mobius band ( an identification space of a
chamber) which is a compact space.

Let

Π =

�

0 1
$F 0

�

, s1 =

�

0 1
1 0

�

and s0 =

�

0 $−1
F

$F 0

�



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 286

be the standard involutions in GL(2). Restricted to the affine line R in the enlarged building
β1GL(2) = βSL(2)×R, Π sends t to t+1. It is a bit hard to calculate the chamber homology
group for GL(2) from a Mobius band but its not that difficult to construct a complex to com-
pute the chamber homology of GL(2) if we restrict to the original quotient space before taking
the real line copy which is an edge of the tree of SL(2). Let I , J0 and J1 are the stabilizer
groups of the edge and the two vertices in the above chamber.

J
0
◦ I • J

1

2.1. The trivial type (I ,1I)

Let

T =

�

F× 0
0 F×

�

be the diagonal subgroup of G = GL(2, F) and let 1 be the trivial representation of T . Then
the pair (T,1) is a cuspidal pair. Let us discuss the special case when s = [T,1]G , the s-type in
this case will be the trivial type (I ,1). We will construct the little complex created by (I ,1I).

Theorem 2. Let I be the Iwahori subgroup of GL(2), and let St2 be the Steinberg representation
of GL(2, F), and let χ1, χ2, χ be unramified unitary characters. Then the unramified unitary
representation of GL(2) can be written as follows:

(i) IndG
B (χ1×χ2)' IndG

B (χ2×χ1).

(ii) χ ⊗ St2.

Proof. see [18]

We have IndJ0
I 1I = 1J0

⊕StJ0
2 and IndJ1

I 1I = 1J1
⊕StJ1

2 . Then the little complex determined
by this type is

0 R(J
0
)⊕R(J

1
)oo

��

R(I)oo

��

0 R(J
0
)⊕R(J

1
)oo R(I)oo

where R(J0)⊕R(J1) is the free abelian group on the two elements

(1J0
,1J1
), (StJ0

2 , StJ1
2 ) ∈R(J

0
)⊕R(J

1
)

and R(I) is the free abelian group on the single generator

1I ∈R(I).

The above bicomplex chain implies that we need to consider only invariant elements. There-
fore, we need to restrict to the invariant elements so we have the following:

1J0
⊕ StJ0

2 ∼ 0 i.e. 1J0
∼−StJ0

2



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 287

1J1
⊕ StJ1

2 ∼ 0 i.e. 1J1
∼−StJ1

2

This means we have one element

(1J0
,1J1
)∼−(StJ0

2 , StJ1
2 ).

By totalizing the above little complex we get

0 R(J
0
)⊕R(J

1
)oo R(I)oo 0oo ,

hence by Theorem 1 we have

H0 = h0, H1 = h0+ h1, H2 = h1.

Now, h0 = Z, h1 = Z, therefore H0 = Z, H1 = Z2 and H2 = Z. i.e.

Heven = Z2 = Hodd .

Let λ be a unitary character of GL(1, F) ' F×, and let τ = λ ◦ det : I → U (1). Theorem
9 in [1] shows that the totalised little complex created by the (I ,τ) is isomorphic to the
totalized little complex created by the trivial type (I ,1I). Therefore, the homology groups
Heven = Z2 = Hodd

2.2. The Irreducible Components of Reducible Principal Series

Let s = [T,σ]G . Consider the s-type (J ,τ) where J is compact open subgroup of G and τ
is an irreducible smooth representation of J .

Lemma 1. Let χ = Ind I
Jτ. Then χ is irreducible.

Proof. See [1]

Lemma 2. We have

IndJ0
I χ = α0⊕ γ0

IndJ1
I χ = α1⊕ γ1.

Let R(J
0
(τ))⊕R(J

1
(τ)) be the free abelian group on the element

(α0,α1) ∼ (γ0,γ1) ∈R(J
0
)⊕R(J

1
) and let R(I(τ)) be the free abelian group on the element

χ. The little complex is then

0 R(J
0
(τ))⊕R(J

1
(τ))oo R(I(τ))oo 0oo .

Hence

H0(τ) = h0(τ) = Z, H1(τ) = h0(τ) + h1(τ) = Z2, H2(τ) = h1(τ) = Z,

i.e. Heven = Z2 = Hodd .



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 288

2.3. Cuspidal Representations

Let s = [G,π], where G = GL(2, F) and π is an irreducible cuspidal representation of G.
Let (J ,σ) be the maximal simple type contained in π [6]. It follows that J(s) = GL(2,OF ) = J0
[1]. By Lemma 1, it follows that λ = IndJ0

J σ is irreducible. Now, the pair (J0,λ) is an s-type.
The restriction of a smooth irreducible representation ρ of G to J0 contains λ if and only if

ρ ∼= π⊗χ ◦ det

where χ is an unramified character of F×, i.e. π contains λ with multiplicity 1. Therefore,
the representation λ is the unique smooth irreducible representation τ of J0 such that (J0,τ)
is an s-type [17]. The little complex determined by λ is

0 C (s)oo 0oo

where C (s) is the free abelian group on the invariant 0-cycle τ. The total homology of the
little complex is given by h0(s) = Z. Therefore,

Heven = Z= Hodd .

2.4. The Principal Series

Let (J ,τ) be s-type, J0 = GL(2,OF ). If J ⊂ J0 then the only double J -coset representative
which G-intertwines τ is 1G . Therefore,

Ind I
Jτ is irreducible

IndJi
J τ is irreducible.

Now, let γ = Ind I
Jτ and σ = IndJi

J τ. Let C (τ) be the free abelian group on the generator σ,
and let C

′
(τ) be the free abelian group on the generator γ. The totalized little complex is

0 C (τ)oo C
′
(τ)oo 0oo .

Then
H0 = Z, H1 = Z2, H2 = Z and so Heven = Z2 = Hodd .

Theorem 3. (i) The base change of a twist of Steinberg representation is again a twist of
Steinberg.

(ii) The base change of a principal series representation is always a principal series.

(iii) The base change of a cuspidal representation will never be a twist of Steinberg representa-
tion. It is possible for the base change of a cuspidal representation to be principal series
representation.



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 289

Proof. Let LF = WF × SL(2,C) and LE = WE × SL(2,C) be the local Langlands groups
and let the two L-parameters corresponding to these groups respectively be

φ :WF × SL(2,C) // G∨ = GL2(C) ,

φ|WE
:WE × SL(2,C) // G∨ = GL2(C) .

(i) Let φF = ψ⊗ St F
2 , where ψ ∈ Ψt(WF ). Since the base change works by restricting the

L-parameter to WE and the restriction is apply only on the Weil group part, therefore
the base change of St F

2 is BC(St F
2 ) = StE

2 and hence the the base change works on the
unitary twist of Steinberg as follows:

BC(φF ) = φE , BC(ψ⊗ St F
2 ) = BC(ψ)⊗ StE

2 =ψ ◦ NE/F ⊗ StE
2 .

(ii) Let φF = (ψ1 ⊗ 1)⊕ (ψ2 ⊗ 1) be the L-parameter, where ψ1, ψ2 ∈ Ψt(WF ) then the
base change map works on the reducible principal series as follows:

BC(φF ) = φE

BC(ψ1⊗1⊕ψ2⊗1) = BC(ψ1⊗1)⊕ BC(ψ2⊗1) = (ψ1 ◦N
E/F
⊗1)⊕ (ψ2 ◦N

E/F
⊗1).

(iii) Let φF =ψσ⊗1, where σ is irreducible representation of WF and ψ ∈Ψt(WF ).

(a) If the L-parameter φE remains irreducible after restriction, then this determines a
cuspidal representation of GL(2, E). Base change in this case will send one cuspidal
representation of GL(2, F) to a cuspidal representation of GL(2, E). Therefore, the
map BC works as follows:

BC(φF ) = φE

BC(ψσ⊗1) = BC(ψ)σ∗⊗1=ψ ◦ NE/Fσ
∗⊗1.

(b) If the L-parameter φE is reducible after restriction, then this representation split
into two one-dimensional representations say σ1 and σ2. This means that the
restriction of the cuspidal representation is a principal series. Therefore, the map
BC works as follows:

BC(φF ) = φE

BC(ψσ⊗1) =ψ ◦ NE/F (σ1⊕σ2)⊗1.

In fact, if π is any irreducible admissible representation of GL(2, F) then one can find an
extension E/F such that BC(π) is either unramified or Steinberg.



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 290

3. K-theory for GL(2)

Let F be a non-archimedean local field with characteristic 0 and p 6= 2. Such a field has
a norm, denoted by modF [19]. The representations in GL(2, F) can be view as a one of the
following:

(i) The irreducible admissible representations of G fall into three classes: principal series,
twists of Steinberg and cuspidal.

(ii) The unramified representations of G are exactly the principal series representations
coming from unramified characters. These are parameterized by (unordered) pairs of
complex numbers.

Let E/F be a finite Galois extension, and let the corresponding Weil groups be denoted
WE ,WF . Let T denote the circle group

T= {z ∈ C :| z |= 1}

and let Ψt(WF ) denote the group of unramified unitary characters of WF . Then we have

Ψt(WF )∼= T, ψ 7→ψ($)

where $F is a uniformizer in F .
Now, letLF denote the local Langlands group: LF :=WF ×SL(2,C). A Langlands param-

eter (or L-parameter) is a continuous homomorphism

φ :LF → GL(2,C),

(GL(2,C) is given the discrete topology) such thatφ(ΦF ) is semisimple, where ΦF is a geomet-
ric Frobenius in WF . Two Langlands parameters are equivalent if they are conjugate under
GL(2,C). The set of equivalence classes of Langlands parameters is denoted by Φ(GL(2)).
Now the base change is defined by the restriction of L-parameter from LF to LE . Consider
first the single L-parameter φ = ρ⊗τ( j1)⊕ρ⊗τ( j2). In this formula, ρ is an irreducible rep-
resentation of WF , τ( j) is the j-dimensional complex representation of SL(2,C). We define
the compact orbit of φ as follows:

Ot(φ) = {
2
⊕

r=1

ψr ⊗ρ⊗τ( jr) : ψr ∈Ψt(WF ), 1≤ r ≤ 2}/∼,

where as before, ∼ denotes the equivalence relation of conjugacy in GL(2,C). Each partition
j1 + j2 = 2 determines an orbit. The disjoint union of these orbits, one of each partition of
2, creates a complex affine algebraic variety with finitely many irreducible components. This
variety is smooth by [4].

Also, let G 0
2 (F) be the set of equivalence classes of irreducible 2-dimensional smooth (com-

plex) representations of WF . Let A 0
2 (F) be the subset of A t

2 (F) consisting of equivalence



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 291

classes of irreducible cuspidal representations of GL(2, F). The local Langlands correspon-
dence gives a bijection,

τ : G 0
2 (F)

//A 0
2 (F) .

We will use the local Langlands correspondence for GL(2) [7, 10, 11, 14]:

πF : Φ(GL(2))→ I r r(GL(2)).

Lemmas 1.1 and 1.2 in [15] explain the formula of the base change. Now let
z j =ψ j($F ), we have the map:

ψ1⊗τ( j1)⊕ψ2⊗τ( j2) 7−→ (z1, z2).

This map gives a bijection
Ot(φ)−→ s ym2(T).

So we will write the L-parameter

φ =ψ1⊗τ( j1)⊕ψ2⊗τ( j2)

as
z1.τ( j1)⊕ z2.τ( j2).

After base change has been applied, this L-parameter becomes

z f
1 .τ( j1)⊕ z f

2 .τ( j2).

The Steinberg representation St2 has L-parameter 1⊗τ(2).

Theorem 4. Let φ = 1⊗τ(2) and let Ot(φ) be the compact orbit of φ. Then we have

BC : T→ T, z 7→ z f .

(i) This map has degree f , and so at the level of the K-theory group K1, BC induces the map

Z→ Z, α1→ f ·α1

of multiplication by the residue degree f , where α1 denotes a generator of the group
K1(T)∼= Z.

(ii) At the level of the K-theory group K0, BC induces the identity map

Z→ Z, α0 7→ α0,

where α0 denotes a generator of K0(T)∼= Z.



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 292

Proof. Since this map has degree f then 1 has been proved. Since α0 is the trivial bundle
of rank 1 over T then 2 has been showed.

1⊗τ(2) 7−→ St F
27−→ 7−→

1⊗τ(2) 7−→ StE
2

We can generalize this in the following form:

σ⊗τ(2) 7−→ St F
27−→ 7−→

σ⊗τ(2) 7−→ StE
2

Next we define the L-parameter φ to be: φ = ρ⊗1⊕ρ⊗1 where ρ is a unitary character of
WF . The unitary characters of WF factor through F× and we have F× ∼= 〈$F 〉 ×UF . We will
take ρ to be trivial on 〈$F 〉, and then regard ρ as a unitary character of UF . The group UF
admits countably many such characters ρ. In this case the compact orbit is symmetric square
of the circle T:

Ot(φ)∼=Ot(BC(φ))∼= S ym2(T) := T2
�

Z/2Z= T2/W.

Lemma 3. The symmetric square T2/W has the homotopy type of a circle

T2/W∼ T

(z1, z2) 7−→ z1z2.

Proof. By sending the pair z = (z1, z2) to a unique monic polynomial

(z1, z2) 7−→ z2+ a1z+ a0, a0 6= 0

with roots z1, z2. It follows that

S ym2(T)∼= {z2+ a1z+ a0 : a0 6= 0} ∼h T,

since the space of coefficients a1, a0 is contractible. Therefore,

S ym2(T)∼h T

using the map (z1, z2) 7→ z1 · z2.

Let πF be the local Langlands correspondence

πF : Φ(GL(2))→ I r rGL(2)



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 293

and let x = diag(t1, t2) be a diagonal element in the standard maximal torus T of GL(2).
Then

χ : x 7→ πF (ρ)(t1, t2)

is a unitary character of T . Let σ be an unramified unitary character of T , and form the
induced representation IndG

T U(σ⊗χ) which is an irreducible unitary representation of G. Let
σ vary over all unramified unitary characters of T , then we obtain a subset of the unitary
dual of G. This subset has the structure of a symmetric square of T. The consequence for UF
admits countably many unitary characters is the unitary dual of G contains countably many
subspaces (in the Fell topology) each with the structure S ym2(T). We are concerned with the
effect of base change E/F on each of these compact spaces.

Theorem 5. Let T2/W denote one of the compact subspaces of the unitary principal series of
GL(2). Then we have

BC : T2/W→ T2/W, (z1, z2) 7→ (z
f
1 , z f

2 )

(i) At the level of the K-theory group K1, BC induces the map

Z→ Z, α1 7→ f ·α1

of multiplication by f , where f is the residue degree and α1 denotes a generator
of K1(T) = Z.

(ii) At the level of the K-theory group K0, BC induces the identity map

Z→ Z, α0 7→ α0,

where α0 denotes a generator of K0(T) = Z.

Proof. From Lemma 3 we have this commutative diagram:

S ym2(T)

BC
��

h // T

BC∗

��

S ym2(T)
h

// T

where BC(z1, z2) = (z
f
1 , z f

2 ), BC∗(z) = z f and h(z1, z2) = z1 · z2. Since

(z1 · z2)
f = z f

1 · z
f
2

we have K j(BC) = K j(BC∗), but BC∗ is a map of degree f . Therefore,

K1(BC)(α1) = f ·α1 and K0(BC)(α0) = α0

where α1 is a generator of K1(T) = Z and α0 is a generator of K0(T) = Z.

Therefore, the K-theory for the trivial type (I ,1I) would be as follows:



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 294

Theorem 6.
K jC

∗
r (s) = K j(S ym2(T)

⊔

T)∼= Z2

Proof. Proof immediately follows from Theorems 4 and 5 .

Definition 3. Let E/F be a quadratic extension and let χ be a character of E×. The pair
ϑ = (E/F,χ) is called admissible if

(i) χ does not factor through the norm map NE/F : E×→ F× and,

(ii) if χ | U 1
E does factor through NE/F , then E/F is unramified.

Let P2(F) be the set of isomorphism classes of admissible pairs ϑ. The map

P2(F)→G 0
2 (F), ϑ 7→ Ind

E/F
χ

is bijection according to [5, p. 215], where χ is a character of WE via the class field theory
isomorphism W ab

E
∼= E× and Ind

E/F
is the functor of induction from representations of WE to

representations of WF .
The tempered dual of GL(2) consists of the cuspidal representations with unitary central

character, the unitary twists of the Steinberg representation, and the unitary principal series.
It is clear that in the admissible pairs we can describe what is happening so we further restrict
ourselves to admissible pairs ϑ for which E/F is totally ramified and χ is a unitary character.
This ensures that π := Ind

E/F
χ is unitary. Therefore det(π) is unitary and τ(π) has unitary

central character. The cuspidal representations of GL(2) with unitary central character ar-
range themselves in the tempered dual as a countable union of circles. For each circle T, we
select an admissible pair ϑ for which τ(π) ∈ T and label this circle as Tϑ.

Theorem 7. Let E′/F be an unramified extension of odd degree. Then we have:

(i) Base change is a proper map.

(ii) When we restrict base change to one circle we get the following:

BC : Tϑ→ T(EE′/E′ ,χ
E′
)
, z 7→ z f (E′/F)

with
χ

E′
= χ ◦ N

EE′/E
.

Proof. Since we are considering circles indexed by characters of ÓUF , then the base change
maps each circle into one precise circle.

Let D be a compact subset of Tχ
E′

which is a closed arc in Tχ
E′

. Then we may write

D= {eiθ ∈ Tχ
E′

: θ0 ≤ θ ≤ θ1, θ ∈ [0, 2π]},



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 295

and we have the pre-image of this arc

BC−1(D) = {eiθ ∈ TχF
: θ0/ f ≤ θ ≤ θ1/ f , θ ∈ [0,2π]}

which is closed arc in TχF
. It follows that BC−1(D) is compact. Therefore, the base change

map BC is a proper map and then (1) has been proved. Now, let ρ ∈ G 0
2 (F), then the order

of the cyclic group of all unramified characters χ such that χρ ' ρ is called a torsion number
of ρ and denotes by ν(ρ). Put

σ = Ind
E/F
χ, π= τ(σ) and σ

E′
= Ind

EE′/E′
χ

E′
= σ|WE′

.

The proof of Theorem 3.3 in [7] shows that the representation σ is totally ramified, in the
sense that ν(σ) = 1. Theorem 4.6 in the same reference shows that the pair (EE′/E′,χE′) is
admissible. Also, we have the map

τ(σE′) = BCE′/Fπ.

By Proposition 7.2 in [16], EE′/E is unramified, whenever the extension E′/F is unramified
and

eEE′/F = eEE′/E′ × eE′/F = eEE′/E × eE/F .

and it follows that
eEE′/E′ = eE/F = 2.

Since EE′/E′ is quadratic extension, EE′/E′ is totally ramified. Therefore σE′ is totally ram-
ified, in another words ν(σE′) = 1. Therefore, the base change maps each circle to another
circle and its given by

z 7−→ z f (E′/F).

If the extension E′/F is a finite unramified Galois extension, then the cuspidal part of the
tempered dual of GL(2) is a countable disjoint union of circles and has the structure of a
locally compact Hausdorff space. The base change map

BC :
⊔

Tϑ→
⊔

Tζ

is a proper map, where ϑ an admissible pair, E/F totally ramified, χ unitary and ζ= (EE′/E′,η).
Therefore, there is a functorial map at the level of K-theory groups

K j(BC) :
⊕

Zζ→
⊕

Zϑ.

Each K-group is a countably generated free abelian group:

K j(
⊔

Tϑ)∼=
⊕

Zϑ, K j(
⊔

Tζ)∼=
⊕

Zζ,

where Zϑ and Zζ denote a copy of Z, j = 0, 1. The base change map selects among the
admissible pairs ζ those of the form (EE′/E′,χ

E′
), where

χ
E′
= χ ◦ NEE′/E .



W. Aeal / Eur. J. Pure Appl. Math, 6 (2013), 282-298 296

Theorem 8. When we restrict K1(BC) to the direct summand Z(EE′/E′,χ
E′
) we get the following

map:

Z(EE′/E′,χ
E′
) −→ Zϑ,

x 7−→ f (E′/F) · x .

On the remaining direct summands, K1(BC) = 0. When we restrict K0(BC) to the direct sum-
mand Z(EE′/E′,χ

E′
) we get the following map:

Z(EE′/E′,χ
E′
) −→ Zϑ,

x 7−→ x .

On the remaining direct summands, K0(BC) = 0.

Here’s a summary of the cases in this work:

(i) On the admissible side, if we have the following:

(a) The twist of Steinberg : {ψ⊗ St(2) :ψ ∈Ψt(WF )} ∼= T.
(b) The cuspidal : {ψ⊗π : π ∈A 0GL(2)} ∼= T.

(c) U.P.S : {IndG
B

�

x ∗
0 y

�

7→ψ1 ·ψ2 :ψ j ∈Ψt(WF )} ∼= T2, when ψ1 6=ψ2.

(d) U.P.S : {IndG
B

�

x ∗
0 y

�

7→ψ1 ·ψ2 :ψ j ∈Ψt(WF )} ∼= T2�(Z/2Z), when ψ1 =ψ2

(ii) Then, the K-theory groups of each of these cases are as follows:

(a) K jC0(T) = Z.
(b) K jC0(T) = Z.

(c) K jC0(T2) = K j(T2) = Z
⊕

Z.

(d) K jC0
�

T2�(Z/2Z)
�

= K jC0(T) = K j(T)∼= Z.

(iii) The homology groups of these cases are as follows:

(a) Heven = Z2 = Hodd .

(b) Heven = Z= Hodd .

(c) Heven = Z2 = Hodd .



REFERENCES 297

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