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Base change and K-theory for orbits of p-adic GL(n).

1 Introduction

The Langlands Program and base change, one of the most studied manifestations of the principle of functoriality, provides a concrete example of the interaction between arithmetic and C*-algebras. In [8], Plymen and the author have studied the effect of base change on the reduced C*-algebra for GL(1) and GL(2) over a finite Galois extension E/F of nonarchimedean local fields. And in [7], base change was interpreted as Galois-fixed points at the level of K-theory groups.

Although a deep and intricate theory in the representation theory of reductive groups, base change [1] has a rather simple interpretation in terms of topological K-theory for C (*)-algebras. Two important arithmetic invariants of the extension E/F play a role in K-theory: the residue degree f = f(E/F) and the Hasse-Herbrand function, see [8] for further details.

Let F be nonarchimedean local field with zero characteristic and let G = GL(n) = GL(n, F). Let [C*.sub.r](G) denote the reduced C*-algebra of G and let [OMEGA] be a component in the Bernstein variety [OMEGA](G), see [2]. The Bernstein decomposition of the Hecke algebra of G

[mathematical expression not reproducible]

determines a canonical decomposition of the reduced C*-algebra

[mathematical expression not reproducible]

Each component in the C*-direct sum is a C*-ideal.

Let Irr(G) denote the smooth dual and let [Irr.sup.t](G) [subset] Irr(G) denote the tempered dual of G. The tempered dual is in bijection with a locally compact Hausdorff space, also called the parameter space of G, which we still denote here by [Irr.sup.t](G). The noncommutative space [C*.sub.r](G) is strongly Morita equivalent to the commutative C*-algebra Co([Irr.sup.t](G)), see [9]. The Bernstein decomposition of [C*.sub.r](G) determines a decomposition of [Irr.sup.t](G):

[mathematical expression not reproducible]

As a consequence of this, we have

[mathematical expression not reproducible]

Let E/F be a finite Galois extension. The domain of an L-parameter is the Weil-Deligne group

[L.sub.F] = [W.sub.F] x SL(2,C)

where [W.sub.F] is the Weil group of F, see [12]. Base change is defined by restriction of an L-parameter of [L.sub.F] to [L.sub.E] . An L-parameter [phi] is tempered if the image of [W.sub.F] under [phi] is bounded, see [3]. Base change, therefore, determines a map of tempered duals

[b.sub.E/F]:[Irr.sup.t]([G.sub.F])[right arrow][Irr.sup.t]([G.sub.E]).

In this article we investigate this map at the level of K-theory for certain components [Irr.sup.t][(G).sub.[OMEGA]].

Brodzki and Plymen [4] proved that there is a complex structure on the smooth dual of GL(n). Each component Irr[(G).sub.[OMEGA]] has the structure of a complex affine variety with finitely many components. The group of unramified characters of [W.sub.F] act in the set of equivalent classes of L-parameters. For each equivalent class [[phi]], the compact orbit [O.sup.t]([phi]) has the structure of a compact orbifold. The parameter space for the tempered representations of GL(n) which have a Iwahori-fixed vector is the extended quotient for the action of the symmetric group [S.sub.n], denoted by [T.sup.n]//[S.sub.n] (see [10, 4]). The reduced Iwahori C*-algebra [C*.sub.r](G//I) is strong Morita equivalent to the commutative C*-algebra C([T.sup.n]//[S.sub.n]). The extended quotient [T.sup.n]//[S.sub.n] contains the ordinary quotient [T.sup.n]/[S.sub.n]. The latter is the parameter space of the reduced spherical C (*)-algebra [C*.sub.r](G//K), where K is a maximal compact subgroup of G. Therefore, we have


After recalling some background in section 2, in sections 3 and 4 we survey part of the results of [8]. Specifically, using the C*-version of Satake isomorphism we show that, at the level of [K.sup.1]-groups, base change is multiplication by the residue degree f = f(E/F) of the extension (Theorem 3.2). To each partition of n corresponds a component of the extended quotient and the induced base change map at the level of K-theory is computed for [C*.sub.r](G//I). The partition is preserved under base change (Theorem 4.5).

Section 5 contains new material. Generalizing the previous results, we consider general abstract orbits for GL(n) and compute the K-theory map associated with base change. We prove that, for this rather general case, base change at the level of [K.sup.1], is multiplication by the residue degree f = f (E/F) (Theorem 5.5). Crucial to the computation is the ring structure of K* = [K.sup.0] [phi] [K.sup.1], and particularly its Z/2Z-grading.

2 Base change as a map of affine varieties

We begin this section with some background on local fields. A nonarchimedean local field with zero characteristic is a finite extension of [Q.sub.p], the field of p-adic numbers. This is a locally compact, non-discrete, totally disconnected topological field with an absolute value x [right arrow] |x|. The ring of integers is denoted by [o.sub.F] = {x [member of] F : |x| [less than or equal to] 1}. The maximal ideal of [o.sub.F] is defined by [p.sub.F] = {x [member of] F : |x| < 1}. [p.sub.F] is principal and we call a uniformizer any element [[??].sub.F] [member of] [p.sub.F]\[p.sup.2.sub.F] such that [p.sub.F] = [[??].sub.F][o.sub.F]. The residue field of F is the finite field [k.sub.F] = [o.sub.F]/[p.sub.F].

Given a Galois extension E/F, there is a corresponding Galois extension of residue fields [k.sub.E]/[k.sub.F]. The residue degree is defined to be f = f (E/F) = [[k.sub.E] : [k.sub.F]]. The ramification index is the integer e = e(E/F) such that [[??].sup.e.sub.F] = [[??].sub.F]. It is a fundamental relation that

[E:F] = n = ef.

The extension is called unramified if [E : F] = [[k.sub.E] : [k.sub.F]] = f, and is called totally ramified if [E : F] = e.

The Galois group Gal([k.sub.E]/[k.sub.F]) is cyclic, generated by the arithmetic Frobenius element [phi] = (x [right arrow] [x.sup.qE]), where [q.sub.E] is the number of element of [k.sub.E]. The geometric Frobenius element [PHI] is, by definition, the inverse of the arithmetic Frobenius, i.e the element [PHI] [member of] Gal([k.sub.E]/[k.sub.F]) such that ([PHI][(x)).sup.qE] = x. In particular, [PHI] is also a generator of Gal([k.sub.E]/[k.sub.F]).

Let E/F be a finite Galois extension of local fields and let the corresponding Weil groups be denoted [W.sub.E], [W.sub.F] (see [12] for an account on Weil groups). We have the following standard short exact sequence

[mathematical expression not reproducible]

The pre-image [I.sub.E] = [d.sup.-1] {0} [congruent to] [W.sub.E]/Z is the maximal compact subgroup of [W.sub.E] and is called the inertia group of E.

Now an unramified quasicharacter [psi] of [W.sub.E] is given by the following simple formula:

[psi](w) = [z.sup.dE(w)]

where z [member of] [C.sup.x]. Note that [W.sub.E] [subset] [W.sub.F]. The base change formula for a quasicharacter X of [W.sub.F] is given by

[b.sub.E/F]([chi]) = [chi]|[W.sub.E]. (2.1)

We now quote the following result, see [8, Lemma 3.5]

Lemma 2.1. Under base change we have

[mathematical expression not reproducible]

for all w [member of] [W.sub.E].

We can remember this result with the (informal) equation

[z.sup.E/F] = [z.sup.f]

The above equation defines the base change map for quasicharacters.

Let [PSI]([W.sub.F]) denote the group of unramified quasicharacters of [W.sub.F]. Then we have

[PSI]([W.sub.F])[congruent to][C.sup.x],[psi][??][psi]([[??].sub.F])

where [W.sub.F] is a uniformizer of F. A unitary quasicharacter is called a character. The group of unramified characters of [W.sub.F] is

[[PSI].sup.t]([W.sub.F])[congruent to]T

where T = {z [member of] C : |z| = 1} is the unit circle. [[PSI].sup.t]([W.sub.F]) is the maximal compact subgroup of [PSI]([W.sub.F]).

Let [L.sub.F] denote the Weil-Deligne group [12]

[L.sub.F] = [W.sub.F] x SL(2,C).

A Langlands parameter (or L-parameter) is a continuous homomorphism

[phi]:[L.sub.F][right arrow]GL(n,C)

where GL(n, C) is given the discrete topology, such that [phi]([[PHI].sub.F]) is semisimple ([[PHI].sub.F] is a geometric Frobenius in [W.sub.F]). The set of equivalence classes of L-parameters is denoted [PHI](GL(n)). We will use the local Langlands correspondence for GL(n) [6]:

[[pi].sub.F][PHI](GL(n))[right arrow]Irr(GL(n)).

Consider the L-parameter

[phi] = 1[cross product][tau]([j.sub.1])[symmetry]***[symmetry]1[cross product][tau]([j.sub.k])

where [tau](j) is the j-dimensional complex representation of SL(2, C) and [j.sub.1] +...+[j.sub.k] = n.

Definition 2.2. We define the orbit of [phi] as follows:

O([phi]) = {[[psi].sub.1][cross product][tau]([j.sub.1])[symmetry]***[symmetry][[psi].sub.k][cross product][tau]([j.sub.k]):[[psi].sub.r][member of][PSI]([W.sub.F]),1[less than or equal to] r [less than or equal to] k}/~

where ~ denotes the equivalent relation of conjugacy in GL(n, C).

The unramified quasicharacters [[psi].sub.1],...,[[psi].sub.k] of [W.sub.F] determine uniquely complex numbers [z.sub.1],...,[z.sub.k] which serve as coordinates on the complex algebraic variety O([phi]). Under base change E/F, these coordinates change precisely in the following way:


Let [phi] be an irreducible representation of the Weil-Deligne group [L.sub.F] = [W.sub.F] x SL(2,C). On restriction to [L.sub.E] = [W.sub.E] x SL(2,C), [phi] will split into irreducible representations:

[[tau].sub.1][symmetry]***[symmetry] [[tau].sub.r].

The unramified twist [psi] * [phi] will then split as follows:


where [chi] is the restriction of [psi] to [L.sub.E]. This implies, writing BC = [b.sub.E/F], that

BC:O([phi])[right arrow]O(BC([phi])).

The coordinate z on the variety O([phi]) is sent, via BC, to the coordinates ([z.sup.f],..., [z.sup.f]) on the variety O(BC([phi])).

If now [phi] is a direct sum of irreducible representations

[phi] = [[phi].sub.1][symmetry]***[symmetry][[phi].sub.k]

then the coordinates ([z.sub.1],...,[z.sub.k]) on the orbit O([phi]) are sent, via BC, to the coordi-nates ([z.sup.f.sub.1],..., [z.sup.f.sub.1],..., [z.sup.f.sub.k],..., [z.sup.f.sub.k]) on the orbit O(BC([phi])).

Our aim is to understand, at the level of K-theory, this map for the underlying compact orbits. This will be achieved in the last section.

Any L-parameter can be written in a canonical form, moving if necessary to another point in the orbit,

[phi] = [[phi].sub.1][symmetry]...[symmetry][[phi].sub.1][symmetry]...[symmetry][[phi].sub.k][symmetry]...[symmetry][[phi].sub.k].(2.2)

Each [[phi].sub.j] is repeated [r.sub.j] times and the representations [[phi].sub.j][|.sub.F]xSL(2,C) are irreducible and pairwise inequivalent. The orbit O([phi]) of [phi] is [4, Th. 1.6]

[mathematical expression not reproducible]

and the associated compact orbit is

[mathematical expression not reproducible]

The quotient [([C.sup.x]).sup.n]/[S.sub.n] is usually denoted [Sym.sup.n]([C.sup.x]) and is called the symmetric product of [([C.sup.x]).sup.n].

An orbit O([phi]) has the structure of a complex algebraic variety with finitely many components, see [4]. Recall that a regular map [phi] : X [right arrow] Y of affine varieties is called finite if C[X] is integral over C[Y]. As a map of algebraic varieties, base change

BC:O([phi])[right arrow]O(BC([phi]))

is a finite morphism, see [8, Th. 3.3].

Example 2.3. Let St denote the Steinberg representation of GL(n). Its Langlands parameter is 1 [cross product] [tau] (n). The orbit (a hyperbola) is given by

O(St) = V(xy - 1).

Note that [C.sup.x] = A\{0} can be seen as an affine variety. In fact, the map

A\{0}[right arrow]V(xy - 1), t[??](t,[t.sup.-1])

is an isomorphism of the (quasi-projective) algebraic variety A\{0} into the affine variety V = V(xy - 1). The correspondent coordinate ring is

C[V] = C[x,y]/(xy - 1)[congruent to]C[x,[x.sup.-1]].

Base change BC induces the pullback

C[t,[t.sup.-1][right arrow]]C[t,[t.sup.-1], t[??][t.sup.f].

3 Spherical representations

Let K be a maximal compact subgroup of GL(n, F). Since they are all conjugate, we may take K = GL(n, [o.sub.F]), where [o.sub.F] is the ring of integers of F. In this section we are mainly interested on the reduced spherical C*-algebra [C*.sub.r](G//K) [10]. This is the sub-C (*)-algebra of [C*.sub.r](GL(n)) whose tempered dual consist of representations which have a K-fixed vector. Our aim is to compute the K-theory map induced by base change. For K-theory of C (*)-algebras we refer the reader to [11].

We call the orbit which contains the point [phi] = 1 [cross product] 1 [symmetry] ... [symmetry] 1 [symmetry] 1 the Borel orbit and we write [O.sub.Borel] = O([phi]). Any L-parameter which corresponds to a spherical representation has the form

[phi] = [[chi].sub.1][cross product]1[symmetry]...[symmetry][[chi].sub.n][cross product]1.

Therefore, it belongs to the Borel orbit and we conclude that

[O.supt]([phi]) = [O.supt](BC([phi])) = [O.supt.sub.Borel].

Lemma 3.1. The symmetric product [T.sup.n]/[S.sub.n] has the homotopy type of a circle.

Proof. Send the unordered n-tuple [z] = [[z.sub.1],...,[z.sub.n]] to the unique polynomial with roots [z.sub.1],...,[z.sub.n] and leading coefficient 1

[z.sub.1],...,[z.sub.n][??][z.sup.n] + [a.sub.n-1][z.sup.n-1] + ...+ [a.sub.1]z + [a.sub.0], [a.sub.0] [not equal to] 0.

We have then

[Sym.sup.n]([C.sup.x])[congruent to]{[z.sup.n] + [a.sub.n-1][z.sup.n-1 + ...+[a.sub.1]z + [a.sub.0]:[a.sub.0] [not equal to] 0}~h[C.sup.x], since the space of coefficients [a.sub.n-1], ...,[a.sub.1] is contractible. Hence

[Sym.sup.n](T)~h T

via the map which sends [[z.sub.1], ...,[z.sub.n]] to the product [z.sub.1]...[z.sub.n].

Theorem 3.2. ([8, Th.4.3]) Let [T.sup.n]/[S.sub.n] denote the compact orbit of a spherical representation in the parameter space [Irr.sup.t](G). Let [phi] be the correspondent L-parameter. Then [phi] and BC([phi]) belong to the Borel orbit [O.sup.t.sub.Borel] and we have

BC:[T.sup.n]/[S.sub.n][right arrow][T.sup.n]/[S.sub.n], ([z.sub.1],...,[z.sub.n])[??]([z.sup.f.sub.1],...,[z.sup.f.sub.n]).

(i) At the level of the K-theory group [K.sup.1], BC induces the map

Z[right arrow]Z,[[alpha].sub.1][??] f ** [[alpha].sub.1]

of multiplication by f, where f is the residue degree and [[alpha].sub.1] denotes a generator of [K.sup.1](T) = Z.

(ii) At the level of the K-theory group [K.sup.0], BC induces the identity map

Z[right arrow]Z,[[alpha].sub.0][??] [[alpha].sub.0]

where [[alpha].sub.0] denotes a generator of [K.sup.0](T) = Z.

For the sake of completeness we include the proof.

Proof. The map T [right arrow] Z, z [??] [z.sup.f] has degree f and so, at the level of the K-theory group [K.sup.1] , base change is multiplication by f

Z[right arrow]Z,[[alpha].sub.1][??] f ** [[alpha].sub.1]

where [K.sup.1](T) = Z = On the other hand, every complex vector bundle over T is trivial and the pullback of a trivial bundle is always trivial and of the same dimension. Therefore, at the level of the K-theory group [K.sup.0], base change induced the map

Z[right arrow]Z,[[alpha].sub.0][??] [[alpha].sub.0]

where [K.sup.0](T) = Z = <[[alpha].sub.0]>. Now, from Lemma (3.1) we have a commutative diagram

[mathematical expression not reproducible]

Here, BC([z.sub.1],...,[z.sub.n]) = ([z.sup.f.sub.1],...,[z.sup.f.sub.n]), h is the homotopy map h([[z.sub.1],...,[z.sub.n]]) = [z.sub.1]...[z.sub.n] and [??] is the map z [??] [z.sup.f]. Since the diagram is commutative, we have [K.sup.j] (BC) = [K.sup.j]([??]). But [??] is a map of degree f. Therefore,

[K.sup.1](BC)([[alpha].sub.1]) = f.[[alpha].sub.1]

where [[alpha].sub.1] is a generator of [K.sup.1](T) = Z and [[alpha].sub.0] is a generator of [K.sup.0](T) = Z.

The C*-version of Satake isomorphism

The reduced spherical C (*)-algebra [C*.sub.r](G//K) is strong Morita equivalent to a commutative C (*)-algebra of continuous functions


In fact, there is an isomorphism of C (*)-algebras [10]:

[C*.sub.r](G//K) [congruent to] C([T.sun.n]/[S.sub.n]).

which can be interpreted as the C (*)-algebra version of Satake isomorphism, see [5]. The group algebra C[[Z.sup.n]] will Fourier transform to a dense subalgebra of C([T.sup.n]/[S.sub.n]), while the [S.sub.n]-invariant subalgebra [mathematical expression not reproducible] will Fourier transform to a dense subalgebra of C([T.sup.n]/[S.sub.n]). In particular, we conclude that the parameter space of [C*.sub.r](G//K) is precisely the compact orbifold [T.sup.n]/[S.sub.n].

4 Representations with a Iwahori-fixed vector

In this section we consider the reduced Iwahori C*-algebra [C*.sub.r](G//I) [10]. The associated parameter space is a finite disjoint union of compact orbifolds and so is compact.

Let [[??].sub.F] be a fixed uniformizer of F. The (standard) Iwahori subgroup of GL(n, F) is the subgroup

[mathematical expression not reproducible]

We have the following classification of the representations of GL(n, F) with a Iwahorifixed vector.

Theorem 4.1. [10, Th. 3.1, p.123] Let I be the Iwahori subgroup of GL(n). Let n = [n.sub.1] + ... + [n.sub.k] a partition of n, let [mathematical expression not reproducible] the Steinberg representation of GL([n.sub.j]) and let [[chi].sub.1], ... ,[[chi].sub.k] be unramified characters of GL(1). Then the representation

[mathematical expression not reproducible]

is unitary, irreducible, tempered and admits I-fixed vectors. Moreover, all such representations are accounted on this way.

Let [GAMMA] be a group and X a space. Suppose that [GAMMA] acts on X. The action is said to be proper if the map [GAMMA] x X [right arrow] X x X, ([gamma], x) [??] (x, [gamma]X) is proper (i.e. the inverse image of compact sets are compact). One important property of proper actions is that equipped with the quotient topology, the space of orbits X/[GAMMA] is Hausdorff. Define

[??] = {([gamma],x)[member of][GAMMA] x X:[gamma]x = x}

and g.([gamma],x) = (g[gamma][g.sup.-1],gx),

for all g [member of] [GAMMA] and ([gamma], x) [member of] [??]. Since (g[gamma][g.sup.-1])(gx) = g([gamma]x) = gx, [GAMMA] acts on [??].

Definition 4.2. The extended quotient associated to the action of [GAMMA] on X is the quotient space X//[GAMMA] = [??]/[GAMMA], where [??]/[GAMMA] is the ordinary quotient.

Let X = [T.sup.n] be the compact n-torus. The symmetric group [GAMMA] = [S.sub.n] acts on X by permuting the coordinates and we can form the extended quotient [T.sup.n]//[S.sub.n]. For [gamma] [member of] [GAMMA] let [X.sup.[gamma]] denote the set of all points in X fixed by [gamma]

[X.sup.[gamma]] = {x[member of]X:[gamma] x = x}

and let [Z.sub.[gamma]] denote the centralizer of [gamma] in [GAMMA]. Then the extended quotient is given by the (finite) disjoint union

[mathematical expression not reproducible]

where one [gamma] is chosen in each [GAMMA]-conjugacy class from Cl([GAMMA]).

The conjugacy class of [gamma] [member of] [S.sub.n] contains all the elements of [S.sub.n] with the same cycle type, and the cycle types are in one-to-one correspondence with partitions of n. The partition

[n.sub.1] + ... +[n.sub.1] + ... +[n.sub.k] + ... [n.sub.k] = [r.sub.1][n.sub.l] + ... +[r.sub.n][n.sub.k] = n, n [greater than or equal to] [n.sub.1] > [n.sub.2] >...> [n.sub.k] [greater than or equal to] 1, corresponds to the elements of [S.sub.n] whose type is characterized by having k different cycles, where [r.sub.i] is the number of cycles of equal size [n.sub.i], i = 1,...,k.

The component [X.sup.[gamma]]/[Z.sub.[gamma]] of [T.sup.n]//[S.sub.n] associated with the partition n = [r.sub.1][n.sub.1] + ... + [r.sub.k][n.sub.k] of n is a product of symmetric products

[mathematical expression not reproducible]

In particular, the components T associated with [gamma] = (123... n) and [Sym.sup.n](T) associated with [gamma] = 1 are always present in the extended quotient

[mathematical expression not reproducible] (4.1)

Theorem 4.3. [10, Th. 3.2, p.124] The parameter space for tempered representations of GL(n) which admit I-fixed vectors is the extended quotient [T.sup.n]//[S.sub.n].

Example 4.4. We illustrate this result for GL(5), by computing the L-parameters and the extended quotient. We have:
Partition   Cycle type   L-parameter [phi]

5 + 0       (12345)      1 [cross product] [tau](5)
4 + 1       (1234)       1 [cross product] [tau](4) [symmetry] 1
                         [cross product] 1
3 + 2       (123)(45)    1[cross product] [tau](3) [symmetry] 1
                         [cross product] [tau](2)
3 + 1 + 1   (123)        1 [cross product] [tau](3) [symmetry] 1
                         [cross product] 1 [symmetry] 1 [cross product]
2 + 2 + 1   (12)(34)     1 [cross product] [tau](2) [symmetry] 1
                         [cross product] [tau](2) [symmetry] 1
                         [cross product] 1
2+1+1+1     (12)         1 [cross product] [tau](2) [symmetry] 1
                         [cross product] 1 [symmetry] 1 [cross product]
                         1 [symmetry] 1 [cross product] 1
1+1+1+1+1   1            1 [cross product] 1 [symmetry] 1
                         [cross product] 1 [symmetry] 1 [cross product]
                         1 [symmetry] 1 [cross product] 1 [symmetry] 1
                         [cross product] 1

The extended quotient is, according to the partition order

[T.sup.5]//[S.sub.5] = T[??][T.sup.2][??][T.sup.2]T x [Sym.sup.2][??]T x [Sym.sup.2][T] [??]T x [Sym.sup.3](T)[??][Sym.sup.5](T).

Let [phi] be the L-parameter associated with a representation with a Iwahori-fixed vector. Then base change is given by

[O.sup.t]([phi])[right arrow][O.sup.t](BC([phi]))[congruent to]([z.sub.1],...,[z.sub.n])[??]([z.sup.f.sub.1],...,[z.sup.f.sub.n])

and we have

[mathematical expression not reproducible]

Now, since [mathematical expression not reproducible] has the homotopy type of a circle (see Lemma 3.1), we conclude that

[mathematical expression not reproducible]

Theorem 4.5. [K.sub.j]([T.sup.n]//[S.sub.n]) is a finite direct sum of copies of Z, labelled by partitions [gamma] of n. Denote by [[alpha].sub.[gamma]] a generator of the abelian group indexed by the partition of type [gamma]. At the level of K-theory groups BC induces the map

(i) [K.sup.1](BC) : [[alpha].sub.[gamma]] [??] f.[[alpha].sub.[gamma]].

(ii) [K.sup.0](BC) : [[alpha].sub.[gamma]] [??] [[alpha].sub.[gamma]].

In particular, [K.sup.j](BC) preserves the type.

Proof. This follows from Lemma 3.1, from Theorem 3.2 and from (4.1).

We give an example showing that base change preserves the type which illustrates the general argument.

Example 4.6. In example 4.4 the Langlands parameters [[phi].sub.(123)] and [[phi].sub.(12)(34)] have different type but isomorphic orbits,

[O.sup.t]([[phi].sub.(123)])[congruent to][O.sup.t]([[phi].sub.(12)(34)])[congruent to]T x [Sym.sup.2](T).

Now, consider the associated L-parameters

[[phi].sub.(123)] = [[phi].sub.1] = [[chi].sub.1][cross product][tau](3)[symmetry][[chi].sub.2][cross product]1[symmetry][[chi].sub.3][cross product]1


[[phi].sub.(12)(34)] = [[phi].sub.2] = [[xi].sub.1][cross product][tau](2)[symmetry][[xi].sub.2][cross product][tau](2)[[xi].sub.3][cross product]1.

They have type (123) and (12)(34), respectively. Base change is given by restricting to the Weil-Deligne group [L.sub.E] = [W.sub.E] X SL(2,C). Then

[[phi].sub.1]|[L.sub.E] = [[chi].sub.1][L.sub.E][cross product][tau](3)[symmetry][[chi].sub.2]|[L.sub.E][cross product]1[symmetry][[chi].sub.3]|[L.sub.3][cross product]1

has the type (123) and

[[phi].sub.2]|[L.sub.E] = [[xi].sub.1]|[L.sub.E][cross product][tau](2)[symmetry][[xi].sub.2]|[L.sub.E][cross product][tau](2)[symmetry][[xi].sub.3]|[L.sub.E][cross product]1

has type of (12)(34). Therefore, the type is preserved.

Remark 4.7. The reduced Iwahori C*-algebra [C*.sub.r](G//I) for GL(n) is strong Morita equivalent to the algebra of continuous functions C([T.sup.n]//[S.sub.n]). Given a finite Galois extension E/F, there is an isomorphism of the corresponding C (*)-algebras

[C*.sub.r]([G.sub.E]//[I.sub.E])[congruent to][C*.sub.r]([G.sub.F]//[I.sub.F]).

From the above Theorem, base change induces a K-theory map

[K.sub.1][C*.sub.r]([G.sub.E]//[I.sub.E])[right arrow][K.sub.1][C*.sub.r]([G.sub.E]//[I.sub.E]), [[alpha].sub.[gamma]][??]f.[[alpha].sub.[gamma]], where f = f (E/F) is the residue degree and [[alpha].sub.[gamma]] is the generator of the component of [T.sup.n]//[S.sub.n] associated with the partition of type [gamma] of n. We conclude that base change is detected by K-theory except if E/F is totally ramified, in which case f (E/F) = 1.

5 K-theory for general orbits

In this section we consider more general compact orbits and compute the K-theory map induced by base change

[O.sup.t]([phi])[right arrow][O.sup.t]([phi])), ([z.sub.1],...,[z.sub.k])[??]([z.sup.f.sub.1],...,[z.sup.f.sub.1],...,[z.sup.f.sub.k],...,[z.sup.f.sub.k]).

The ring structure of K (*)([T.sup.n]) will play a fundamental role in our computations, see [11]. We start with the map

BC:[O.sup.t]([phi])[right arrow][O.sup.t](BC([phi])), z[??]([z.sup.f],...,[z.sup.f]) (5.1)

where [O.sup.t]([phi]) [congruent to] T and [O.sup.t](BC([phi])) [congruent to] [T.sup.k]. Define a pair of maps

[mathematical expression not reproducible]

where g(z) = [z.sup.f] and [DELTA](z) = (z, ...,z) is the diagonal. Then BC = [DELTA] o g and we have

[K.sup.j](BC) = [K.sup.j](g)[??][K.sup.j]([DELTA]):[K.sup.j]([T.sup.j])[right arrow][K.sup.j](T)

Now, [K.sup.j](g) is simply multiplication by f, [beta] [??] f.[beta], where [beta] is the generator of [K.sup.1](T) and [K.sup.0](g) is the identity map [alpha] [??] [alpha], where [alpha] is a generator of [K.sup.0](T). Hence,

[mathematical expression not reproducible]

and we only need to compute [K.sup.j]([DELTA]) for j = 0,1.

Computation of the map [K.sup.j]([DELTA])

[K.sub.1](C(T)) [congruent to] Z is generated by the class [u] [member of] [K.sub.1](C(T)), where u [member of] C(T) is the unitary element u(z) = z. We shall denote this class by [beta] = [u].

[K.sub.0](C(T)) = Z is generated by the class [1], where 1 is the identity of C(T) (which is clearly a projection). Denote this class by [alpha] = [1].

Let :[P.sub.j] [T.sup.2] [right arrow] T be the [] projection (j = 1, 2). Then, at the level of K-theory, we have [P*.sub.j] : [K.sub.1](C(T)) [right arrow] [K.sub.1](C([T.sup.2])) and each generator of [K.sub.1](C([T.sub.2])) is the pullback of the canonical generator of the circle

[e.sub.1] = [P*.sub.1]([beta]),[e.sub.2] = [P*.sub.2]([beta]),[beta][member of][K.sub.1](C(T)).

(here, * denotes the induced map at the level K-theory).

We may interpret [e.sub.j] as the class of the unitary [u.sub.j] ([z.sub.1], [z.sub.1]) = [z.sub.j] (j = 1, 2). Now, composing the diagonal map [DELTA] : T [right arrow] [T.sup.2] with each projection [P.sub.j] we get the identity [id.sub.T]. Therefore,

[DELTA]*([P*.sub.j]([beta])) = [DELTA]*([e.sub.j]) = [beta], for j = 1,2.

It is elementary to see that [DELTA]* : [K.sup.1]([T.sup.2]) [right arrow] [K.sup.1](T) sends an element (a, b) [member of] [Z.sup.2] into a + b [member of] Z:

[DELTA]*(a[symmetry]b) = [DELTA]*([P*.sub.1](a.[beta]))[symmetry][DELTA]*([P*.sub.2](b.[beta])) = [DELTA]*([P*.sub.1](a.[beta])) + [DELTA]*([P*.sub.2](b.[beta])) = (a + b).[beta]

More generally, take the n canonical generating unitaries [u.sub.1], ...,[u.sub.n], defined by [u.sub.j]([z.sub.1],...,[z.sub.n]) = [z.sub.j]. Let [P.sub.j] : [T.sup.n] [right arrow] T be the [] projection (j = 1,...,n). Then, each generator [e.sub.j] = [[u.sub.j] is the pullback of the generator of the circle, [e.sub.j] = [P*.sub.j]([beta]).

In order to simplify notation we will work with K-theory of spaces instead of K-theory of C (*)-algebras. We still denote the image of a generator [e.sub.j] under the isomorphism [K.sup.1](C([T.sup.n])) [K.sub.1]([T.sup.n]) by [e.sub.j]. The ring K*(C([T.sup.n]) = K*([T.sup.n]) is generated by the classes e1 = [[u.sub.1]], ... , [e.sub.n] = [[u.sub.n]] as the exterior ring over Z with basis [e.sub.1], ... , [e.sub.n]:

K*([T.sup.n])[congruent to][[LAMBDA].sub.Z][[e.sub.1],...,[e.sub.n]].

Moreover, [K.sup.0]([T.sup.n]) is generated (as a group) by the exterior product of an even number of elements from {[e.sub.1], ... , [e.sub.n]}, with no two indices repeated:

[K.sup.0]([T.sup.n])[congruent to]Z[1,[e.sub.1][LAMBDA][e.sub.j],[e.sub.i][LAMBDA][e.sub.j][LAMBDA][e.sub.k][LAMBDA][e.sub.l],...],

while [K.sup.1] ([T.sup.n]) is generated (as a group) by the exterior product of an odd number of elements, with no two indices repeated:

[K.sup.1]([T.sup.n])[congruent to]Z[[e.sub.i],[e.sub.i][LAMBDA][e.sub.j][LAMBDA][e.sub.k],...].

Example 5.1. We have

[K.sup.0]([T.sup.3])[congruent to]Z[1,[e.sub.1][LAMBDA][e.sub.2],[e.sub.1][LAMBDA][e.sub.3],[e.sub.2][LAMBDA][e.sub.3]]; [K.sup.1]([T.sup.3])[congruent to]Z[[e.sub.1],[e.sub.2],[e.sub.3],[e.sub.1][LAMBDA][e.sub.2][LAMBDA][e.sub.3]].

To compute the map [DELTA] (*) : K (*)([T.sup.3]) [right arrow] K (*)(T), we need to determine the ring structure of K*(T) [congruent to] Z.[alpha] [phi] Z.[beta]. Since K*(T) is a graded ring, we have deg([alpha]) = 0 and deg([beta]) = 1. Therefore,

[alpha].[beta] = [beta], [alpha].[alpha] = [beta].[beta] = [alpha].

We may now compute the image of the generators.

[DELTA]* (1) = [alpha], [DELTA]* ([e.sub.i]) = [beta], [DELTA]*([e.sub.i] [and] [e.sub.j]) = [DELTA]*([e.sub.i]). [DELTA]*([e.sub.j]) = [[beta].sup.2] = [alpha] [DELTA]* ([e.sub.i] [and] [e.sub.j] [and] [e.sub.k]) = [DELTA]* ([e.sub.i]). [DELTA]*([e.sub.j]). [DELTA]* ([e.sub.k]) = [[beta].sup.3] = [beta]

Therefore, we have

(a, b, c, d) [member of] [K.sup.j] ([T.sup.3]) [??] a + b+ c + d [member of] [K.sup.j] (T), j = 0, 1.

The above example generalizes to the n-torus as follows.

Proposition 5.2. The map [DELTA]* : [K.sup.j] ([T.sup.n]) [right arrow] [K.sup.j](T) induced by the diagonal [DELTA] : T [right arrow] [T.sup.n] is given by

[mathematical expression not reproducible]

where [mathematical expression not reproducible] is an element of [mathematical expression not reproducible].

Now we consider the general case where we have a map made with several diagonals.

[mathematical expression not reproducible] (5.2)

To elucidate the general case we begin with an example.

Example 5.3. Let [DELTA] : [T.sup.2] [right arrow] [T.sup.3] be the map ([z.sub.1], [z.sub.2]) [??] ([z.sub.1], [z.sub.1], [z.sub.2]). We want to compute explicitly

[DELTA]* : [K.sup.j] ([T.sup.3]) [right arrow] [K.sup.j] ([T.sup.2]), for j = 0, 1. We have

[K.sup.0] ([T.sup.2]) = Z[1, [e.sub.1][and][e.sub.2]], [K.sup.1] ([T.sup.2]) = Z[[e.sub.1], [e.sub.2]] [K.sup.0] ([T.sup.3]) = Z[1, [f.sub.1][and][f.sub.2], [f.sub.1][and][f.sub.3], [f.sub.2][and][f.sub.3]] [K.sup.1] ([T.sup.3]) = Z[[f.sub.1], [f.sub.2], [f.sub.3], [f.sub.1][and][f.sub.2][and][f.sub.3]]

It is clear that [DELTA]*(1) = 1. What are the [DELTA]*([f.sub.i])? The composition

[mathematical expression not reproducible]

where [P.sub.j] : [T.sup.3] [right arrow] T is the jth projection (j=1,2,3), define two projections

[P'.sub.i] : [T.sup.2] [right arrow] T. In fact, [P.sub.1] [??] [DELTA] = [P.sub.2] [??] [DELTA] = [P'.sub.1] and [P.sub.3] [??] [DELTA] = [P'.sub.2]. Now, the generators

[f.sub.j] are defined as the pullback [P.sup*.sub.j]([beta]) (j = 1, 2, 3). Accordingly, the generators [e.sub.i] are defined as the pullback [P'.sup.*.sub.i]([beta]) (i = 1, 2). We have

[DELTA]* ([f.sub.1]) = [DELTA]* ([P.sup.*.sub.1]([beta])) = [P'.sup.*.sub.1]([beta]) = [e.sub.1]

[DELTA]* ([f.sub.2]) = [DELTA]* ([P.sup.*.sub.2]([beta])) = [P'.sup.*.sub.1]([beta]) = [e.sub.1]

[DELTA]* ([f.sub.3]) = [DELTA]* ([P.sup.*.sub.3]([beta])) = [P'.sup.*.sub.2]([beta]) = [e.sub.2]

We are now able to compute the image of all generators.

[DELTA]* ([f.sub.1] [and] [f.sub.2]) = [DELTA]* ([f.sub.1]) [and] [DELTA]* ([f.sub.2]) = [e.sub.1] [and] [e.sub.1] = 0

[DELTA]* ([f.sub.2] [and] [f.sub.3]) = [DELTA]* ([f.sub.2]) [and] [DELTA]* ([f.sub.3]) = [e.sub.1] [and] [e.sub.2]

[DELTA]* ([f.sub.1] [and] [f.sub.2] [and] [f.sub.3]) = [DELTA]* ([f.sub.1]) [and] [DELTA]* ([f.sub.2]) [and] [DELTA]* ([f.sub.3]) = [e.sub.1] [and] [e.sub.2]

[DELTA]* ([f.sub.1] [and] [f.sub.2] [and] [f.sub.3]) = [DELTA]* ([f.sub.1]) [and] [DELTA]* ([f.sub.2]) [and] [DELTA]* ([f.sub.3]) = 0

Finally, we have

[DELTA]* = [K.sup.0] ([DELTA]) : [K.sup.0]([T.sup.3]) [right arrow] [K.sup.0]([T.sup.2])

[K.sup.0] ([DELTA]) ([a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4]) = ([a.sub.1], [a.sub.3] + [a.sub.4]), and

[DELTA]* = [K.sup.1] ([DELTA]) : [K.sup.1]([T.sup.3]) [right arrow] [K.sup.1]([T.sup.2])

[K.sup.1] ([DELTA]) ([a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4]) = ([a.sub.1] + [a.sub.2], [a.sub.3])

The previous example should give some intuition for the general case. We will change notation for a more natural one. Denote the generators as follows:

[mathematical expression not reproducible]

Once again, the difficult step is to decide what is the image of each generator, [DELTA]*([f.sup.j.sub.i]). Let us rewrite the previous example using this new notation. We have the following correspondence [f.sub.1] = [f.sup.1.sub.1], [f.sub.2] = [f.sup.2.sub.1], [f.sub.3] = [f.sup.1.sub.2]. Then, we have

[DELTA]* ([f.sub.1]) = [DELTA]* ([f.sup.1.sub.1]) = [e.sub.1]

[DELTA]* ([f.sub.2]) = [DELTA]* ([f.sup.2.sub.1]) = [e.sub.1]

[DELTA]* ([f.sub.3]) = [DELTA]* ([f.sup.1.sub.2]) = [e.sub.2]

Proposition 5.4. [DELTA] (*)([f.sup.j.sub.i]) = [e.sub.i].

Proof. Let [mathematical expression not reproducible] be the projection

[mathematical expression not reproducible]

Then, composing with the diagonal

[mathematical expression not reproducible]

we have

[P.sub.ij] [??] [DELTA] ([z.sub.1], ..., [z.sub.k]) = [z.sub.i], for all j = 1, ..., [n.sub.i].

Hence, [P.sub.ij] [??] [DELTA] gives the ith projection [P.sub.i] : [T.sup.k] [right arrow] T and it follows that

[DELTA]*([f.sup.j.sub.i]) = [e.sub.i], for i = 1, ..., k.

We are now able to describe the maps [DELTA] (*) and BC (*).

Theorem 5.5. Let [mathematical expression not reproducible] be a generator of [mathematical expression not reproducible], where [i.sub.1] < ... < [i.sub.r] and * denotes any index l occurring in [mathematical expression not reproducible]. Then

[mathematical expression not reproducible]

for j = 0, 1. Moreover,

[mathematical expression not reproducible]

Proof. It follows from Proposition 5.4 that we can form a basis of

[mathematical expression not reproducible]

by considering only sequences of lower indices [i.sub.1] < ... < [i.sub.r]. Note that if j = 0 then we will have sequences with an even number of lower indices and if j = 1 we will have sequences with an odd number of lower indices.

Acknowledgement. The author wish to thank the organizers of the 5th workshop on Functional Analysis and Applications at Aveiro University.


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Sergio Mendes

Department of Mathematics, ISCTE-Lisbon University Institute, Lisbon, Portugal E-mail:
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