EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 1, 2014, 45-54 ISSN 1307-5543 – www.ejpam.com Cycles in the Chamber Homology for SL(2, F) Wemedh Aeal School of Mathematics,The University Of Manchester, Greater Manchester, United Kingdom Abstract. We emphasized finding the explicit cycles in the chamber homology groups and the K-theory groups in term of each representation for SL(2, F). This led to an explicit computing of chamber homology and the K-theory groups. We have identified the base change effect on each of these cycles. The base change map on the homology group level works by sending a generator of the homology group of SL(2, E) labeled by a character of E× to the generator of the homology group of SL(2, F) labeled by a character of F× multiplied by the residue field degree. Whilst, it works by sending the K-theory group generator of the reduce C∗-algebra of SL(2, E) labeled by the 1-cycle (resp. 0-cycle) to the multiplication of the residue field degree with a generator of the K-theory group of SL(2, F) labeled by the base changed effect on 1-cycle (resp. 0-cycle). 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, Non-commutative Geometry, Number Theory. 1. Introduction Let F be a p-adic non-archimedean local field with p 6= 2 and G = SL(2, F). We have F× ∼= UF × Z, where UF is the group of p-adic units, and the dual of F× is ÓF× ∼= ÓUF × T, where T is the circle group. In this paper we emphasized finding the explicit cycles in the chamber homology groups and the K-theory groups in term of each representation for SL(2, F). This led to an explicit computing of chamber homology and the K-theory groups. We have identified the base change effect on each of these cycles. The base change map on the homology group level works by sending a generator of the homology group of SL(2, E) labeled by a character of E× to the generator of the homology group of SL(2, F) labeled by a character of F× multiplied by the residue field degree. Whilst, it works by sending the K-theory group generator of the reduce C∗-algebra of SL(2, E) labeled by the 1-cycle (resp. 0-cycle) to the multiplication of the residue field degree with a generator of the K-theory group of SL(2, F) labeled by the base changed effect on 1-cycle (resp. 0-cycle). Email address: wemedh@hotmail.com http://www.ejpam.com 45 c© 2014 EJPAM All rights reserved. W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 46 Consequently, we showed that the base change of Steinberg is again a Steinberg, the base change of a principal series is always a principle series and the base change of a cuspidal can certainly be either another cuspidal or a principal series. We have found that whilst the Baum-Connes correspondence takes the homology group generator of SL(2, E) to a generator of the K-theory group of C∗-algebra of SL(2, E) by induction, it takes the effect of the base change map on the homology side to the base change effect on the K-theory side by induction as well. 2. Local Langlands Correspondence and Base Change Let F be a non-archimedean local field, and G = SL(2, F). Let LF be the local Langlands group: LF :=WF × SL(2,C). A Langlands parameter is a continuous homomorphism φ :LF → G∨ = PGL(2,C), where G∨ = PGL(2,C) is the Langlands dual group. We say that two Langlands parameters are equivalent if they are conjugate under the group PGL(2,C). Let Φ(G) be the set of equivalence classes of the Langlands parameters. Now, the Local Langlands correspondence is defined to be the surjective map I r r(G)−→ Φ(G), Aφ 7−→ φ where Aφ is the pre-image of φ which is called the L-packet. The base change map is defined by the restriction of L-parameter from LF to LE , where E is a finite extension of F φ|WE :WE × SL(2,C)→ PGL(2,C). Lemma 1. Let α E = γ E ◦ β E :WE → E×, where γ E :W ab E → E× and β E :WE →W ab E then we have: i) NE/F (αE(w)) = αF (w), w ∈WE ⊂WF . ii) f .valE = valF ◦ NE/F . iii) dE =−valE ◦αE . iv) Let w ∈WE ⊂WF . Then we have f .dE(w) = dF (w). Proof. See [2, 1.2.2] for 1, [10, p. 139] for 2, and see [6] for 3 and 4. Now, an unramified character ψ of WE is given by the following simple formula: ψ(w) = zdE(w), z ∈ C×. The base change formula for a character χ of WF is given by BC(χ) = χ |WE . Lemma 2. Under base change we have BC(ψ)(w) = (z f )dE(w) for all w ∈WE . W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 47 Proof. The result follows directly from part 4 of lemma 1. Lemma 3. Let φ = 1⊗τ(2) and φ ′ =ψ⊗τ(2) be two L-parameters, whereψ is an unramified character of WF . Then φ = φ ′ in PGL(2,C). Proof. Let dF : WF // W ab F ' F× valF // Z . We have ψ(w) = zd(w) where z ∈ C×, ψ unitary character if and only if z ∈ T. Let φ = 1⊗τ(2) :WF × SL(2,C)→ PGL2(C) and φ ′ =ψ⊗τ(2) :WF × SL(2,C)→ PGL2(C) such that φ(w, A) = 1 ·τ(2) � A � = τ(A) and φ ′ (w, A) =ψ(w) ·τ(2) � A � = zd(w) ·τ(A). We see that τ(A) and (z d(w) ·τ(A)) are both in the same group PGL(2,C) and this means thatφ = φ ′ . Theorem 1. Letφ = 1⊗τ(2) be the L-parameter of the Steinberg representation, then we have BC(StG(F)) = StG(E). Proof. Let LF =WF × SL(2,C) and LE =WE × SL(2,C) be the local Langlands groups and let φ : LF 1WF⊗τ(2) // PGL(2,C) be the L-parameter, this parameter works as follows (w, Y ) 7−→ [Y ]. We know that LE ⊂ LF . The base change works by restriction the L-parameter to WE , in another words φ|WE : LE 1WE⊗τ(2) // PGL(2,C) . Since the restriction works only on the Weil group side which in our case is the trivial representation of WF and since the restriction of the trivial representation of WF is also the trivial representation of WE , then the resulting representation is also the Steinberg representation, i.e BC(StG(F)) = StG(E). φ :WF × SL(2,C) �� // PGL2(C) ‖ �� φ|WE :WE × SL(2,C) // PGL2(C) W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 48 Theorem 2. Let T be one of the circles in the unitary principal series of SL(2, F), then we have T→ T, z 7→ z f , under base change E/F. i) At the level of the K-theory group K1, BC induces the map Z→ Z, α1 7→ f .α1, where f is the residue field degree and α1 denotes a generator of K1(T) = Z. ii) At the level of K-theory group K0, BC induces the identity map Z→ Z, α0 7→ α0, where α0 denotes a generator of K0(T) = Z. Proof. We know that the principal series of SL(2, F) can be defined as follows: IndSL(2,F) B (χ) where χ � x y 0 x−1 � = χ(x). Now we have WF × SL(2,C) �� // PGL2(C) ‖ �� F× × SL(2,C) φ // PGL2(C) This means the above map φ works as follows: (x ,τ) 7−→ � χ(x) 0 0 1 � . Here � χ(x) 0 0 1 � is the coset of � χ(x) 0 0 1 � ∈ PGL2(C). If we twist χ by an unramified character we get a circle T embedded in PGL2(C). Also, the Weyl group Z/2Z acts on character of F×, character of F× =UF × 〈$F 〉 splits into {ramified character of UF say χ 1 } and {an unramified character of 〈$F 〉 say χ 0 ($) = z ∈ T}. The generator w of Z/2Z sends z to z−1, it sends χ1 to χ−1 1 . Suppose that χ1 6= χ−1 1 , i.e. χ2 1 6= 1. For such χ, the representation IndSL(2,F) B χ is irreducible. Define the L-parameter φ as follows: φ = ρ⊗1 where ρ is a unitary character of WF such that ρ :WF // W ab F ' F× χ // T . Also, we have ρ 7−→ IndSL(2,F) B χ The unitary characters (ρ2 6= 1) of WF factor through F× and we have ÓF× =Ô〈$〉 ×ÓUF , ρ is a unitary character of ÓUF . The group ÓUF admits countably many such characters ρ. Therefore, the compact orbit is the circle T: Ot(φ)∼=Ot(BC(φ))∼= T. After restriction and using the local class functions theory we get that this map has degree f . Therefore, if χ2 6= 1 this means by Lemma 1 and Theorem 2 in each circle the base change formula is z 7→ z f . W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 49 3. Representatives in the Chamber Homology H0 In this section we will investigate the case H0. Since we have two types of representations for SL(2, F) which are: the discrete series and the principal series representations, so we need to describe each case individually. The unitary principal series representation are as same as described in H1. We need to deal with reducible principal series, the special representation and the discrete series. Let’s start with the special representation. This means we are going to deal with the Steinberg representation. We recall the maximal compact subgroups J0 and J1, which were described in the previous section as the stabilizer subgroups of the vertices of the edge of the tree βSL(2, F). Theorem 3. Let J0 and J1 be the two maximal compact open subgroups of SL(2) and let I the Iwahori sub- group of SL(2). There are only three generators for H0 which are 1J0 , 1J1 , and the induced representation of 1I to J0 or J1. Proof. Let 1J0 (resp. 1J1 ) be a representation in R(J0) (resp. R(J1)), so [1J0 , 0] and [0,1J1 ] ∈ H0. We have [IndJ0 I 1I , 0] = [0, IndJ1 I 1I]⇐⇒∃v ∈R(I) such that (IndJ0 I 1I ,−IndJ1 I 1I) = ∂ (v). This means we have only one possibility which is v = 1I . Therefor three possibilities for H0- generators are 1J0 , 1J1 , and IndJ0 I 1I (resp. IndJ1 I 1I ). The question here is which combination of these three generators correspond to the Steinberg representation StG of SL(2)? Theorem 4. The 0-cycle corresponding to StG of SL(2) in K0 is (IndJ0 I 1I −1J0 , 0). Proof. Let G = SL(2, F) and J0 = SL(2,O ). According to the Anh Reciprocity Theorem in [5, p. 57], if dµ is a Haar measure then we have the following: i) IndG I 1I = ∫ X πdµ(π), X = {π ∈ bGr : π|I ⊃ 1I}. ii) IndG J0 1J0 = ∫ Y πdµ(π), Y = {π ∈ bGr : π|J0 ⊃ 1J0 }. Now, IndG I 1I =IndG J0 1J0 ⊕ StG ⇐⇒IndG J0 (IndJ0 I 1I)− IndG J0 1J0 = StG ⇐⇒IndG J0 (IndJ0 I 1I −1J0 ) = StG . Therefore the 0-cycle corresponding to StG is (IndJ0 I 1I − 1J0 , 0). We also see that the Baum-Connes conjecture (map) in this case is IndG J0 . The proof of the above theorem shows that the map IndG J0 takes [IndJ0 I 1I −1J0 , 0]F 7→ [StG]F , i.e. it takes the generator of H F 0 to the generator of K F 0 labeled by StG . This means we have three independent elements. In the same way this map works on the E-sides by taking the [IndJ0 I 1I −1J0 , 0]E 7→ [StG]E . From now on we will replace the notation of StG by St F 2 and StE 2 to refer for the Steinberg representa- tion of SL(2, F) and SL(2, E) respectively. W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 50 Theorem 5. The base change on K0-theory level takes the K0-generator of the reduce C∗-algebra of SL(2, E) labeled by StE 2 to the K0-generator of the reduce C∗-algebra of SL(2, F) labeled by St F 2 and the K-theory group K0 C∗r SL(2, F) = Z3. Proof. From Theorems 3 and 4 we have only three generators and this implies that K0 C∗r SL(2, F) = Z3. H0(SL(2, E)) BC �� µE 0 // K0C∗r SL(2, E) K0(BC) �� H0(SL(2, F)) µF 0 // K0C∗r SL(2, F) Figure 1: The base change for SL(2) On the chamber homology level, the base change map works by taking the generator of the group H0 of SL(2, E) to the generator of H0 of SL(2, F); see Figure 1. On the other hand, if we deal with the reducible principal series (the intervals), this means we are going to induce the Legendre character to one of the maximal compact subgroups J0, J1 or both. Theorem 6. There are three generators for H0 which they are constructed by inducing a representation of the Legendre character from I to the maximal subgroups J0 and J1. Proof. We know that if λ2 = 1 then Ind SL(2,Fp) B λ= λ+B ⊕λ − B . This means our induced representation can be written as decomposition of two representations. So if we induced to the maximal compact subgroups J0, J1 we would have three multiple choices. Let λI be any representation in R(I), then IndJ0 I λI (resp. IndJ1 I λI ) is the induced representation of the Legendre character from I to J0 (resp. J1). Now, we have IndJ0 I λI = λ + J0 ⊕λ−J0 and IndJ1 I λI = λ + J1 ⊕λ−J1 . This means we have three generators for H0 which are: λ+J1 , λ+J0 and λ−J0 or λ−J1 , λ+J0 and λ−J0 . This also shows that the assembly map IndG J0 works as follows [IndJ0 I λI −λ+J0 , 0]F 7→ [λ−J0 ]F i.e. it takes the generator of H F 0 to the generator of K F 0 labeled by λ−J0 . In the same way this map works on the E-sides by taking the [IndJ0 I λI −λ+J0 , 0]E 7→ [λ−J0 ]E . This means we have three independent elements. Theorem 7. The base change on K0-theory level takes the K0-generator of the reduced C∗-algebra of SL(2, E) labeled by λ−J0(E) to the K0-generator of the reduce C∗-algebra of SL(2, F) labeled by λ−J0(F) . The K-theory group K0 C∗r SL(2, F) in this case is Z3. W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 51 Proof. Theorem 6 shows that we have three generators for K0 and this means K0 = Z3. On the other hand, we introduce the cuspidal representations as follows: let ℵ= ρ⊗1 :WF × SL(2,C)→ PGL(2,C) then we have ℵ |WE :WE × SL(2,C)→ PGL(2,C). Now let ℵ be an irreducible representation. i) If ℵ |WE remains irreducible after restriction, then this determines a cuspidal representation of SL(2, E). Base change in this case, will send one cuspidal representation of SL(2, F) to a cuspidal representation of SL(2, E). ii) If ℵ |WE is reducible, then this representation split into two 1-dimensional representations. i.e. ℵ= ℵ1 ⊕ℵ2 = (ρ1 ⊗1)⊕ (ρ2 ⊗1), where ρ1 and ρ2 are two characters of WE . This means on the K-theory level there is one generator for each cuspidal representation and the K0 = Z. 4. Representatives in the Chamber Homology H1 A description of the cycles in the group H1 will be introduced in this section. Let G = SL(2, F) be the group of unimodular 2 × 2 matrices with entries in the field F . It is a locally compact totally disconnected topological group [8, 9]. Let I = � O O $O O � ∩SL(2). This is a compact open subgroup of G, called the Iwahori subgroup. Let w0 = � 0 −1 1 0 � and w1 = � 0 −$−1 $ 0 � . These elements appear in the Tits system associated to G, which plays an important role in what follows [3, 7]. Let J0 = I ∪ Iw0 I = � O O O O � ∩ SL(2) and J1 = I ∪ Iw1 I = � O $−1O $O O � ∩ SL(2), these are compact open subgroups of G, we have J0 ∩ J1 = I . The tree for G = SL(2) is the graph βG, the group G acting on βG by multiplication on the left. We see that I is the stabilizer of the fundamental edge, and that J0, J1 are the stabilizer of the vertices of this edge, respectively. Now if I , J0 and J1 are the compact subgroups of G = SL(2) defined in the previous two paragraphs, then we have this chain complex 0 R(J0)⊕R(J1)oo R(I) IndJ0 I ⊕−IndJ1 I =∂oo 0oo So that H0 = R(J0)⊕R(J1) ∂R(I) and H1 = ker∂ . For more details see [1]. W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 52 Definition 1. A character χ of F× is called tame character if χ|U 1 F is trivial. Lemma 4. Let χ be a tame character of I then χ −χ−1 ∈ H1. Proof. Let χ be character of I , i.e. I modp // B ⊂ SL2(Fp) χ : SL(2)∩ � O O pO O � −→ T, � x y 0 x−1 � 7−→ χ(x). Now let w = � 0 −1 1 0 � ∈ SL(2), w ∈W whereW is the Weyl group of SL(2). We have wχ(x) = χ(wxw−1). To prove that χ − wχ ∈ H1 it is enough to show that χ − wχ ∈ R(I), i.e. ∂ (χ − wχ) = 0. In other words we need to show that IndJ0 I (χ −wχ) = 0 and IndJ1 I (χ −wχ) = 0. Now choose χ 6= wχ, so wχ � x y 0 x−1 � := χ � x−1 0 −y x � . This means wχ = χ−1. Therefore we only need to prove IndJ0 I (χ −χ −1) = 0 and IndJ1 I (χ − χ −1) = 0. But IndJ0 I (χ − χ −1) = 0 if and only if IndJ0 I χ ∼= IndJ0 I χ −1. Since χ and χ−1 are distinct, then they are determine the the same representation and this representation is irreducible if and only if χ2 6= 1 [4]. This means IndJ0 I χ ∼= IndJ0 I χ −1. Therefore IndJ0 I (χ −χ −1) = 0. Same results will be shown if we take J1. This means we have χ −χ−1 ∈ H1. Now let k be a positive integer, I(k) = � O O $kO O � ∩ SL(2, F). Consider now the subgroup I(k)w = I(k)∩w−1 I(k)w = � O $kO $kO O � ∩ SL(2). Let ψ be an invariant function on I and let α : SL(2)→ SL(2), α : � a b c d � 7→ � d $−1c $b a � . Define ψα(g) = ψ(α(g)), then ψ induces to zero on J1(resp. J0) if and only if ψα induces to zero on J0(resp. J1). Therefore ψ ∈ H1 if and only if ψα ∈ H1. Fix a character χ : UF → T not of order two, and let k be the least positive integer such that χ[1+$kO ] = 1. The character χ extends to the group I(k) using the formula χ : � a b $kc d � 7→ χ(a). Lemma 5. Let ψ χ = Ind I I(k)χ then i) ψ χ is an irreducible character. W. Aeal / Eur. J. Pure Appl. Math, 7 (2014), 45-54 53 ii) ψα χ =ψ χ , where α is the automorphism of SL(2). Now let cχ = ψχ −ψ χ . Then cχ ∈ H1, and the cycle cχ , one selected from each pair of characters {χ,χ}, constitute a basis for H1. Therefore all cycles will be of this form. Theorem 8. i) The base change on K1-theory level works as follows: K1C∗r SL(2, E) K1(BC) // K1C∗r SL(2, F) , α σ◦NE/F 7−→ f ·ασ ii) The base change on H1-level works as follows: (βSL(2, E)) 1 BC // H1(βSL(2, F)) , c σ◦NE/F 7−→ f · cσ where α σ◦NE/F (resp.α σ ) and c σ◦NE/F (resp.cσ) are the K1 and H1 generator of SL(2, E) � SL(2, F) � respectively. Proof. By Theorem 2 this map has degree f . The base change on the K1-theory level takes the K1-generator of the reduce C∗-algebra of SL(2, E) to the K1-generator of the reduce C∗-algebra of SL(2, F) multiplying by the residue field degree f , so (1) has been proved. We also know that the base change on the chamber homology side works by sending each unramified unitary character of the Iwahori subgroup of SL(2, F) to itself composed with the norm map. So, the base change map on the chamber homology side takes the generator of the chamber homology group of SL(2, E) (labeled by this composite) to the generator of the chamber homology group of SL(2, F) (labeled by unramified unitary character) multiplying by the residue field degree f . Corollary 1. The assembly map H1(SL(2, F)) µF 1 // K1C∗r SL(2, F) under the base change works as follows: f · cσ 7−→ f ·ασ where cσ and ασ are H1 and K1 generators for SL(2, F) respectively. 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