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Intertwining Operators, L–Functions, and Representation Theory*

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Intertwining Operators, L–Functions, and Representation Theory* Intertwining Operators, L{Functions, and Representation Theory* by Freydoon Shahidi** Purdue University *Lecture Notes of the Eleventh KAIST Mathematics Workshop 1996, Taejon, Korea; Edited by Ja Kyung Koo **Partially supported by NSF Grant DMS9622585 1 2 Table of Contents Introduction 1. Notation 2. Modular forms as forms for GL2 3. Maass forms and Ramanujan{Petersson's Conjecture 4. Automorphic cuspidal representations 5. L{functions for GL2 6. Cuspidal representations for GLn 7. Rankin{Selberg L{functions for GLn 8. Applications of properties of L(s; π π0) × 9. The method 9.1 L{groups 9.2 Parabolic subgroups 9.3 Cusp forms and Eisenstein series 9.4 Constant term and intertwining operators 9.5 L{functions in the constant term 9.6 Generic representations 9.7 Local coefficients 9.8 Main global results 9.9 Local factors and functional equations 10. Some local applications 3 Introduction These notes are based on a series of twelve lectures delivered at the Korea Ad- vanced Institute of Science and Technology (KAIST), Taejeon, during their bian- nual workshop in algebra that took place over the period of July 29{August 4, 1996. As its main goal, I have chosen to sketch a proof of the estimate 1=5 1=5 q− < αv < q v j j v for Hecke eigenvalues of a cusp form on GL2(AF ), where F is an arbitrary number field, using the method initiated by Langlands and developed by myself. The first half of the notes (Sections 1{8) is devoted to developing the problem and introducing some important L{functions (symmetric power L{functions for GL(2) and Rankin{ Selberg product L{functions for GL(m) GL(n)). The second half (Section 9) is × spent to explain the method in the split case and finally prove the estimate. The notes are concluded by describing a number of local results in harmonic analysis and representation theory of local groups that can be proved using this method (Section 10). It is a pleasure to thank Professor Ja Kyung Koo, chairman of the Department of Mathematics at KAIST, for his wonderful and memorable hospitality, as well as his mathematical interest. Many of the initial efforts in arranging this visit was done by Dr. Hi{joon Chae of KAIST to whom I would like to extend my thanks and appreciation. I should also thank the Department of Mathematics, especially the algebra group, at KAIST for their hospitality. Last but not least, I should thank my audience. It was a pleasure to deliver these talks to such a pleasant and enthusiastic audience, a number of whom traveled long distances to participate in this workshop. 4 1. Notation x Throughout these lectures F will be either a local or global field of characteristic zero. More precisely, F is a global field of characteristic zero if it is a number field, i.e. a finite extension of Q. A local field of characteristic zero is either R, C, or a finite extension of Qp, the field of p{adic numbers, i.e. the completion of Q with m m respect to p{adic absolute value p; rp =s p = p− ; p - rs. Then p extends j j j j j j to an absolute value = v, on F where v is the place lying over p. j j j j Suppose F is a p{adic local field (non{archimedean). Then O = x F x 1 f 2 jj j ≤ g is a ring, called ring of integers. The set P = x O x < 1 f 2 jj j g is the maximal ideal of O and O∗ = O P is the group of units. The field F = O=P n is called the residue field of F whose cardinality which is finite is denoted by q = pf . e If e is the ramification index of p in F , i.e. pO = P , then [F : Qp] = ef. The ideal P is principal and one usually fixes a uniformizing parameter $ P for P . Then 2 1 $ v = q− . j j Now let F be a number field. For each place v of F , let Fv be its completion with respect to v. A place is an equivalence class of absolute values for F . We will say v < if v p for some rational prime p. Otherwise Fv = R or C and we 1 j ∼ say v = . In either case F Q Qp = v pFv;F Q R = v Fv. If v < , let 1 ⊗ ⊕ j ⊗ ⊕ j1 1 Ov;Pv; F v = Ov=Pv; qv; $v, and Ov∗ be as before. Let A = AF Fv be the subring of Fv defined by ⊂ Qv Qv A = (xv)v xv Ov; 0v ; f j 2 8 g i.e. the direct limit of Fv with respect to Ov. With direct limit topology, A Qv v<Q becomes a locally compact ring, called the ring1 of adeles of F . The group of units in A; A∗ = I will be A∗ = (xv)v xv F ∗; xv O∗; 0v : f j 2 v 2 v 8 g 5 Considered as the direct limit of Fv∗ with respect to Ov∗ and with direct limit Qv v<Q 1 topology, A∗ is called the group of ideles of F . Then one can define (x ) = x v v Y v v k k v j j 1 1 whose kernel I = (A∗) contains F ∗ by diagonal imbedding. Then F ∗ is a lattice 1 1 1 in I , i.e. F ∗ is discrete in I and I =F ∗ is compact. So is F in A. The finiteness of class number, the number of ideals (fractional) modulo the principal ones, is 1 equivalent to the compactness of I =F ∗. Fix an integer n > 0 and let GLn(R) be the group of invertible matrices with R{entries, where R is a commutative ring with 1. Let R = A. Then GLn(A) is called the adelization of GLn over F . If g GLn(A), 2 then g = (gv)v; gv GLn(Fv) with gv GLn(Ov); 0v < . 2 2 8 1 Let G be a Zariski{closed subgroup of a GLn(Fa), where Fa is an algebraic closure of F . Assume G is defined over F . Then it is defined over every Fv. Let Gv = G(Fv). Then g GLn(A) belongs to G(A), adelization of G=F , if and only 2 if each gv Gv. For almost all v < ; G is defined over Ov and therefore G(Ov) 2 1 makes sense and Gv GLn(Ov) = G(Ov). Consequently (gv)v G(A) implies that \ 2 gv G(Ov) for almost all v < . 2 1 2. Modular forms as forms for GL2 x In this section we shall briefly review how classical modular forms and automor- phic forms on GL2(AQ) are related. Let Γ be a congruence subgroup for SL2, i.e. a subgroup of SL2(Z), containing a principal congruence subgroup a b ΓN = g = SL2(Z) g I(mod N) c d 2 j ≡ for some N. The group SL2(Z) and therefore every congruence subgroup Γ acts on upper half plane h h = z C Im(z) > 0 f 2 j g by a b az + b z = : c d · cz + d 6 2 In fact Im(g z) = Im(z)= cz + d as long as g SL2(R). Let h∗ be the com- · j j 2 pactification of h by adding cusps of Γ to h. Roughly speaking, a modular form of weight k > 0, k an integer, is a complex function on h∗ which is holomorphic there and satisfies a b (2.1) f(γ z) = (cz + d)kf(z)(γ = Γ): · c d 2 We may fix an integer N > 0 and consider the Hecke subgroup a b Γ0(N) = c 0(N) SL2(Z): c d ≡ ⊂ Given a character χ mod N of (Z=NZ)∗, extended to all of Z, we may consider holomorphic forms f such that 1 k a b f(γ z) = χ(a)− (cz + d) f(z); γ = Γ0(N): · c d 2 Then given any congruence subgroup Γ and a holomorphic form with respect to Γ, one can find a Γ0(N) and a χ such that f becomes a χ{modular form with respect to Γ0(N). Thus one may only consider forms with respect to Γ0(N) but with respect to arbitrary characters of (Z=NZ)∗. A modular form f is a cusp form if it vanishes on every cusp of Γ (or Γ0(N)). The best reference for this is [Shi1]. Let us show how modular cusp forms on h are in fact functions on GL2(Q) GL2(AQ), n i.e. an automorphic form. We refer to [B,G] for details. First observe that every g SL2(R) can be written as 2 1=2 1=2 y xy− g = 1=2 k(θ)(y > 0) 0 y− with cos θ sin θ k(θ) = : sin θ −cos θ Thus g (z = x + iy h; θ) 7! 2 and g i = z since k(θ) i = i. Given a modular cusp form f, define: · · k φf (g) = f(g i)(ci + d)− · k iθk = f(g i)y 2 e− : · 7 a b If j(g; z) = cz + d for g = SL2(Z), then c d 2 j(g1g2; z) = j(g1; g2 z)j(g2; z): · Thus j(γg; i) = j(γ; g i)j(g; i) · from which we get k φf (γg) = f(γg i)j(γg; i)− · k k = f(g i)j(γ; g i) j(γg; i)− · · = φf (g) It is then clear that φf is bounded as a function of Γ G; G = SL2(R), and by the n 2 finiteness of the volume of Γ G; φf L (Γ G).
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