THE PLANCHEREL FORMULA OF L2 (N0 \ G; ψ) WHERE G IS A p-ADIC GROUP TANG U-LIANG (BSc (Hons), NUS ) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgments The author wishes to thank Professor Gordon Savin, Professor Gan Wee Teck and Associate Professor Loke Hung Yean for initiating this project, suggestions, comments and guidance i Contents Acknowledgments i Summary v Chapter 1.1 Introduction and statement of main results 1.2 The Cartan and Iwasawa decompositions of G 1.3 Parabolic subgroups and Schwartz spaces Chapter 2.1 Whittaker functions 2.2 The Harish-Chandra transform 13 14 17 Chapter 3.1 The Plancherel measure of L2 (G) 3.2 The discrete spectrum of L2 (N0 \ G; ψ) 3.3 The proof of Lemma 3.2.1.2 21 22 24 25 Chapter 4.1 The Whittaker transform 4.2 Proof of Theorem 4.1.3.2 4.3 The Plancherel formula for L2 (N0 \ G; ψ) 31 32 36 40 Bibliography 45 iii Summary We study the right regular representation on the space L2 (N0 \ G; ψ) where G is a quasi-split p-adic group and ψ a non-degenerate unitary character of the unipotent subgroup N0 of a minimal parabolic subgroup of G We obtain the direct integral decomposition of this space into its constituent representations In particular, we deduce that the discrete spectrum of L2 (N0 \ G; ψ) consists precisely of ψ generic discrete series representations and derive the Plancherel formula for L2 (N0 \ G; ψ) v CHAPTER 1.1 INTRODUCTION AND STATEMENT OF MAIN RESULTS 1.1 Introduction and statement of main results 1.1.1 Let ψ be a nondegenerate unitary character of the unipotent radical N0 of a minimal standard parabolic subgroup of a connected quasi-split p-adic group, G Define L2 (N0 \ G; ψ) as the space of functions on G which transform according to ψ, i.e f (ng) = ψ(n)f (g) and are square integrable modulo N0 This space becomes a unitary representation of G via right translation The purpose of this work is to obtain the Plancherel formula for this unitary representation Dinakar Ramakrishnan first studied the case for GL(2) in [Ram2] obtaining a Plancherel formula for the archimedean and non-archimedean group Nolan Wallach then proved this result for arbitrary real reductive groups (see [Wa, Chapter 14]) Indeed, we have found out that much Wallach’s arguments can be adapted for the p-adic case There are two crucial steps in proving the Plancherel formula for L2 (N0 \ G; ψ) One of them is to prove the surjectivity of a certain map from the space of Schwartz functions on G to the space of Schwartz functions on N0 \ G This is the author’s original contribution Secondly we must define a Whittaker transform which transforms certain smooth functions on orbits of discrete series representations to Schwartz functions on N0 \ G analogous to the Harish-Chandra wave packet map We refer the reader to Section 4.1.3 for the precise definition of this transform In order to define this map, we require the a certain Jacquet integral (this integral is defined later in the paper) extend to a holomorphic function This required fact is a consequence of the results of Casselman in [C-S] (see also [Jac] and [Shah]) 1.1.2 Now to state our main result Let P = M N be a standard parabolic subgroup of G with M and N its Levi and unipotent subgroup respectively Let ψ M denote the restriction of ψ to M ∩ N0 We take a ψ M generic discrete series G representation (σ, Hσ ) and consider the unitarily induced representation IP (σ ⊗ ν) where ν ∈ Im(Xur (M )) = iaM /L runs over all unramified unitary characters of M We refer the reader to section 1.3.4 for the definitions of the relevant notations Now let W hψM (Hσ ) denote the (one dimensional) space of Whittaker functionals on σ Let G Hσ,ν = IP (σ, ν) ⊗ W hψM (Hσ ) and consider the direct integral Iσ,M = Z ⊕ iaM /L Hσ,ν µ(σ, ν) dν ˜ where µ(σ, ν) is a certain normalization of the Plancherel measure on ia/L ˜ Let W (G|M ) := {w ∈ W G | w.M = M }/W M where W G and W M denote the Weyl group of G and its Levi M respectively If σ ∈ E2 M , then w.σ is defined and denote E2 M (M )/W (G|M ) to be the set of ψ ψ isomorphism classes of square integrable representations of M which are ψ M generic modulo the action of W (G|M ) We prove that 1.1 INTRODUCTION AND STATEMENT OF MAIN RESULTS Theorem 1.1.2.1 There exists a unitary linear isomorphism from X X M ⊂G σ∈E2 ψM Iσ,M (M )/W (G|M ) onto L2 (N0 \ G; ψ) where each of the Iσ,M is a G-module This is a refinement of the Plancherel formula for a particular symmetric space (i.e N0 \ G) of “polynomial growth” studied in [B] We also compute the explicit normalization of µ(σ, ν) If µ(σ, ν) dν denotes the ˜ Plancherel measure on iaM /L, then µ(σ, ν) dν = ˜ µ(σ, ν) dν |W (G|M )|γ(G|M )c2 (G|M ) It is worth noting that the Plancherel formula for L2 (G) is used in the proof of the Plancherel formula for L2 (N0 \ G; ψ) This explains the formal similarity with the two formulas 1.1.3 This paper is organized as follows From Section 1.2.1 until the end of Chapter we give a simplified exposition of Bruhat-Tits theory adequate for our purposes and prove Lemma 1.2.2.1 This is a key lemma required to prove the crucial Lemma 3.2.1.2 We also describe the notations and conventions for parabolic subgroups and tori needed to describe unitary parabolic induction Finally, we end by giving a description of the Schwartz spaces on G and N0 \ G needed later In Chapter we discuss the general theory of Whittaker functions for discrete series representations and tempered representations, the Harish-Chandra transform for functions in C ∗ (N0 \ G; ψ) and conclude with an application of this theory to a result of Savin, Khare and Larsen Chapter is where we discuss aspects of the Placherel measure on G, the multiplicity one property for L2 (N0 \ G; ψ) and prove Lemma 3.2.1.2 Finally, we set the stage for deriving the full Plancherel formula in Chapter Our main result is Theorem 4.1.3.2 Theorem 1.1.2.1 stated in this introduction is precisely Corollary 4.3.2.1 1.1.4 While this paper was being written, it was brought to the author’s attention that Erez Lapid and Mao Zhengyu had obtained an explicit form of the Whittaker function and its asymptotics on a split group G in [L-Z] Theorem 2.1.3.3 is a direct corollary of their results They conjectured the following: Let W (π) denote the Whittaker model of a R generic representation and suppose that ZG N0 \G |W (g)|2 dg is finite for all W (g) ∈ W (π), then π is , square integrable By Theorem 2.2.1.5 and Theorem 3.2.1.3 we conclude that this conjecture is true Sakellaridis and Venkatesh has also announced a proof of this conjecture when G is a split group Patrick Delorme has obtained the results of this work independantly in [D1] and [D2] However, our approach differs slightly from his treatment We also thank Professor Delorme for pointing out a gap in the previous version of Proposition 2.2.1.3 CHAPTER 4.1 THE WHITTAKER TRANSFORM 4.1 The Whittaker transform 4.1.1 In this final chapter, we derive the full Plancherel formula for L2 (N0 \ G; ψ) The main result we are aiming at is Theorem 4.1.3.2 which is essentially the Plancherel formula The significance of this theorem is that allows us to decompose the space into an orthogonal sum of G-modules each of which is is indexed by a standard parabolic subgroup and an orbit of ψ-generic discrete series It turns out that the formal resemblance of this decomposition to the Plancherel decomposition for L2 (G) stems from the fact that L2 (N0 \ G; ψ) can be thought of as the L2 completion of the space of ψ-Fourier coefficients of Schwartz functions in C ∗ (G) (see Lemma 3.2.1.2 (1)) 4.1.2 Suppose Pθ = Mθ Nθ is a standard parabolic subgroup of a quasi-split group G and (σ, Hσ ) a discrete series representation of Mθ Assume that σ is ψ θ generic and let W hψθ (Hσ ) be the space of its Whittaker functionals As Mθ is also quasi-split, a result of Shalika assures us that this space is one-dimensional We recall the following construction used in the proof of Lemma 3.2.1.2(2) As before let N∗ = Mθ ∩ N0 Then N0 = Nθ N∗ Let w, v ∈ Hσ and consider the transform: Z λv (w) := N∗ ψ −1 (n∗ ) σ(n∗ )w, v σ dn∗ This is well defined as σ(m)w, v σ ∈ C ∗ (Mθ ) The integral above converges absolutely for any w ∈ Hσ and defines a Whittaker functional λv ∈ W hψθ (Hσ ) One observes that λv (w) = λw (v) Choose any λ ∈ W hψθ (Hσ ) and w ∈ Hσ satisfying λ(w) = Then for u ∈ Hσ , we have λu = λu (w)λ = λw (u)λ We recall the definition of f[α,O,Pθ ];w,v (g) in (3.1.2.2) ¯ Lemma 4.1.2.1 Let α ∈ C ∞ (O), w, v ∈ I(σ) We define η = λw Then Z (f[α,O,Pθ ];w,v )ψ (g) = ¯ O Jσ,ν (η)(w)Jσ,ν (λ)(πPσ ,σ,ν (g)v)α(ν) dµO (ν) ¯ Proof We already know that Z (f[α,O,Pθ ];w,v )ψ (g) = ¯ O Jσ,ν (λEσ,ν (w) )(πPθ ,σ,ν (g)v)α(ν) dµO (ν) ¯ However we may write λEσ,ν (w) = η(Eσ,ν (w))λ = Jσ,ν (η)(w)λ from which the lemma follows immediately Now fix a base point σ0 ∈ O and for every σ ∈ O identify Hσ = Hσ0 Then without loss of generality, W hψθ (Hσ0 ) = W hψθ (Hσ ) Thus, there is an unambiguous choice of basis λ ∈ W hψθ (Hσ ) for each σ ∈ O The space C ∞ (O; W hψθ (Hσ )) is then the space of all functions of the form {(ν → α(ν)λ) | α ∈ C ∞ (O)} Define a map Λ : C ∞ (O) ⊗ Iσ → C ∞ (O; W hψθ (Hσ )) by the formula Λ(α ⊗ w)(ν) = α(ν)Jσ,ν (η)(w)λ Lemma 4.1.2.2 The linear map Λ defined above is surjective 32 4.1 THE WHITTAKER TRANSFORM Proof Now we see easily that λw (v) = (ϕv,w )ψθ (1) This implies that we may as well assume that η is not the zero functional We must prove this assertion for (ν → α(ν)λ) for any α ∈ C ∞ (O) and a fixed λ ∈ W hψθ (Hσ ) By Theorem 3.3.2.1 Jσ,ν (ηj ) is a nonzero functional Thus one may find w ∈ I(σ) such that Jσ,ν (η)(wν ) = As Jσ,ν is continuous in ν, Jσ,µ (η)(wµ ) is nonzero as long as we vary µ around a small enough neighborhood Uν of ν ∈ O Since O is compact, one may find a finite covering {Ul }p of the support of l=1 α ∈ C ∞ (O) subject to the condition that for each of these open sets Ul , one finds vectors wl ∈ I(σ) such that on each of these open neighborhoods Ul , l l Jσ,µ = Jσ,µ (η)(wµ ) is nonzero for all µ ∈ Ul Now choose a partition of unity, {ϕl } subordinate to the cover {Ul } and define for each open set Ul a (smooth) function gl (ν) which vanishes outside Ul and is equals to l (g l (ν)) = ϕl (ν)(Jσ,ν )−1 Thus one clearly has X l Λ(gl α ⊗ wν ) = l X l = l gl (ν)α(ν)Jσ,ν (η)(wν )λ X ! ϕl (ν) α(ν)λ = α(ν)λ l This proves the lemma Consider the following Hermitian inner product on the discrete series representation Hσ and the image of σ in ◦ C ∗ (N2 \ Mθ ; ψ θ ) : v, w σ and Z (4.1.2.1) N2 \Mθ λ(σ(m)v)η(σ(m)w) dm for λ, η ∈ W hψθ (Hσ ) respectively By Schur’s lemma, the Hermitian form on an irreducible unitary representation is unique up to scalar Thus, if we set (λ, η)σ to be the ratio of (4.1.2.1) and v, w σ , this pairing is sesquilinear and positive definite Lemma 4.1.2.3 For any two vectors w, v ∈ Hσ , d(σ)(λv , λw )σ = λv (w) where d(σ) is the formal degree of σ Proof Let Mθ = M and by abuse of notation write ψ θ as ψ By definition Z (λv , λw )σ x, y σ = = Z N∗ \M N∗ \M λv (σ(m)x)λw (σ(m)y) dn Z N∗ ×N∗ ψ −1 (n1 n−1 ) σ(n1 m)x, v σ σ(n2 m)y, w σ dn1 dn2 dm 33 4.1 THE WHITTAKER TRANSFORM By change of variable n1 n2 to n1 , this equals Z Z ZN∗ \M Z N∗ ×N∗ ψ −1 (n) σ(m)x, σ(n)−1 v = M ψ −1 (n1 ) σ(n2 m)x, σ(n1 )−1 v N∗ σ σ σ(m)y, w σ(n2 m)y, w σ σ dn1 dn2 dm dndm Using Lemma 3.2.1.1 applied to M , we may reverse the order of integration to integrate over M first So we obtain d(σ)−1 x, y Z σ ψ −1 (n) σ(n)w, v N∗ σ dn = d(σ)−1 x, y σ λv (w) By canceling away x, y σ , we obtain our stated identity Lemma 4.1.2.4 Let β ∈ C ∞ (O; W hψθ (Hσ )), α ∈ C ∞ (O) and v ∈ I(σ) Then (Λ(α ⊗ v), β )σ = d(σ)−1 α(ν)β(ν)Jσ,ν (λ)(v) where β ∈ C ∞ (O) defines β (i.e β (ν) = β(ν)λ) Proof To justify this identity, we need only to observe that by definition, (Λ(α ⊗ v), β )σ = α(ν)β(ν)Jσ,ν (η )(v)(λ , λ)σ = α(ν)β(ν)Jσ,ν (η (λ, λ )σ )(v) where λ is a choice of basis for W hψθ (Hσ ) satisfying (λ , λ )σ = and η = λw where w ∈ Hσ is chosen so that λ (w ) = However, we note that λv = η (v)λ = d(σ)(λv , η )λ for any v ∈ I(σ) by Lemma 4.1.2.3 In other words η = d(σ)−1 λ This implies the result 4.1.3 Recall the unitary intertwining operator defined in [Wal, pg 295] (w ∈ W (G|M ) and σ ∈ O) ◦ cP |P (w, σ) G G G G : IP σ ⊗ IP σ ∨ → IP wσ ⊗ IP wσ ∨ for two parabolic subgroups P and P with the same Levi subgroup M It is known that ◦ cP |P (w, σ) is regular on O and that we may express this map in the following form: u ⊗ v → JP |w.P (wσ)L(kw )u ⊗ Jw.P |P (wσ ∨ )−1 L(kw )v where L denotes left translation See [Wal, proof of Lemme V.3.1] We define Aw (σ) := JP |w.P (wσ)L(w) and for a fixed σ ∈ O, consider Aw (σ χv ) as a function on O in which case we write Aw (σ χν ) as Aw (ν) For α ∈ C ∞ (O; W hψθ (Hσ )) and v ∈ I(σ) we define Z W[α,O,Pθ ];v (g) := ¯ O Jσ,ν (α(ν))(πPθ ,σ,ν (g)v) dµO (ν) ¯ Let β ∈ C ∞ (O; W hψθ (Hσ )) be defined by β ∈ C ∞ (O) 34 4.1 THE WHITTAKER TRANSFORM Lemma 4.1.3.1 Let Aw (ν) be the intertwining operator defined by the unitary map ◦ cPθ |Pθ (w, σ) Then there exists a map ¯ ¯ Mw (ν)−1 : W hψθ (Hσ ) → W hψθ (Hwσ ) smooth on O so that, (4.1.3.1) Z W[β ,O,Pθ ];v (g) = ¯ O β(ν)Jwσ,wν (Mw (ν)−1 λ)(πPθ ,wσ,wν (g)Aw (ν)v) dµO (ν) ¯ for any w ∈ W (G|Mθ ) Proof Indeed, if λ ∈ W hψθ (Hwσ ), then Jwσ,wν (λ) ◦ Aw (ν) ∈ Jσ,ν (W hψθ (Hσ )) Let Z denote the poles of Aw (ν) on O and let O = O − Z Then as Jσ,ν is an isomorphism (Theorem 3.3.2.1) and Aw (ν) is an isomorphism wherever it is regular, we define Mw (ν) : W hψθ (Hwσ ) → W hψθ (Hσ ) by Jwσ,wν (λ) ◦ Aw (ν) = Jσ,ν (Mw (ν)λ) for all ν ∈ O Since the representation spaces of wσ and σ are the same, without loss of generality we may assume that W hψθ (Hwσ ) = W hψθ (Hσ ) and we consider Mw (ν) as a rational function on O Let Mw (ν)−1 denote its reciprocal It is regular on O and vanishes on the set Z If λ ∈ W hψθ (Hσ ), then (4.1.3.2) Jwσ,wν (Mw (ν)−1 λ) ◦ Aw (ν) = Jσ,ν (λ) on O The lemma follows immediately from definition We remark that if P = G, then we take O = {e} as a singleton so that α is identified with λ ∈ W hψ (Hσ ) where σ is a discrete series representation of G Then W[α,{e},G];v (g) is W (v, λ)(g) Theorem 4.1.3.2 (1) If α ∈ C ∞ (O; W hψθ (Hσ )), then ∗ W[α,O,Pθ ];v ∈ C (N0 \ G; ψ) for all v ∈ I(σ) ¯ ¯ (2) The span of W[α,O,Pθ ];v over all v ∈ I(σ) and datum [α, O, Pθ ] such that σ ∈ O is ¯ ψ θ -generic is equal to C ∗ (N0 \ G; ψ) In particular, the span is dense in L2 (N0 \ G; ψ) ¯ ¯ (3) Given two datum [α, O, Pθ ] and [β, P, Pθ ], suppose either θ = θ or O = wP for any w ∈ W (G|Mθ ), then W[α,O,Pθ ];v , W[β,P,Pθ ];w = ¯ ¯ Otherwise, if θ = θ and O = P, then W[α,O,Pθ ];v , W[β,O,Pθ ];w = |W (G|Mθ )|γ(G|Mθ )c(G|Mθ )2 ¯ ¯ Z ‹ O (α(ν), β(ν))σ dµO (ν) v, w I(σ) Proof of Theorem 4.1.3.2 (1) This is an immediate consequence of Lemma 4.1.2.1 and Lemma 4.1.2.2 One first expresses α ∈ C ∞ (O; W hψθ (Hσ )) as a finite combination of Λ(β ⊗ w) After expanding out W[α,O,Pθ ];v , this shows that it is a finite ¯ combination of (f[β,O,Pθ ];w,v )ψ hence the result ¯ 35 4.2 PROOF OF THEOREM 4.1.3.2 Proof of Theorem 4.1.3.2 (2) We know from the main theorem of the Plancherel formula for L2 (G) that C ∗ (G) is the span over all f = f[α,O,Pθ ];w,v for all ¯ ¯θ ] (including those orbits of discrete series which are not ψ θ generic) datum [α, O, P (When P = G, f reduces to matrix coefficients of a discrete series representation of G) If σ ∈ O we let β (ν) = Λ(α ⊗ w) where α ∈ C ∞ (O), w ∈ I(σ) Then β is identically the zero function if σ is not ψ θ generic Otherwise, if it is nonzero, Lemma 4.1.2.1 implies that (f[α,O,Pθ ];w,v )ψ = W[β ,O,Pθ ];v ¯ ¯ Since Λ is surjective by Lemma 4.1.2.2 and we also know that taking ψ-coinvariants of C ∗ (G) is surjective onto C ∗ (N0 \ G; ψ) by Lemma 3.2.1.2 (1), this proves (2) 4.2 Proof of Theorem 4.1.3.2 4.2.1 In this section we present the proof of Theorem 4.1.3.2 (3) Given two standard parabolic subgroups of G, Pθ and Pϑ we say that (Pθ , Aθ ) dominates (Pϑ , Aϑ ) if and only if Pθ ⊃ Pϑ and Aθ ⊂ Aϑ Lemma 4.2.1.1 Consider two standard parabolic pairs (Pϑ , Aϑ ) and (Pθ , Aθ ), a ¯ datum [α, O, Pθ ] and suppose that ((f[α,O,Pθ ];w,v )ψ )Pϑ = Then (Pϑ , Aϑ ) dominates ¯ (Pθ , Aθ ) Proof Recall that Z f (g) = f[α,O,Pθ ];v,w (g) = ¯ O πPθ ,σ,ν (g)vν , wν α(ν) dµO (ν) ¯ The value of (fψ )Pθ at m ∈ Mθ is given by the integral Z Z δPϑ (m) ¯ Nϑ N0 ψ −1 (n0 )f (n0 nm)dn0 d¯ ¯ n ¯ By Lemma 2.2.1.1 we may switch the order of integration and integrate over Nϑ first Consider the following version of the Iwasawa decomposition, ¯ ¯ G = K Pϑ = KMϑ Nϑ Thus writing n0 = k(n0 )m(n0 )¯ (n0 ) we compute n Z Z Z ¯ Nϑ f (n0 nm) d¯ = ¯ n ¯ Nϑ O πPθ ,σ,ν (k(n0 )m(n0 )¯ m)vν , wν α(ν) dµ(ν) d¯ n n ¯ = δPϑ (m(n0 )) (4.2.1.1) Z Z ¯ Nϑ O ϕv,π(k(n0 ))w [¯ m(n0 )m](ν)α(ν) dµ(ν)d¯ n n −1 = δPϑ (m(n0 ))δPϑ2 (m)(f[α,O,Pθ ];v,π(k(n0 ))w )Pϑ (m(n0 )m) ¯ To ensure that the right hand side is not equivalently zero, it is necessary that (Pϑ , Aϑ ) dominates (Pθ , Aθ ) by [Wal, Proposition VI.4.1] Now recall that , is the L2 inner product on the space L2 (N0 \ G; ψ) 36 4.2 PROOF OF THEOREM 4.1.3.2 Lemma 4.2.1.2 If two parabolic subgroups, Pθ and Pϑ are not equal, then for any ¯ ¯ datum [β, P, Pϑ ], [α, O, Pθ ] and any discrete series representation τ ∈ P and σ ∈ O, (f[β,OQ ,Q];w,v )ψ , (f[α,O,P ];w ,v )ψ = ¯ ¯ for all w, v ∈ I(τ ) and w , v ∈ I(σ) Proof Set f = f[β,P,Pϑ ];w,v and ϕ = (f[α,O,Pθ ];w ,v )ψ Then B(f, ϕ) is equals to the ¯ ¯ inner product we wish to compute above However we know from (3.3.2.3) and Lemma 4.2.1.1 that B(f, ϕ) = unless (Pϑ , Aϑ ) dominates (Pθ , Aθ ) Reversing the roles of f and ϕ, we also conclude that B(f, ϕ) = unless (Pθ , Aθ ) dominates (Pϑ , Aϑ ) Combining these two statements gives the lemma 4.2.2 We wish to compute ((f[α,O,P ];w,v )ψ )P Recall from [Wal, Proposition VI.4.1] ¯ P that (f[α,O,P ];w,v ) is equals to ¯ (4.2.2.1) γ(G|M )c(G|M )2 Z O X α(ν) χwν ⊗ wσ(m).Aw (ν)uν (1), Aw (ν)vν (1) wσ dν w∈W (G|M ) where Aw (ν) are the intertwining operators defining the unitary intertwining map ◦ cP |P (w, χν ⊗ σ) We remind the reader that , wσ denotes the G-invariant Hermitian ¯ ¯ inner product on wσ Proposition 4.2.2.1 The Harish-Chandra transform, ((f[α,O,P ];u,v )ψ )P ¯ (4.2.2.2) = γ(G|M )c(G|M )2 Z O α(ν) X χwν (m)λEwσ,wν (Aw (ν)v) (wσ(m)Aw (ν)uν (1)) dν w∈Wθ Proof Combining the expressions in the right hand side of (4.2.1.1) with (4.2.2.1), we have that ((f[α,O,P ];u,v )ψ )P ¯ Z Z = δP (m) ¯ N N0 ψ −1 (n0 )f (n0 nm) dn0 d¯ ¯ n By (4.2.2.1), this equals to = γ(G|M )c(G|M )2 δP (m(n0 )) Z Z N0 O ψ −1 (n0 ) α(ν) X χwν (m(n0 )m) wσ(m(n0 )m).Aw (ν)uν (1), Aw (ν)vν (k(n0 )−1 ) w∈Wθ 37 wσ dν 4.2 PROOF OF THEOREM 4.1.3.2 By moving δP (m(n0 )), wσ(m(n0 )) and χwν(m(n0 )) into the second variable of , obtain Z γ(G|M )c(G|M ) Z O N0 α(ν) X w∈Wθ O α(ν) X we ψ −1 (n0 ) χwν (m) wσ(m).Aw (ν)uν (1), δP χwν ⊗ wσ(m(n0 )−1 ).Aw (ν)vν (k(n0 )−1 ) ¯ Z = γ(G|M )c(G|M ) Z wσ , N0 ψ −1 (n0 ) χwν (m) wσ(m).Aw (ν)uν (1), Aw (ν)vν ((¯ (n0 )m(n0 )k(n0 ))−1 ) n wσ dν w∈Wθ G since Aw (ν)vν ∈ IP wσ ⊗ χwν Continuing, this expression ¯ = γ(G|M )c(G|M ) Z N0 Z O α(ν) X χwν (m) w∈Wθ ψ −1 (n0 ) wσ(m).Aw (ν)uν (1), Aw (ν)vν (n−1 ) wσ dν Arguing as in the proof of Lemma 3.2.1.2 (1), this expression is equals to γ(G|M )c(G|M )2 Z O α(ν) X χwν (m)λEwσ,wν (Aw (ν)v) (wσ(m)Aw (ν)uν (1)) dν w∈Wθ Proof of Theorem 4.1.3.2 (3) Assume Pθ = Pϑ As we may express W[α,O,P ];v ¯ in terms of fψ (c.f proof of Theorem 4.1.3.2 (1)), the orthogonality statement follows immediately from Lemma 4.2.1.2 Now assume that θ = ϑ and that we have two orbits O and P containing discrete series representations σ and respectively Consider functions W[α,O,Pθ ];v and W[β,P,Pθ ];v1 ¯ ¯ where α = Λ(α ⊗ u) and β = Λ(β ⊗ u1 ) with u, v ∈ I(σ) and u1 , v1 ∈ I( ) We observe that by Lemma 4.1.2.1, W[Λ(α ⊗v),O,Pθ ];u (g) = (f[α,O,Pθ ];u,v )ψ (g) ¯ ¯ P Since Lemma 4.1.2.2 tells us that Λ is surjective what must be shown is that if W (G|Mθ )O, then W[β ,P,Pθ ];v1 , W[α ,O,Pθ ];v = ¯ ¯ To this end, it suffices to show that B(f[β,P,Pθ ];u1 ,v1 , (f[α,O,Pθ ];u,v )ψ ) = ¯ ¯ λE We begin from the expression of (3.3.2.3) We write the expression ,ν (v1 ) ( (m)u1 (k)) there as W (u1 (k), λE ,ν (v1 ) )(m) =: W u1 (k),λEv1 (m) ,ν 38 wσ dν 4.2 PROOF OF THEOREM 4.1.3.2 In the same way λEwσ,wν (Aw (ν)v) (wσ(m)Aw (ν)uν (1)) in (4.2.2.2) is written as Au(1),λ W (Aw (ν)uν (1), λEwσ,wν (Aw (ν)v) )(m) =: Wwσ,wν EAv (m) If we fix a χν ∈ Im Xur (M ), then these are square integrable Whittaker functions on the Levi, M and realizes the representations and wσ in ◦ C(N∗ \ M ; ψ θ ) Now we let ϕ = (f[α,O,Pθ ];u,v )ψ and f = f[β,P,Pθ ];u1 ,v1 If we substitute this into ¯ ¯ (3.3.2.3), and use (4.2.2.2) then the expression becomes γ(G|M )c(G|M )2 Z N∗ \M ×K „Z O Z X α(ν) β(ν1 )χν1 (m)W P Au(k),λEAv χwν (m)Wwσ,wν u1 (k),λEv1 (m) dµP (ν1 ) ,ν1 Ž (m) dν dm dk w∈Wθ (4.2.2.3) = γ(G|M )c(G|M )2 Z Z Z P β(ν1 ) X O α(ν) χν1 (m)W N∗ \M ×K w∈W θ u1 (k),λEv1 Au(k),λ (m)χwν (m)Wwσ,wν EAv (m) dm dk ,ν1 dν dµP (ν1 ) As is not equivalent to wσ for any w ∈ Wθ , the integral over N∗ \ M vanishes Now we come to the final assertion of the theorem In this we will assume that Pθ = Pϑ and because of Lemma 4.1.3.1, P = O However we will consider the function W[β ,O,Pθ ];v where β is replaced by ¯ β = (ν → β(ν)λ) where β ∈ C ∞ (O) and λ ∈ W hψθ (Hσ ) Now from (3.3.2.3) and (4.1.3.1) W[β,O,Pθ ];u1 ,W[Λ(α⊗v),O,Pθ ];u = ¯ ¯ Z Z Z N0 \G = O Jσ,ν (β(ν)λ)(πPθ ,σ,ν (g)u1 )(f[α,O,Pθ ];u,v )ψ (g) dµO (ν) dg ¯ ¯ Z N2 \M ×K O β(ν)λ(χν ⊗ σ(m)u1 (k))((R(k)(f[α,O,Pθ ];u,v )ψθ )Pθ (m) dµO (ν) dm dk ¯ = γ(G|M )c(G|M ) Z X N2 \M ×K w∈W θ Z Z O β(ν) O α(ν1 ) Au (k),M −1 λ χwν (m)Wwσ,wν Au(k),λ (m)χwν1 (m)Wwσ,wν1 EAv (m) dm dk dν1 dµO (ν) by (4.2.2.2) and (4.1.3.1) The integral over N2 \ M vanishes unless ν = ν1 in which case we are simply integrating on the image of the diagonal embedding O → O × O 39 4.3 THE PLANCHEREL FORMULA FOR L2 (N0 \ G; ψ) It is clear that the measure on the image is simply dµO (ν) Then the expression above reduces to (4.2.2.4) γ(G|M )c(G|M )2 X Z w∈Wθ Z O α(ν)β(ν) Au (k),M −1 λ N2 \M ×K Wwσ,wν Au(k),λ (m)Wwσ,wν EAv (m) dm dk dµO (ν) Recall that λ ∈ W hψθ (Hσ ) is the functional used to define (ν → β(ν)λ) For any ν ∈ O, let v ∈ Hwσ be chosen to satisfy Mw (ν)−1 λ(v ) = and set η = λv Then we have that Au(k),λ Wwσ,wν EAv (m) = λwσ(m)Au(k) (EAv) = Jwσ,wν (λwσ(m)Au(k) )(Av) and λwσ(m)Au(k) = η(wσ(m)Au(k))Mw (ν)−1 λ so that Au(k),λ Wwσ,wν EAv (m) = η(wσ(m)Au(k))Jwσ,wν (Mw (ν)−1 λ)(Av) Now, Z Au (k),M −1 λ N2 \M ×K Wwσ,wν Au(k),λ (m)Wwσ,wν1 EAv (m) dm dk = Jwσ,wν (Mw (ν)−1 λ)(Aw (ν)v)(η, Mw (ν)−1 λ)wσ = d(wσ)−1 Jwσ,wν (Mw (ν)−1 λ)(Aw (ν)v) u, u1 Z K Aw (ν)u(k), Aw (ν)u1 (k) wσ I(σ) by definition of the canonical Hermitian form on I(σ) and Lemma 4.1.2.3 Continuing from (4.2.2.4), this expression simplifies to γ(G|M )c(G|M )2 u, u1 X I(σ) d(wσ)−1 w∈Wθ Z O α(ν)β(ν)Jwσ,wν (Mw (ν)−1 λ)(Aw (ν)v) dµO (ν) and by (4.1.3.1) and the well known fact that d(wσ) = d(σ), this equals = γ(G|M )c(G|M )2 u, u1 = γ(G|M )c(G|M ) u, u1 I(σ) |Wθ | I(σ) |Wθ | Z Z O O d(σ)−1 α(ν)β(ν)Jσ,ν (λ)(v) dµO (ν) (α (ν), β (ν))σ dµO (ν) where α (ν) = Λ(α ⊗ v) and β (ν) = β(ν)λ for α, β ∈ C ∞ (O) Lemma 4.1.2.4 justifies the last step The proof is now complete 4.3 The Plancherel formula for L2 (N0 \ G; ψ) 4.3.1 We first begin by defining the direct integral of Hilbert spaces This material is a simplified exposition from [Wa, pg 312] Let us assume that S is a topological space equipped with a σ-finite Borel measure ds and that we are given family of Hilbert spaces indexed by points in S i.e {Hs }s∈S The collection of all functions x : s → x(s) ∈ Hs is said to be a section of {Hs } and if it satisfies these three axioms i If x, y ∈ F, then s → x(s), y(s) Hs is measurable 40 4.3 THE PLANCHEREL FORMULA FOR L2 (N0 \ G; ψ) ii If z is a section of {Hs } and if s → z(s), x(s) Hs is measurable for all x ∈ F, then z ∈ F iii There exists a countable subset {xj }j∈N of F such that if s ∈ S then {xj (s) | j ∈ N} is dense in Hs is said to be measurable sections We denote the set of all measurable sections of {Hs } as F Axiom ii ensures that F forms a vector space Note that because of Axiom i if x ∈ F, it makes sense to integrate s → ||x(s)||2 s A section is said to be square integrable if H || x|| := Z || x(s)||2 s ds < ∞ H S We identify two elements of x, y ∈ F if σ({s ∈ S | || x(s) − y(s)||Hs > 0}) = By abuse of notation we use F to denote the quotient space of square integrable measurable sections by this equivalence relation We define an inner product on F by Z x, y = x(s), y(s) ds S The completion of F to a Hilbert space is denoted Z ⊕ S Hs ds 4.3.2 Fix an open compact subgroup Hm and ν ∈ ImXur (Mθ ) Consider the following finite-dimensional space Hm G Hσ,ν := (IPθ σ ¯ χν )Hm ⊗ W hψθ (Hσ ) with Hermitian inner product w1 ⊗ λ1 , w2 ⊗ λ2 Hm Hσ,ν = w1 , w2 I(σ) (λ1 , λ2 )σ Observe that for a fixed f ∈ C ∗ (N0 \ G; ψ)Hm the integral in Corollary 3.3.2.2 Hm defines a conjugate linear functional on Hσ,ν considered as a Hilbert space By the Reisz Representation theorem there exists a unique element in I(σ) ⊗ W hψθ (Hσ ) denoted WPθ ,σ (f )(ν) such that ¯ WPθ ,σ (f )(ν), w ⊗ λ ¯ Z Hm Hσ,ν = N0 \G f (g)Jσ (λ)(πPθ ,σ,ν (g)w) dg ¯ As we vary σ χν over O, we may consider WPθ ,σ (f ) as an element of ¯ ∞ I(σ) ⊗ C (O; W hψθ (Hσ )) Define the following linear operator on I(σ) ⊗ C ∞ (O; W hψθ (Hσ )) to C ∗ (N0 \ G; ψ), TPθ ,σ (v ⊗ α) := |W (G|Mθ )|−1 γ(G|Mθ )−1 c(G|Mθ )−2 W[α,O,Pθ ];v ¯ ¯ Let E2 θ (Mθ ) denote the set of isomorphism classes of ψ θ generic discrete series ψ representations on Mθ By E2 θ (Mθ )/W (G|Mθ ) we mean the set of equivalence classes of ψ discrete series representations modulo the natural action of the Weyl group W (G|Mθ ) 41 4.3 THE PLANCHEREL FORMULA FOR L2 (N0 \ G; ψ) Theorem 4.3.2.1 (Plancherel Formula for L2 (N0 \ G; ψ)) Consider f ∈ C (N0 \ G; ψ), then ∗ (4.3.2.1) X f= X TPθ ,σ (WPθ ,σ (f )) ¯ ¯ (Pθ ,Aθ ) (P0 ,A0 ) σ∈E2 θ (Mθ )/W (G|Mθ ) ψ S Proof Set Hσ,ν = K K Hσ,ν and consider the Hilbert space defined by Iσ,Mθ = Z ⊕ O Hσ,ν d˜O (ν) µ where d˜O = |W (G|Mθ )|−1 γ(G|Mθ )−1 c(G|Mθ )−2 dµO (ν) µ Then we consider I(σ) ⊗ C ∞ (O; W hψθ (Hσ )) as a dense subspace of Iσ,Mθ The linear operator TPθ ,σ extends to Iσ,Mθ by continuity ¯ We check that for any v1 ⊗ α1 , v2 ⊗ α2 ∈ I(σ) ⊗ C ∞ (O; W hψθ (Hσ )), TPθ ,σ (v1 ⊗ α1 ), TPθ ,σ (v2 ⊗ α2 ) ¯ ¯ L2 = |W (G|Mθ )|−2 γ(G|Mθ )−2 c(G|Mθ )−4 W[α,O,Pθ ];v1 , W[α,O,Pθ ];v2 ¯ ¯ by definition of TPθ ,σ ¯ = |W (G|Mθ )|−1 γ(G|Mθ )−1 c(G|Mθ )−2 Z ‹ O (α1 (ν), α2 (ν))σ dµO (ν) v1 , v2 by Theorem 4.1.3.2(3) Z = ‹ O (α1 (ν), α2 (ν))σ d˜O (ν) µ = v1 ⊗ α1 , v2 ⊗ α2 v1 , v2 I(σ) Iσ,Mθ Thus we see that TPθ ,σ extends to a unitary operator from Iσ,Mθ into ¯ L2 (N0 \ G; ψ) By Theorem 4.1.3.2 (2) and Lemma 4.1.3.1, one sees that X T := X TPθ ,σ ¯ (Pθ ,Aθ ) (P0 ,A0 ) σ∈E2 θ (Mθ )/W (G|Mθ ) ψ is a linear map from X X I(σ) ⊗ C ∞ (O; W hψθ (Hσ )) (Pθ ,Aθ ) (P0 ,A0 ) σ∈E2 θ (Mθ )/W (G|Mθ ) ψ onto C ∗ (N0 \ G; ψ) By the previous paragraph, T is extends to unitary linear operator on X X Iσ,Mθ (Pθ ,Aθ ) (P0 ,A0 ) σ∈E2 θ (Mθ )/W (G|Mθ ) ψ 42 I(σ) 4.3 THE PLANCHEREL FORMULA FOR L2 (N0 \ G; ψ) As before, let α(ν) = α (ν)λ for any α ∈ C ∞ (O) and a fixed λ ∈ W hψθ (Hσ ) Then, we see that TP ,σ (WP ,σ (f )), TP ,σ (w ⊗ α) ¯ ¯ ¯ Z L2 = = ZO α (ν) WP ,σ (f )(ν), w ⊗ λ ¯ ZN0 \G = N0 \G Z f (g) O µ Hσ,ν d˜ O (ν) Jσ,ν (α (ν)λ)(πPθ ,σ,ν (g)w) d˜O (ν) dg µ ¯ f (g)|W (G|Mθ )|−1 γ(G|Mθ )−1 c(G|Mθ )−2 W[α,O,Pθ ];v (g) dg ¯ = f, TP ,σ (w ⊗ α) ¯ L2 It is clear that if P = P or if σ and σ are not Weyl group conjugates of each other, then TP ,σ (WP ,σ (f )), TP ,σ (w ⊗ α) L2 = ¯ ¯ ¯ Let R(f ) denote the right hand side of (4.3.2.1) Then the considerations of the previous two paragraphs imply that f, h = R(f ), h for all h ∈ C ∗ (N0 \ G; 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ψ) where G is a quasi-split p- adic group and ψ a non-degenerate unitary character of the unipotent subgroup N0 of a minimal parabolic subgroup of G We obtain the direct... nondegenerate unitary character of the unipotent radical N0 of a minimal standard parabolic subgroup of a connected quasi-split p- adic group, G Define L2 (N0 \ G; ψ) as the space of functions on G which