Functional equation for GL 2 and cuspidal local constants

Transcription

1 Functional equation for GL 2 and cuspidal local constants Johan M. Commelin November 12, 2014 Let π be an irreducible smooth representation of G = GL 2 (F ). We want to attach an L-function L(π, s), and a so called local constant ε(π, s, φ) to π. We give the definition for general smooth representations of G. Further, we compute these invariants in the cuspidal case. The non-cuspidal case will be done by Julius. The main results of this talk are: (1) If π is cuspidal, the L-function is constant 1. (2) Let (, J, Λ) be a cuspidal type. Set π = c-ind G J Λ. The epsilon factor is determined by ε(π, 1 2, ψ) = qa trλ (cx)ψ (cx), x for some c J and q a that will be described later. To arrive at our destination, we will have to (i) define Fourier transforms of coefficients of G; (ii) define L-functions/functional equation/local constants (including a review of the GL 1 case); (iii) work our way through some technical results with strata and cuspidal types. 1 Fourier transforms We simultaneously define the Fourier transform for the case GL 1 and GL 2. Notation: n Either 1 or 2. F be a non-archimedean local field. be the ring M n (F ). M M n (o), the integral part of. G be the group GL n (F ) =. µ be a Haar measure on. ψ ˆF, ψ 1, a non-trivial character of F. ψ = ψ tr. 1

2 C c () be the space of locally constant functions C that have compact support. Given Φ C c (), the Fourier transform ˆΦ (relative to µ and ψ) is given by ˆΦ = Φ(y)ψ (xy) dµ(y). Lemma 1 (a) The function ˆΦ is an element of C c (). (b) There is a positive real number c (depending on µ and ψ) such that ˆΦ = cφ( x), Φ C c (), x. (1) (c) Fixing ψ, there is a unique µ ψ so that this c is equal to 1. This measure satisfies µ ψ (M) = q ln2 /2, where l is the level of ψ. (d) For a F, we have µ aψ = a n2 /2 µ ψ. Proof First note that Cc () is spanned by the characteristic functions (indicator functions) of the sets a + p j M (for a and j Z). One then proves the first two items for the characteristic functions of the sets p j M. The next step is proving it for their translations. Consequently the first two items hold for all Φ Cc (). For these characteristic functions one computes c = µ(m) 2 q ln2. It is immediate from (1) that scaling µ with b > 0 has the effect of scaling c with b 2. Hence, it is clear that there exists a measure with c = 1, and since µ is determined by µ(m), it is unique. This proves the third item. The fourth item is an immediate consequence. The measure µ ψ is the self-dual Haar measure (with respect to ψ). 2 Introducing the main actors 2.1 The L-function Let (π, V ) be an irreducible smooth representation of G. Let C(π) be the space of coefficients of π. It is the space of smooth functions on G generated by functions of the form γˇv v : G C g ˇv, π(g)v Given Φ C c () and f C(π), we define a function on the complex plane ζ(φ, f, s) = Φ(x)f(x) det x s dµ (x) G where dµ is a Haar measure on G. There is one small little problem: the integral does not converge in general. Lemma 2 (p148) There exists an s 0 R such that the integral converges, absolutely and uniformly in vertical strips in the region Rs > s 0, for all Φ and f. The integral represents a rational function in q s. 2

3 Next we define Z(π) = {ζ(φ, f, s ) Φ C c (), f C(π)} C(q ±s ). It turns out that Z(π) is a principal fractional ideal. Lemma 3 (p148) There is a unique polynomial P π (X) C[X], with P π (0) = 1, and Z(π) = P π (q s ) 1 }{{} The L-function! C[q ±s ]. Moreover, this polynomial does not depend on µ. Finally, we define the L-function L(π, s) = P π (q s ) The local constant and the functional equation So far we haven t applied the Fourier transform yet. It is about time to change that. Observe that if f C(π), then lies in C(ˇπ). Moreover, ˇf : G C g f(g 1 ) C(π) C(ˇπ) f ˇf is a linear isomorphism. We use this duality to obtain a relation between the L-function of π and ˇπ. Fix a non-trivial ψ ˆF, and use the Haar measure that is self-dual with respect to ψ. Theorem 1 (p148) There is a unique rational function γ(π, s, ψ) C(q s ) such that ζ(ˆφ, ˇf, 3 2 s) = γ(π, s, ψ)ζ(φ, f, s), for all Φ and f. This rational function allows us to define the local constant. It is L(π, s) ε(π, s, φ) = γ(π, s, ψ) L(ˇπ, 1 s). Observe that ε(π, s, ψ)ε(ˇπ, 1 s, ψ) = γ(π, s, ψ)γ(ˇπ, 1 s, ψ). Using the theorem, we compute ζ(ˆφ, f, s) =γ(ˇπ, 1 s, ψ)ζ(ˆφ, ˇf, 3 2 s) =γ(ˇπ, 1 s, ψ)γ(π, s, ψ)ζ(φ, f, s) and since we are dealing with the self-dual Haar measure, we derive ε(π, s, ψ)ε(ˇπ, 1 s, ψ) = γ(ˇπ, 1 s, ψ)γ(π, s, ψ) = ω π ( 1) 3

4 Lemma 4 There exist a C and b Z such that ε(π, s, ψ) = aq bs. We now investigate how the γ- and ε-factors depend on ψ. We compute ζ(ˆφ, ˇf, 3 2 s) = ˆΦ(x) ˇf(x) det x 3 2 s dµ (x) G = Φ(y)aψ (xy) dµ aψ (y) ˇf(x) det x 3 2 s dµ (x) G = Φ(y)ψ (axy) a 2 dµ ψ (y) ˇf(x) det x 3 2 s dµ (x) G = Φ(y)ψ (ty) a 2 dµ ψ (y) ˇf(a 1 t) det ta s dµ (t) G = a 2 Φ(y)ψ (ty) dµ ψ (y)ω π (a) ˇf(t) a 2s 3 det t 3 2 s dµ (t) G =ω π (a) a 2s 1 Φ(y)ψ (ty) dµ ψ (y) ˇf(t) det t 3 2 s dµ (t) from which we derive G γ(π, s, aψ) =ω π (a) a 2s 1 γ(π, s, ψ), ε(π, s, aψ) =ω π (a) a 2s 1 ε(π, s, ψ). Using certain truncations of ζ(φ, f, s) we build approximations Z(Φ, f, X) which are formal Laurent series. With these the book tackles the following proposition. Proposition 1 Let (π, V ) be an irreducible cuspidal representation of G. Then Z(π) = C[X, X 1 ]. s an immediate consequence, the L-function of an irreducible cuspidal representation is the constant function 1. 3 Technical results concerning the local constants So, the upshot of Maarten s talks was the following theorem. Theorem 2 (p108) The map (, J, Λ) π Λ = c-ind G J Λ induces a bijection between the set of conjugacy classes of cuspidal types in G and the set of equivalence classes of irreducible cuspidal representations of G. In a similar spirit is the following proposition. Proposition 2 (p110) The map (, Ξ) π Ξ = c-ind G K Ξ induces a bijection between the set of G-conjugacy classes of cuspidal inducing data in G and the set of equivalence classes of irreducible cuspidal representations of G. 4

5 Recall that a cuspidal inducing datum in G is a pair (, Ξ), where is a chain order, and Ξ an irreducible smooth representation of K of the form Ξ = Ind K J Λ for some cuspidal type (, J, Λ). (Oh, and K is the group {g G gg 1 = }; J is a subgroup of K ; and Λ a representation of J.) Now that everyone is back on track (including me), we can introduce the main tool for the calculations of this section. It is the so-called non-abelian Gauss sum. Write P for the Jacobson radical of. Let W be the space underlying the representation Ξ. Define an element of End C ( ˇW ), T (Ξ, ψ) = ˇΞ(cx)ψ (cx), where c K satisfies c = P n. x U /U n+1 Lemma 5 The sum T (Ξ, ψ) does not depend on the choices of coset representatives x, nor on the choice of c. Moreover, the endomorphism is scalar. Write τ(ξ, ψ) for the complex number satisfying T (Ξ, ψ) = τ(ξ, ψ)1 ˇW. The number τ(ξ, ψ) C is the non-abelian Gauss sum of Ξ. Observe that τ(ξ, ψ) = 1 tr (T (Ξ, ψ)) dim(ξ) = 1 dim(ξ) x U /U n+1 tr(ˇξ(cx))ψ (cx). We formulate the first computational results with τ(ξ, ψ). Let (π, V ) be an irreducible cuspidal representation of G. From now on, fix ψ ˆF, a character of level one. Recall that l(π) denotes { } there exists a chain order such that min n/e π contains the trivial character of U n+1 Theorem 3 We have ε(π, s, ψ) =(P n : ) ( 1 2 s)/2 τ(ξ, ψ) ( : P n+1 ) 1 2 =q 2l(π)( 1 2 s) τ(ξ, ψ) ( : P n+1 ) 1. 2 The next thing we want to do is translate the formalism of Gauss sums from inducing data back to cuspidal types. Let (, J, Λ) be a cuspidal type that induces Ξ. Lemma 6 The Gauss sum τ(ξ, ψ) is the unique eigenvalue of the scalar operator ˇΛ(cx)ψ (cx), x J U /U n+1 for any c J such that c = P n. 5

6 For some reason (currently unknown to me) we can choose c J such that Λ [n/2]+1 U is a multiple of ψ c. Proposition 3 The Gauss sum τ(ξ, ψ) is the unique eigenvalue of the scalar operator ( : P (n+1)/2 ) ˇΛ(cy)ψ (cy). y U (n+1)/2 /U n/2+1 The theorem, lemma and proposition together give the following corollary. Corollary 1 Let (, J, Λ) be a cuspidal type in G. Let n be the least integer 0 such that U n+1 KerΛ. Choose c J such that Λ [n/2]+1 U is a multiple of ψ c. If π = c-ind G J Λ, then ε(π, 1 2, ψ) = qa x trˇλ(cx)ψ (cx), where x ranges over U (n+1)/2 /U n/2+1 and { q a 1 if n is odd, dim Λ = ( : P) 1/2 if n is even. 6