A correction to Conducteur des Représentations du groupe linéaire

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1 A correction to Conducteur des Représentations du groupe linéaire Hervé Jacquet December 5, 2011 Nadir Matringe has indicated to me that the paper Conducteur des Représentations du groupe linéaire ([JPSS81a], citeerror2) contains an error. I correct the error in this note. The correct proof is actually simpler than the erroneous proof. Separately, Matringe has given a different, interesting proof of the result in question ([Mat11]). We recall the result in question. Let F be a local field. We denote by α the absolute value, by q the cardinality of the residual field and finally by v the valuation of F. Thus α(x) = x = q v(x). Let ψ be an additive character of F whose conductor is the ring of integers O F. Let G r be the group GL(r) regarded as an algebraic group. We denote by dg the Haar measure of G r (F ) for which the compact group G r (O F ) has volume 1. Let N r be the subgroup of upper triangular matrices with unit diagonal. We define a character θ r,ψ : N r (F ) C by the formula θ r,ψ (u) = ψ u i,i+1 We denote by du the Haar measure on N r (F ) for which N r (O F ) has measure 1. We have then a quotient invariant measure on N r (F )\G r (F ). Let S r be the algebra of symmetric polynomials in. (X 1, X 1 1, X 2, X 1 2,... X r, X 1 r ). Let H r be the Hecke algebra. Let S r : H r S r be the Satake isomorphism. Thus for any r tuple of non-zero complex numbers (x 1, x 2,..., x r ) we have an homomorphism H r C, defined by φ S(φ)(x 1, x 2,..., x r ). 1

2 There is a unique function W : G r (F ) C satisfying the following properties: W (gk) = W (g) for k G r (O F ), W (ug) = θ ψ (u)w (g) for u inn r (F ), for all (x 1, x 2,..., x r ) and all φ H r, W (gh)φ(h)dh = S(φ)(x 1, x 2,..., x r ) W (g), W (e) = 1. G r(f ) We will denote this function by W (x 1, x 2,... x r ; ψ) and its value at g by W (g; x 1, x 2,... x r ; ψ). Let (π, V ) be an irreducible admissible representation of G r (F ). We assume that π is generic, that is, there is a non-zero linear form λ : V C such that λ(π(u)v) = θ r,ψ (u) λ(v) for all u N r (F ) and all v V. Recall that such a form is unique, within a scalar factor. We denote by W(π; ψ) the space of functions of the form g λ(π(g)v), with v V. It is the Whittaker model of π. On the other hand, we have the L factor L(s, π) ([GJ72]). We denote by P π (X) the polynomial defined by L(s, π) = P π (q s ) 1. The main result of [JPSS81a] is the following Theorem. Theorem 1 There is an element W W(π; ψ) such that, for all r 1-tuple of non zero complex numbers (x 1, x 2,..., x r 1 ), ( ) g 0 W W (g; x 1, x 2,... x r 1 ; ψ) det g s 1/2 dg N r 1 (F )\G r 1 (F ) = P π (q s x i ) 1. In [JPS] it is shown that if we impose the extra condition ( ) ( ) gh 0 g 0 W = W 2

3 for all h G r 1 (O F ) and g G r 1 (F ) then W is unique. The vector W is called the essential vector of π and further properties of this vector are obtained in [JPSS81a]. The proof of this theorem is incorrect in [JPS]. We give a simple proof here. 1 Review of some properties of the L factor Let r 2 be an integer. Let (t 1, t 2,..., t r 1 ) be a r 1-tuple of complex numbers. We assume that Re(t 1 ) Ret 2 Re(t r 1 ). We denote by π(t 1, t 2,... t r 1 ) the corresponding principal series representation. It is the representation induced by the characters α t 1, α t 2,... α t r 1. Its space I(t 1, t 2,... t r 1 ) is the space of smooth functions φ : G r 1 (F ) C such that a φ 0 a g = a r a 1 t 1+ r 2 2 a2 t 2+ r a r 1 t r 1 r 2 2 φ(g). The space I(t 1, t 2,... t r 1 ) contains a unique vector φ 0 equal to 1 on G r 1 (O F ) and thus invariant under G r 1 (O F ). We recall a standard result. Lemma 1 The vector φ 0 is a cyclic vector for the representation π(t 1, t 2,... t r 1 ). Proof: Indeed, if Re(t 1 ) = Ret 2 = = Re(t r 1 ), the representation is irreducible and our assertion is trivial. If not, we use Langlands construction ([Sil78]). There is a certain intertwining operator N defined on the space of the representation and the kernel of N is a maximal invariant subspace. By direct computation Nφ 0 0 and our assertion follows. The representation I(t 1, t 2,... t r 1 ) admits a non-zero linear form λ such that, for u N r 1 (F ), λ(π(u)φ) = θ r 1,ψ (u)λ(g). We denote by W(t 1, t 2,..., t r 1 ; ψ) the space spanned by the functions of the form g W φ (g), W φ (g) = λ(π(t 1, t 2,... t r 1 )(g)φ), with φ I(t 1, t 2,... t r 1 ) We recall the following result ([JS83]) 3

4 Lemma 2 The map φ W φ is injective. It follows that the image W 0 of φ 0 is a cyclic vector in W(t 1, t 2,..., t r 1 ; ψ). Up to a multiplicative constant, W 0 is equal to the function W (x 1, x 2,..., x r 1 ; ψ). Now let π be a generic representation of G r (F ). For W W(π, ψ) and W W(t 1, t 2,..., t r 1 ; ψ) we consider the integral Ψ(s, W, W ) = N r 1 \G r 1 W ( g 0 ) W (g) det g s 1/2 dg The integral converges absolutely if Res >> 0 and extends to a meromorphic function of s. In any case, it has a meaning as a formal Laurent series in the variable q s (see below). We recall a result from [JPSS83] Lemma 3 There are functions W j W(π; ψ) and W j W(t 1, t 2,..., t r 1 ; ψ), 1 j k, such that Ψ(s, W j, W j) = L(s + t i, π). 1 j k Since W 0 is a cyclic vector we see, after a change of notations, that there are functions W j W(π; ψ) and integers n j, 1 j k, such that q nis Ψ(s, W i, W (x 1, x 2,... x r 1 ; ψ)) = i L(s + t i, π). In our discussion x 1 x 2 x r 1. However, the functions W (x 1, x 2..., x r 1 ; ψ) are symmetric in the variables x i. Thus we have the following result. Lemma 4 Given a r 1-tuple of non-zero complex numbers (x 1, x 2,... x r 1 ) there are functions W j W(π; ψ) and integers n j, 1 j k, such that q njs Ψ(s, W j, W (x 1, x 2,... x r 1 ; ψ)) = j P π (q s x i ) 1. 2 The ideal I π First, we can define a function W (X 1, X 2,... X r 1 ; ψ) with values in S r 1 such that,n for every g and every r 1-tuple (x 1, x 2,..., x r 1 ), the scalar W (g; x 1, x 2,... x r 1 ) is the value of the polynomial W (g; X 1, X 2,... X r 1 ; ψ) at the point (x 1, x 2,..., x r 1 ). For g in a set compact modulo N r 1 (F ) the 4

5 polynomials W (g; X 1, X 2,... X r 1 ; ψ) remain in a finite dimensional vector subspace of S r 1. We have the relation det g s W (g; x 1, x 2,..., x r 1 ; ψ) = W (g; q s x 1, q s x 2,..., q s x r 1 ; ψ). It follows that if det g = q n then the polynomial W (g; X 1, X 2,... X r 1 ; ψ) is homogeneous of degree n. For each integer n define the integral det g =q n W Ψ n (W ; X 1, X 2,... X r 1 ; ψ) = ( ) g 0 W (g, X 1, X 2,... X r 1 ; ψ) det g 1/2 dg The support of the integrand is contained in a set compact modulo N r 1 (F ), which depends on W. In addition, there is an integer N W such that the support of the integrand is empty if n < N(W ). The polynomial is homogeneous of degree n, that is, Ψ n (W ; X 1, X 2,... X r 1 ; ψ X n Ψ n (W ; X 1, X 2,... X r 1 ; ψ) = Ψ n (W ; XX 1, XX 2,... XX r 1 ; ψ). We consider the formal Laurent series Ψ(X; W ; X 1, X 2,... X r 1 ; ψ) = n X n Ψ n (W ; X 1, X 2,... X r 1, ψ), or, more precisely, Ψ(X; W ; X 1, X 2,... X r 1 ; ψ) = n N W X n Ψ n (W ; X 1, X 2,... X r 1 ; ψ) If we multiply, this Laurent series by P π(xx i ) we obtain a new Laurent series Ψ(X; W, X 1, X 2,... X r 1 ; ψ) P π (XX i ) = n N 1 (W ) X n a n (X 1, X 2,... X r 1 ; ψ). where N 1 (W ) is another integer depending on W. We can replace π by its contragredient representation π, W by W, ψ by ψ. Recall that W is defined by W (g) = W (w r t g 1 ) 5

6 where w r is the permutation matrix whose non-zeero entries are on the antidiagonal. The function W belongs to W( π, ψ). We define similarly Ψ( W ; X 1, X 2,..., X r 1 ; ψ). We have then the following functional equation Ψ(q 1 X 1 ; W ; X 1 1, X 1 2,... X 1 r 1 r 1 ; ψ) i=1 r 1 = ɛ π (XX i, ψ) Ψ(X, W, X 1, X 2,... X r 1 ; ψ) i=1 The ɛ factors are monomials. that in fact Ψ(X, W, X 1, X 2,..., X r 1 ; ψ) N 2 (W ) n N 1 (W ) P π (q 1 X 1 X 1 i ) P π (XX i ). Thus there is another integer N 2 (W ) such P π (XX i ) = X n a n (X 1, X 2,... X r 1 ). From now on we drop the dependence on ψ from the notation. Hence the product Ψ(X, W, X 1, X 2,... X r 1 ) P π(xx i ) is in fact a polynomial in X with coefficients in S r 1. Moreover, because the a n are homogeneous of degree n, there is a polynomial Ξ(W ; X 1, X 2,... X r 1 ) S r 1 such that Ψ(X; W ; X 1, X 2,... X r 1 ) P π (XX i ) = Ξ(W ; XX 1, XX 2,... XX r 1 ). In a precise way, let us write P π (X i ) = R P m (X 1, X 2,..., X r 1 ) m=0 where P m is homogeneous of degree m. Then Ψ(X; W ; X 1, X 2,... X r 1 ) n X n R m=0 P π (XX i ) =. Ψ n m (W ; X 1, X 2,..., X r 1 )P m (X 1, X 2,... X r 1 ). 6

7 The polynomial Ξ(W ; X 1, XX 2,... X r 1 ) is then determined by the condition that its homogeneous component of degree n, Ξ n (W ; X 1, X 2,..., X r 1 ) be given by Ξ n (W ; X 1, X 2,..., X r 1 ) = R Ψ n m (W ; X 1, X 2,..., X r 1 )P m (X 1, X 2,... X r 1 ). m=0 Let I π be the vector space spanned by the polynomials Ξ(W ; X 1, X 2,... X r 1 ). Lemma 5 In fact I π is an ideal. Proof: Let Q be an element of S r 1. Let φ be the corresponding element of H r 1. Then W (gh; X 1, X 2,..., X r 1 )φ(h)dh = W (g; X 1, X 2,..., X r 1 )Q(X 1, X 2,... X r 1 ). by Let W be an element of W(π, ψ). Define another element W 1 of W(π, ψ) We claim that W 1 (g) = G r 1 W [ ( h 1 0 g )] φ(h) det h 1/2 dh. Ξ(W 1 ; X 1 X 2,... X r 1 ) = Ξ(W ; X 1 X 2,..., X r 1 )Q(X 1, X 2,..., X r 1 ). This will imply the Lemma. By linearity, it suffices to prove our claim when Q is homogeneous of degree t. Then φ is supported on the set of h such that det h = q t. We have then, for every n, det g =q n = = det g =q n+t det g =q n+t W Ψ n (W 1 ; X 1, X 2,..., X r 1 ) = ( ) gh 1 0 W W (g; X 1, X 2,..., X r 1 )φ(h) det h 1/2 dh det g 1/2 dg ( ) g 0 W W (gh; X 1, X 2,..., X r 1 )φ(h)dh det g 1/2 dg ( ) g 0 W (g; X 1, X 2,..., X r 1 ) det g 1/2 dgq(x 1, X 2,..., X r 1 ) = Ψ n t (W ; X 1, X 2,..., X r 1 ) Q(X 1, X 2,..., X r 1 ). 7

8 Hence = Ξ n (W 1 ; X 1, X 2,... X r 1 ) = R Ψ n m (W 1 ; X 1, X 2,..., X r 1 )P m (X 1, X 2,..., X r 1 ) m=0 R Ψ n m t (W ; X 1, X 2,..., X r 1 )P m (X 1, X 2,..., X r 1 )Q(X 1, X 2,..., X r 1 ) m=0 = Ξ n t (W ; X 1, X 2,... X r 1 ) Q(X 1, X 2,..., X r 1 ). Since Q is homogeneous of degree t our assertion follows. 3 Conclusion Given a r 1-tuple of non-zero complex numbers (x 1, x 2,..., x r 1 ), Lemma 4 shows that we can find W j and integers n j such that (q s ) n j Ξ(W j, q s x 1, q s x 2,..., q s x r 1 ) = 1. In particular, 1 j k Thus the element 1 j k Q(X 1, X 2,... X r 1 ) = Ξ(W j, x 1, x 2,..., x r 1 ) = 1. 1 j k Ξ(W j ; X 1, X 2,... X r 1 ) of I π does not vanish at (x 1, x 2,..., x r 1 ). By the Theorem of zeros of Hilbert I π = S r 1. In particular, there is W such that This implies the Theorem. Ξ(W ; X 1, X 2,... X r 1 ) = 1. Remark 1: The proof in [JPSS81a] is correct if L(s, π) is identically 1. In general, the proof there only shows that the elements of I π cannot all vanish on a coordinate hyperplane X i = x. Remark 2 : Consider an induced representation π of the form π = I(σ 1 α s 1, σ 2 α s 2,..., σ k α s k ) 8

9 where the representations σ 1, σ 2,... σ k are tempered and s 1, s 2,... s + k are real numbers such that s 1 > s 2 > > s k. The representation π may fail to be irreducible. But, in any case, it has a Whittaker model ([JS83]) and Theorem 1 is valid for the representation π. Remark 3: The proof of Matringe uses the theory of derivatives of a representation. The present proof appears simple only because we use Lemma 3, the proof of which is quite elaborate (and can be obtained from the theory of derivatives)., References [BZ77] I. N. Bernstein and A. V. Zelevinsky. Induced representations of reductive p-adic groups. I. Ann. Sci. École Norm. Sup. (4), 10(4): , [CPS11] J.W. Cogdell and I.I Piatetski-Shapiro. Derivatives and l- functions for GL(n), [GJ72] Roger Godement and Hervé Jacquet. Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin, [JPSS81a] H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika. Conducteur des représentations du groupe linéaire. Math. Ann., 256(2): , [JPSS81b] Hervé Jacquet, Ilja Piatetski-Shapiro, and Joseph Shalika. Conducteur des représentations génériques du groupe linéaire. C. R. Acad. Sci. Paris Sér. I Math., 292(13): , [JPSS83] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika. Rankin- Selberg convolutions. Amer. J. Math., 105(2): , [JS83] Hervé Jacquet and Joseph Shalika. The Whittaker models of induced representations. Pacific J. Math., 109(1): , [JS85] Hervé Jacquet and Joseph Shalika. A lemma on highly ramified ɛ-factors. Math. Ann., 271(3): ,

10 [Mat11] [Sil78] [Wal88] Nadir Matringe. Essential whitttaker functions for GL(n) over a p-adic field, Allan J. Silberger. The Langlands quotient theorem for p-adic groups. Math. Ann., 236(2):95 104, Nolan R. Wallach. Real reductive groups. I, volume 132 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, [Zel80] A. V. Zelevinsky. Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4), 13(2): ,