On the support of matrix coefficients of supercuspidal representations of the general linear group over a local non-archimedean field
aa r X i v : . [ m a t h . R T ] F e b ON THE SUPPORT OF MATRIX COEFFICIENTS OFSUPERCUSPIDAL REPRESENTATIONS OF THE GENERAL LINEARGROUP OVER A LOCAL NON-ARCHIMEDEAN FIELD
EREZ LAPID
Abstract.
We derive an upper bound on the support of matrix coefficients of suprecus-pidal representations of the general linear group over a non-archimedean local field. Theresults are in par with those which can be obtained from the Bushnell–Kutzko classifica-tion of supercuspidal representations, but they are proved independently. Introduction
Throughout, let F be a local non-archimedean field, O its ring of integers, and let G = GL r ( F ) be the general linear group of rank r with center Z . Let K = GL r ( O ) bethe standard maximal compact subgroup of G and for any n ≥ K ( n ) = K r ( n ) be theprincipal congruence subgroup, i.e., the kernel of the canonical map K → GL r ( O /̟ n O )where ̟ is a uniformizer of F . Denote by B ( n ) the ball B ( n ) = { g ∈ G : k g k , k g − k ≤ q n } where q is the size of the residue field of F and k g k = max | g i,j | F where g i,j are the entriesof g . Thus, {B ( n ) : n ≥ } is an open cover of G by compact sets.The purpose of this short paper is to give a new proof of the following result. Theorem 1.1.
Let ( π, V ) be a supercuspidal representation of G and let ( π ∨ , V ∨ ) be itscontragredient. Let v ∈ V , v ∨ ∈ V ∨ and assume that v and v ∨ are fixed under K ( n ) forsome n ≥ . Then the support of the matrix coefficient g ( π ( g ) v, v ∨ ) is contained in Z B ( c ( r ) n ) where c ( r ) is an explicit constant depending on r only. As explained in [FLM12], the theorem is a direct consequence of the classification ofirreducible supercuspidal representations by Bushnell–Kutzko [BK93], or more precisely,of the fact that every such representation is induced from a representation of an opensubgroup of G which is compact modulo Z . This is in fact known for many other casesof reductive groups over local non-archimedean fields and in these cases it implies theanalogue of Theorem 1.1. We refer the reader to [FLM12] for more details.In contrast, the proof given here is independent of the classification. It is based on two in-gredients. The first, which is special to the general linear group, is basic properties of localRankin–Selberg integrals for G × G which were defined and studied by Jacquet–Piatetski-Shapiro–Shalika. In particular, we use an argument of Bushnell–Henniart, originally used Date : October 19, 2018. Throughout, by a representation of G we always mean a complex, smooth representation. to give an upper bound on the conductor of Rankin–Selberg local factors [BH97]. Thesecond ingredient is Howe’s result on the integrality of the formal degree with respect to asuitable Haar measure [How74], a result which was subsequently extended to any reductivegroup [HC99, Vig90, SS97]. The two ingredients are linked by the fact, which also followsfrom properties of Rankin–Selberg integrals, that the formal degree is essentially the con-ductor of π × π ∨ , a feature that admits a conjectural generalization for any reductive group[HII08]. (For another relation between formal degrees and support of matrix coefficientssee [Key92].)As explained in [FL17a, FL17b], Theorem 1.1 is of interest for the problem of limitmultiplicity.I would like to thank Stephen DeBacker, Tobias Finis, Atsushi Ichino and Julee Kim foruseful discussions and suggestions. I am especially grateful to Guy Henniart for his inputleading to Remark 2.4.2. A variant for Whittaker functions
It is advantageous to formulate a variant of Theorem 1.1 for the Whittaker model.Throughout, fix a character ψ of F which is trivial on O F but non-trivial on ̟ − O F . Let N be the subgroup of upper unitriangular matrices in G . If π is a generic irreduciblerepresentation of G , we write W ψ ( π ) for its Whittaker model with respect to the character ψ N of N given by u ψ ( u , + · · · + u r − ,r ). Recall that every irreducible supercuspidalrepresentation of G is generic [GK75].Let A be the diagonal torus of G . For all n ≥ A ( n ) be the open subset A ( n ) = { diag( t , . . . , t r ) ∈ A : q − n ≤ | t i /t i +1 | ≤ q n , i = 1 , . . . , r − } . Clearly, ZA ( n ) = A ( n ) and A ( n ) is compact modulo Z . Theorem 2.1.
There exists a constant c = c ( r ) with the following property. Let π bean irreducible supercuspidal representation of G with Whittaker model W ψ ( π ) and n ≥ .Then the support of any W ∈ W ψ ( π ) K ( n ) is contained in N A ( cn ) K . In order to prove Theorem 2.1 we set some more notation. For any function W on G let M W = sup { val(det g ) : W ( g ) = 0 and k g r k = 1 } ,m W = inf { val(det g ) : W ( g ) = 0 and k g r k = 1 } , (including possibly ±∞ ) where g r is the last row of g and k ( x , . . . , x r ) k = max | x i | .Recall that by a standard argument (cf. [CS80, Proposition 6.1]) for any right K ( n )-invariant and left ( N, ψ N )-equivariant function W on G , if W ( tk ) = 0 for some t =diag( t , . . . , t n ) ∈ A and k ∈ K then | t i /t i +1 | ≤ q n , i = 1 , . . . , r −
1. Hence, m W ≥ − (cid:0) r (cid:1) n .For any W ∈ W ψ ( π ) let f W ∈ W ψ − ( π ∨ ) be given by f W ( g ) = W ( w r t g − ) where w r = (cid:18) . .. (cid:19) .Let t = t ( π ) be the order of the group of unramified characters χ of F ∗ such that π ⊗ χ ≃ π . Clearly, t divides r . UPPORT OF MATRIX COEFFICIENTS 3
Let f = f ( π × π ∨ ) ∈ Z be the conductor of the pair π × π ∨ (see below).Theorem 2.1 is an immediate consequence of the following two results. Proposition 2.2.
For any = W ∈ W ψ ( π ) we have M W + m f W = r − t − f. In particular, if W ∈ W ψ ( π ) K ( n ) then M W ≤ (cid:18) r (cid:19) n + r − t − f. Proposition 2.3.
We have f ≥ ( r + 1) r − t + v q ( t )) where v q ( t ) is the maximal power of q dividing t . Moreover, f is even if q is not a square.Proof of Proposition 2.2. The argument is inspired by [BH97].We may assume without loss of generality that π is unitarizable. For any Φ ∈ S ( F r )consider the local Rankin–Selberg integral A ψ ( s, W, Φ) = Z N \ G | W ( g ) | Φ( g r ) | det g | s dg, a Laurent series in x = q − s which represents a rational function in x [JPSS83]. Note thatif Φ(0) = 0 then A ψ ( s, W, Φ) is a Laurent polynomial in x since W is compactly supportedmodulo ZN . Also note that for any λ ∈ F ∗ we have A ψ ( s, W, Φ( λ · )) = | λ | − rs A ψ ( s, W, Φ) . Recall the functional equation ([JPSS83, Theorem 2.7 and Proposition 8.1] together with[BH99]) q f ( − s ) (1 − q − ts ) A ψ ( s, W, Φ) = (1 − q t ( s − ) A ψ − (1 − s, f W , ˜Φ)where ˜Φ( x ) = Z F n Φ( y ) ψ ( x t y ) dy is the Fourier transform of Φ and f ∈ Z is the conductor.Now let Φ be the characteristic function of the standard lattice { ξ ∈ F r : k ξ k ≤ } andset A ψ ( s, W ) = A ψ ( s, W, Φ ). Then ˜Φ = Φ and we obtain q f ( − s ) (1 − q − ts ) ) A ψ ( s, W ) = (1 − q t ( s − ) A ψ − (1 − s, f W ) . Let Φ = Φ − Φ ( ̟ − · ) be the characteristic function of the K -invariant set { ξ ∈ F r : k ξ k = 1 } of primitive vectors. Then A ψ ( s, W ) := A ψ ( s, W, Φ ) = (1 − q − rs ) A ψ ( s, W ) andthus, q f ( − s ) − q − ts − q − rs A ψ ( s, W ) = 1 − q t ( s − − q r ( s − A ψ − (1 − s, f W ) . EREZ LAPID
Recall that A ψ ( s, W ) is a Laurent polynomial P ψw ( x ) in x = q − s . We get an equality ofLaurent polynomials(1) q f x f − x t − x r P ψW ( x ) = 1 − y t − y r P ψ − f W ( y )where y = q − x − . Note that the fact that P ψW ( x ) is divisible by − x r − x t amounts to sayingthat the integral Z g ∈ N \ G : k g r k =1 , val(det g ) ≡ a (mod r/t ) | W ( g ) | dg is independent of a , which in turn follows from (and in fact, equivalent to) the fact that W is orthogonal to W | det | π i jr log q unless j is divisible by r/t . Also note that P ψW has non-negative coefficients and the degree of P ψW is M W . Likewise, the degree of P ψW ( y ) as aLaurent polynomial in x (i.e., the order of pole of P ψW at 0) is − m W . Comparting degreesin (1) we obtain Proposition 2.2. (cid:3) Proof of Proposition 2.3.
By an argument based on properties of Rankin-Selberg integrals,the formal degree, with respect to a suitable choice of Haar measure, is related to f by theformula d π = tr q f − q − − q − t ([ILM17, Theorem 2.1]). Comparing it to the formal degree of the Steinberg representationSt with respect to the same measure we get d π d St = t · q t − ( r +12 ) + f · q r − q t − . On the other hand, d π d St (which is independent of the choice of Haar measure) is a (positive)integer (cf. [How74], [Rog81], [Hen84, Appendice 3]). The lemma follows. (cid:3) Remark . As explained to me by Guy Henniart, the lower bound in Proposition 2.3is not sharp. In fact, a precise formula for f ( π × π ∨ ), and more generally for f ( π × π )for an arbitrary pair of irreducible supercuspidal representations π i of GL n i ( F ), i = 1 , f ( π × π ∨ ) is insensitive to twisting π by a character. Suppose that π is minimal under twists, i.e., f ( π × χ ) ≥ f ( π ) for any character χ of F ∗ where f ( π ) isthe conductor of π . Then by [BH17, Lemma 3.5] and its proof we have f ( π × π ∨ ) ≥ r − t + r ( f ( π ) − r ) with equality if f ( π ) = r , i.e., if π has level 0 (that is, if π has anon-zero vector invariant under K (1), in which case t = r ). Thus, if f ( π ) > r + 1 thenwe get f ( π × π ∨ ) ≥ r ( r + 1) − t . On the other hand, we always have f ( π ) ≥ r [Bus87,(5.1)] and if f ( π ) = r + 1, i.e., if π is epipelagic then t = 1 and f ( π × π ∨ ) = ( r − r + 2).More generally, if f ( π ) is coprime to r , i.e., if π is a Carayol representation, then t = 1 and f ( π × π ∨ ) = ( r − f ( π ) + 1). For instance, this follows from [BHK98, (6.1.1),(6.1.2)] and UPPORT OF MATRIX COEFFICIENTS 5 [ibid., Theorem 6.5(i)] where in its notation we have n = e = d = r and c ( β ) = m ( r − k = m , e ( γ ) = d = r .To conclude, for any supercuspidal π we have(2) f ( π × π ∨ ) ≥ r ( r + 1) − t with equality if and only if π is a twist of either a representation of level 0 or an epipelagicrepresentation.Alternatively, one could infer (2) and the conditions for equality from the local Langlandscorrespondence for G (cf. [GR10]). Details will be appear in the upcoming thesis of Kilic.The results of [BHK98] also give that f ( π × π ∨ ) is even. (Details will be given elsewhere.)In the Galois side this follows from a result of Serre [Ser71].I am grateful to Guy Henniart for providing me this explanation and allowing me toinclude it here.For our purpose, the precise lower bound on f ( π × π ∨ ) is immaterial – it is sufficient tohave the inequality f ≥ f ≥ c ′ n for some fixed c ′ depending only r ). The pointis that we do not use either the local Langlands correspondence or the classification ofsupercuspidal representations (but of course, we do use the non-trivial analysis of [How74]which depends on [How77]).Nonetheless, it would be interesting to prove (2) (and perhaps the evenness of f ( π × π ∨ ))without reference to the classification or to the local Langlands correspondence.3. Proof of main result
In order to deduce Theorem 1.1 from Theorem 2.1 we make the argument of [LM15,Proposition 2.11] effective in the case of G = GL r .For any t = diag( t , . . . , t r ) ∈ A consider the compact open subgroup N ( t ) = N ∩ t K t − = { u ∈ N : val( u i,j ) ≥ val( t i ) − val( t j ) for all i < j } of N . Set t = diag( ̟, ̟ , . . . , ̟ r − ) ∈ A. Proposition 3.1.
Let f be a compactly supported continuous funciton on G . Assume that f is bi-invariant under K r − ( n ) for some n ≥ . Then Z N f ( u ) ψ N ( u ) du = Z N ( t n ) f ( u ) ψ N ( u ) du. In particular, R N f ( u ) ψ N ( u ) du = 0 if f vanishes on B ((2 r − − n ) . We first need some more notation. Write N = U ⋉ V where U = N r − is the group ofunitriangular matrices in GL r − embedded in GL r in the upper left corner and V is theunipotent radical of the parabolic subgroup of type ( r − , r − V = { u ∈ N : u i,j = 0 for all i < j < r } . EREZ LAPID
For any α = ( α , . . . , α r − ) ∈ F r − let x ( α ) be the r × r -matrix that is the identity except forthe ( r − α , . . . , α r − + 1 , x is not a group homomorphismbecause of the diagonal entry.For t ∈ A write U ( t ) = U ∩ N ( t ) and V ( t ) = V ∩ N ( t ) so that N ( t ) = U ( t ) ⋉ V ( t ). Thefollowing is elementary. Lemma 3.2.
For any n ≥ let L ( n ) be the lattice of F r − given by L ( n ) = { ( α , . . . , α r − ) : val( α j ) ≥ n (2 r − − j − ) , j = 1 , . . . , r − } . Then (1) x ( α ) , x ( α ) u ∈ K r − ( n ) for any α ∈ L ( n ) and u ∈ U ( t n ) . (2) For any v ∈ V we have vol( L ( n )) − Z L ( n ) ψ N ( v x ( α ) ) dα = ( ψ N ( v ) v ∈ V ( t n ) , otherwise.Proof of Proposition 3.1. We prove the statement by induction on r . The case r = 1is trivial. For the induction step, note that if f is bi-invariant under K r − ( n ) then thefunction h = R V f ( · v ) ψ N ( v ) dv on GL r − is bi- K r − ( n )-invariant (since GL r − normalizesthe character ψ N | V ). Therefore, by induction hypothesis we have Z N f ( u ) ψ N ( u ) du = Z U h ( u ) ψ N ( u ) du = Z U ( t n ) h ( u ) ψ N ( u ) du = Z V Z U ( t n ) f ( uv ) ψ N ( uv ) du dv. Now we use Lemma 3.2. By part 1, Since f is bi- K r − ( n )-invariant, for any α ∈ L ( n ) wecan write the above as Z V Z U ( t n ) f ( ux ( α ) vx ( α ) − ) ψ N ( uv ) du dv = Z V Z U ( t n ) f ( uv ) ψ N ( uv x ( α ) ) du dv. Averaging over α ∈ L ( n ) and using part 2, we may replace the integration over V byintegration over V ( t n ). This yields the induction step. (cid:3) Let Π ψ = ind GN ψ . For ϕ ∈ Π ψ and ϕ ∨ ∈ Π ψ − let( ϕ, ϕ ∨ ) N \ G = Z N \ G ϕ ( g ) ϕ ∨ ( g ) dg. Also set, A ◦ ( n ) = A ∩ B ( n ) = { diag( t , . . . , t r ) ∈ A : q − n ≤ | t i | ≤ q n , i = 1 , . . . , r } . Proposition 3.1, together with the argument of [LM15, Proposition 2.12], which was com-municated to us by Jacquet, yield the following.
UPPORT OF MATRIX COEFFICIENTS 7
Corollary 3.3.
There exists a constant c , depending only on r with the following property.Assume that ϕ ∈ Π K ( n ) ψ and ϕ ∨ ∈ Π K ( n ) ψ − are both supported in N A ◦ ( n ) K for some n ≥ .Then (Π ψ ( · ) ϕ, ϕ ∨ ) N \ G is supported in the ball B ( cn ) . Indeed, the function M ( g ) = (Π ψ ( · ) ϕ, ϕ ∨ ) N \ G is clearly bi- K ( n )-invariant. Let f be acompactly supported bi- K ( n )-invariant function on G . By Fubini’s theorem Z G M ( g ) f ( g ) dg = Z G Z N \ G ϕ ( xg ) ϕ ∨ ( x ) f ( g ) dg dx = Z N \ G Z G ϕ ( xg ) ϕ ∨ ( x ) f ( g ) dg dx = Z N \ G Z G ϕ ( g ) ϕ ∨ ( x ) f ( x − g ) dg dx = Z N \ G Z N \ G ϕ ∨ ( x ) ϕ ( y ) K f ( x, y ) dy dx where K f ( x, y ) = Z N f ( x − ny ) ψ N ( n ) dn. From Proposition 3.1 we infer that there exists c , depending only on r , such that if f vanishes on B ( cn ) then K f ( x, y ) = 0 for all x, y ∈ B ( n ). The corollary follows.Finally, we prove Theorem 1.1. Proof of Theorem 1.1.
Let π be a supercuspidal irreducible representation of G . Supposethat W ∈ W ψ ( π ) K ( n ) and W ∨ ∈ W ψ − ( π ∨ ) K ( n ) . By Theorem 2.1, both W and W ∨ are supported in N A ( cn ) K for suitable c . Upon modifying c , we may write W ( g ) = R Z W ( zg ) ω − π ( z ) dz (with vol( Z ∩ K ) = 1) where W ∈ Π K ( n ) ψ is supported in N A ◦ ( cn ) K .For instance we can take W = W X where X is the set X = { g ∈ G : 0 ≤ val(det g ) < r } . Similarly, write W ∨ ( g ) = R Z W ∨ ( zg ) ω π ( z ) dz where W ∨ ∈ Π K ( n ) ψ − is supported in N A ◦ ( cn ) K .Then up to a scalar( π ( g ) W, W ∨ ) = Z ZN \ G W ( xg ) W ∨ ( x ) dx = Z Z (Π ψ ( zg ) W , W ∨ ) N \ G dz. The result therefore follows from Corollary 3.3. (cid:3)
References [BH97] C. J. Bushnell and G. Henniart,
An upper bound on conductors for pairs , J. Number Theory (1997), no. 2, 183–196. MR 1462836[BH99] Colin J. Bushnell and Guy Henniart, Calculs de facteurs epsilon de paires pour GL n sur un corpslocal. I , Bull. London Math. Soc. (1999), no. 5, 534–542. MR 1703873[BH17] C. J. Bushnell and G. Henniart, Strong exponent bounds for the local Rankin-Selberg convolution ,Bull. Iranian Math. Soc. (2017), no. 4, 143–167. MR 3711826[BHK98] Colin J. Bushnell, Guy M. Henniart, and Philip C. Kutzko, Local Rankin-Selberg convolutions for GL n : explicit conductor formula , J. Amer. Math. Soc. (1998), no. 3, 703–730. MR 1606410[BK93] Colin J. Bushnell and Philip C. Kutzko, The admissible dual of
GL( N ) via compact open sub-groups , Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ,1993. MR 1204652 EREZ LAPID [Bus87] Colin J. Bushnell,
Hereditary orders, Gauss sums and supercuspidal representations of GL N , J.Reine Angew. Math. (1987), 184–210. MR 882297[CS80] W. Casselman and J. Shalika, The unramified principal series of p -adic groups. II. The Whittakerfunction , Compositio Math. (1980), no. 2, 207–231. MR 581582 (83i:22027)[FL17a] Tobias Finis and Erez Lapid, An approximation principle for congruence subgroups II: applicationto the limit multiplicity problem , Math. Z. to appear (2017), arXiv:1504.04795.[FL17b] ,
On the analytic properties of intertwining operators II: local degree bounds and limitmultiplicities , 2017, arXiv:1705.08191.[FLM12] Tobias Finis, Erez Lapid, and Werner M¨uller,
On the degrees of matrix coefficients of intertwiningoperators , Pacific J. Math. (2012), no. 2, 433–456. MR 3001800[GK75] I. M. Gel ′ fand and D. A. Kajdan, Representations of the group
GL( n, K ) where K is a local field ,Lie groups and their representations (Proc. Summer School, Bolyai J´anos Math. Soc., Budapest,1971), Halsted, New York, 1975, pp. 95–118. MR 0404534 (53 Arithmetic invariants of discrete Langlands parameters ,Duke Math. J. (2010), no. 3, 431–508. MR 2730575 (2012c:11252)[HC99] Harish-Chandra,
Admissible invariant distributions on reductive p -adic groups , University Lec-ture Series, vol. 16, American Mathematical Society, Providence, RI, 1999, Preface and notes byStephen DeBacker and Paul J. Sally, Jr. MR 1702257 (2001b:22015)[Hen84] Guy Henniart, La conjecture de Langlands locale pour
GL(3), M´em. Soc. Math. France (N.S.)(1984), no. 11-12, 186. MR 743063[HII08] Kaoru Hiraga, Atsushi Ichino, and Tamotsu Ikeda,
Formal degrees and adjoint γ -factors , J.Amer. Math. Soc. (2008), no. 1, 283–304. MR 2350057 (2010a:22023a)[How74] Roger Howe, The Fourier transform and germs of characters (case of Gl n over a p -adic field) ,Math. Ann. (1974), 305–322. MR 0342645 (49 Some qualitative results on the representation theory of Gl n over a p -adic field ,Pacific J. Math. (1977), no. 2, 479–538. MR 0492088 (58 On the formal degrees of square-integrable rep-resentations of odd special orthogonal and metaplectic groups , Duke Math. J. (2017), no. 7,1301–1348. MR 3649356[JPSS83] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika,
Rankin-Selberg convolutions , Amer. J.Math. (1983), no. 2, 367–464. MR 701565 (85g:11044)[Key92] C. David Keys,
A bound for the formal degree of a supercuspidal representation , Amer. J. Math. (1992), no. 6, 1257–1268. MR 1198303 (94a:22034)[LM15] Erez Lapid and Zhengyu Mao,
A conjecture on Whittaker-Fourier coefficients of cusp forms , J.Number Theory (2015), 448–505. MR 3267120[Rog81] Jonathan D. Rogawski,
An application of the building to orbital integrals , Compositio Math. (1980/81), no. 3, 417–423. MR 607380 (83g:22011)[Ser71] Jean-Pierre Serre, Conducteurs d’Artin des caract`eres r´eels , Invent. Math. (1971), 173–183.MR 0321908 (48 Representation theory and sheaves on the Bruhat-Tits build-ing , Inst. Hautes ´Etudes Sci. Publ. Math. (1997), no. 85, 97–191. MR 1471867 (98m:22023)[Vig90] Marie-France Vign´eras,
On formal dimensions for reductive p -adic groups-adic groups