AARCHIMEDEAN NEWFORM THEORY FOR GL n PETER HUMPHRIES
Abstract.
We introduce a new invariant, the conductor exponent, of a generic irreducibleCasselman–Wallach representation of GL n that quantifies the extent to which this representationmay be ramified. We also determine a distinguished vector, the newform, occurring with multi-plicity one in this representation, with the complexity of this vector measured in a natural wayby the conductor exponent. Finally, we show that the newform is a test vector for GL n × GL n and GL n × GL n − Rankin–Selberg integrals when the second representation is unramified. Thistheory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, andShalika; combined, this completes a global theory of newforms for automorphic representationsof GL n over number fields. By-products of the proofs include new proofs of Stade’s formulæ anda new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals. Introduction
Let M k ( q, χ ) denote the finite-dimensional vector space of holomorphic modular forms ofweight k , level q , and nebentypus χ , where χ is a primitive Dirichlet character of conductor q χ | q . The classical theory of newforms due to Atkin and Lehner [AL70] states that for each q (cid:48) | q with q (cid:48) (cid:54) = q and q (cid:48) ≡ q χ ) and for each (cid:96) | qq (cid:48) , the function ( ι (cid:96) f )( z ) := f ( (cid:96)z )defines an element of M k ( q, χ ) whenever f ∈ M k ( q (cid:48) , χ ). We call ι (cid:96) f an oldform. Moreover, theorthogonal complement with respect to the Petersson inner product of the vector subspace ofoldforms has an orthonormal basis consisting of newforms, which are eigenfunctions of the n -thHecke operator not just for each positive integer n for which ( n, q ) = 1 but for all n ∈ N .Casselman [Cas73], building on the seminal work of Jacquet and Langlands [JL70], gave anad`elic reformulation of the Atkin–Lehner theory of newforms in terms of distinguished vectorsin certain classes of representations of GL ( Q p ). Such a theory of newforms has been extendedto the setting of generic irreducible admissible smooth representations of GL n ( F ), where F isa nonarchimedean local field [JP-SS81]. This allows one to generalise Atkin–Lehner theory toautomorphic forms on GL n over any number field. Furthermore, this theory has blossomedto include a well-developed theory of oldforms and conductor exponents associated to suchrepresentations, coupled with the local theory of test vectors for certain families of GL n × GL m Rankin–Selberg integrals [Jac12, JP-SS81, Mat13, Ree91].While this nonarchimedean theory has been well understood since the seminal work of Jacquet,Piatetski-Shapiro, and Shalika [JP-SS81], there has been little development — even conjecturally— of the corresponding archimedean theory beyond the case n = 2. For n = 2, such anarchimedean theory for classical automorphic forms involves Maaß raising and lowering operators,which raise and lower the weight of the automorphic form; the ad`elic theory of newforms forrepresentations of GL ( R ) and GL ( C ), concerning distinguished vectors in finite-dimensionalrepresentations of the maximal compact subgroups O(2) and U(2), is due to Popa [Pop08].In this article, we put the archimedean setting on an equal footing with the nonarchimedeansetting by developing such a theory for GL n ( R ) and GL n ( C ). We summarise our main resultsconcerning generic irreducible Casselman–Wallach representations π of GL n as follows; precisestatements are given in more detail in Section 4. • Among the K n -types τ of π whose restriction to K n − contains the trivial representation,there exists a unique K n -type τ ◦ of lowest Howe degree, which occurs with multiplicity Mathematics Subject Classification.
Key words and phrases.
Casselman–Wallach representation, Godement–Jacquet zeta integral, Rankin–Selbergintegral, test vector, Whittaker function.Research partially supported by the European Research Council grant agreement 670239. a r X i v : . [ m a t h . N T ] A ug PETER HUMPHRIES one in π ; moreover, the subspace of K n − -invariant τ ◦ -isotypic vectors in π is one-dimensional (Theorem 4.7). We call the distinguished nonzero vector lying in thissubspace, unique up to scalar multiplication, the newform of π , and we define theconductor exponent c ( π ) of π to be the Howe degree of τ ◦ (Definition 4.8). • For each nonnegative integer m ≥ c ( π ), the dimension of the subspace of K n − -invariantvectors in π that are τ -isotypic for some K n -type τ of Howe degree m is given explicitlyas a certain binomial coefficient (Theorem 4.12). We call vectors of this form oldforms. • The epsilon factor ε ( s, π, ψ ) of π is equal to i − c ( π ) (Theorem 4.14); additionally, theconductor exponent c ( π ) of π is additive with respect to isobaric sums of representations(Theorem 4.15) and is inductive (Theorem 4.16). • When viewed in the Whittaker model and appropriately normalised, the newform isa test vector for the GL n × GL n − (Theorem 4.17) and GL n × GL n (Theorem 4.18)Rankin–Selberg integrals whenever the second representation is spherical. • The newform is a test vector for the Godement–Jacquet zeta integral (Theorem 4.22).Some aspects of this archimedean theory have long been expected. In particular, there hasbeen a flurry of work in recent years studying the problem of finding test vectors for localGL n × GL m Rankin–Selberg integrals involving ramified representations over archimedean fields;see, for example, [HIM12, HIM16, HIM20, IM20, Miz18, Pop08]. With the exception of [Pop08],previous work has invariably involved choosing test vectors that are associated in some way tothe minimal K -type (in the sense of Vogan [Vog81]), whereas we propose a theory of newformswithout such a direct relation to minimal K -types. Apart from the recent work [IM20], theseresults have also been confined to low rank, namely n ≤ m ≤ n ( F ) that in a certain sense quantifies the extent of ramification of such arepresentation. Somewhat surprisingly, there seems to have been no previous considerations inthe literature of such a theory of the conductor exponent, in spite of the fact that it has manyproperties that mirror those of the nonarchimedean conductor exponent introduced by Jacquet,Piatetski-Shapiro, and Shalika [JP-SS81].The structure of this article is the following. Section 2 contains a brief review of the theoryof induced representations of Whittaker and Langlands types, GL n × GL m Rankin–Selbergintegrals, Godement–Jacquet zeta integrals, and L -functions and epsilon factors. We survey thenonarchimedean theory of newforms, oldforms, conductor exponents, and test vectors for Rankin–Selberg integrals and Godement–Jacquet zeta integrals in Section 3. This serves to motivate theresults stated in Section 4, where we present an analogous theory in the archimedean setting;these results are all essentially new, although several of the results for n = 2 are implicit in thework of Popa [Pop08]. We discuss this theory further in Section 5. The remaining sections aredevoted to the proofs of the theorems stated in Section 4.2. Preliminaries
Groups and Haar Measures.
Local Fields and Absolute Values.
Let F be a local field, and denote by | · | F the absolutevalue on F : for nonarchimedean F with ring of integers O , maximal ideal p , and uniformiser (cid:36) ,so that (cid:36) O = p and O / p ∼ = F q for some finite field of order q , this absolute value is normalisedsuch that | (cid:36) | F = q − , while for archimedean F , this is normalised such that | x | F = (cid:26) max { x, − x } if F = R , xx if F = C .When the local field is clear from context, we write | · | in place of | · | F . We also let (cid:107) · (cid:107) := | · | / C denote the standard modulus on C . RCHIMEDEAN NEWFORM THEORY FOR GL n Haar Measures on F and F × . We let dx denote the Haar measure on F normalisedsuch that it is self-dual with respect to a fixed nontrivial additive character ψ = ψ F of F . For F = R , we choose ψ ( x ) := exp(2 πix ), so that dx is the Lebesgue measure; for F = C , we choose ψ ( x ) := exp(2 πi ( x + x )), so that dx is twice the Lebesgue measure; finally, we choose ψ to beunramified when F is nonarchimedean, in which case dx gives O volume 1. The multiplicativeHaar measure d × x for F × is defined to be ζ F (1) | x | − dx , where ζ F ( s ) := π − s Γ (cid:16) s (cid:17) if F = R ,2(2 π ) − s Γ( s ) if F = C ,11 − q − s if F is nonarchimedean.2.1.3. Subgroups of GL n ( F ) and the Iwasawa Decomposition. For each r -tuple of positive integers( n , . . . , n r ) ∈ N r for which n + · · · + n r = n , let P( F ) = P ( n ,...,n r ) ( F ) denote the associatedstandard upper parabolic subgroup of GL n ( F ) containing the standard Borel subgroup ofupper triangular matrices. This has the Levi decomposition P( F ) = N P ( F )M P ( F ), wherethe block-diagonal Levi subgroup M P ( F ) is isomorphic to GL n ( F ) × · · · × GL n r ( F ), whilethe unipotent radical N P ( F ) of P( F ) consists of upper triangular matrices with block-diagonalentries (1 n , . . . , n r ); here we have written 1 n to denote the n × n identity matrix. When P( F ) isthe standard Borel (and minimal parabolic) subgroup P (1 ,..., ( F ), we write N P ( F ) =: N n ( F ) ∼ = F n ( n − / , the subgroup of unipotent upper triangular matrices, and M P ( F ) =: A n ( F ) ∼ = ( F × ) n ,the subgroup of diagonal matrices.The maximal compact subgroup K n of GL n ( F ), unique up to conjugacy, is K n = O( n ) if F = R ,U( n ) if F = C ,GL n ( O ) if F is nonarchimedean.When the context is clear, we write K in place of K n . Given a standard parabolic subgroupP( F ), we have the Iwasawa decomposition GL n ( F ) = P( F ) K n = N P ( F )M P ( F ) K n . Note thatthe Iwasawa decomposition is not unique since M P ( F ) intersects K n nontrivially.2.1.4. Haar Measures on GL n ( F ) and Its Subgroups. We normalise the Haar measure dg onGL n ( F ) (cid:51) g via the Iwasawa decomposition g = uak for the standard Borel subgroup, so that dg = δ − n ( a ) du d × a dk . Here du = (cid:81) n − j =1 (cid:81) n(cid:96) = j +1 du j,(cid:96) for u ∈ N n ( F ) with upper triangularentries u j,(cid:96) ∈ F , d × a = (cid:81) nj =1 d × a j for a ∈ A n ( F ) with diagonal entries a j ∈ F × , δ n ( a ) = (cid:81) nj =1 | a j | n − j +1 denotes the modulus character of the Borel subgroup, and dk is the Haarmeasure on the compact group K n (cid:51) k normalised to give K n volume 1 (so that when F = R and n = 1, in which case K = O(1) = {± } ∼ = Z / Z , this is just half the counting measure).More generally, given a standard parabolic subgroup P( F ) = N P ( F )M P ( F ) of GL n ( F ), theHaar measure dg on GL n ( F ) (cid:51) g is given by dg = δ − ( m ) du d × m dk with respect to the Iwasawadecomposition g = umk , where for m = blockdiag( m , . . . , m r ), the modulus character is δ P ( m ) = r (cid:89) j =1 | det m j | n − n + ··· + n j − ) − n j and d × m = (cid:81) rj =1 dm j with dm j the Haar measure on GL n j ( F ) (cid:51) m j normalised via the Iwasawadecomposition for the standard Borel subgroup of GL n j ( F ).2.2. Representations.
Isobaric Sums.
Given representations ( π , V π ) , . . . , ( π r , V π r ) of GL n ( F ) , . . . , GL n r ( F ),where F is a local field and n + · · · + n r = n , we form the representation π (cid:2) · · · (cid:2) π r of M P ( F ),where (cid:2) denotes the outer tensor product and M P ( F ) denotes the block-diagonal Levi subgroupof the standard (upper) parabolic subgroup P( F ) = P ( n ,...,n r ) ( F ) of GL n ( F ). We then extend PETER HUMPHRIES this representation trivially to a representation of P( F ). By normalised parabolic induction, weobtain an induced representation ( π, V π ) of GL n ( F ), π := Ind GL n ( F )P( F ) r (cid:2) j =1 π j , where V π denotes the space of smooth functions f : GL n ( F ) → V π ⊗ · · · ⊗ V π r that satisfy f ( umg ) = δ / ( m ) π ( m ) ⊗ · · · ⊗ π r ( m r ) f ( g ) , for any u ∈ N P ( F ), m = blockdiag( m , . . . , m r ) ∈ M P ( F ), and g ∈ GL n ( F ), and the action of π on V π is by right translation, namely ( π ( h ) · f )( g ) := f ( gh ). We call π the isobaric sum of π , . . . , π r , which we denote by π = r (cid:1) j =1 π j . Induced Representations of Whittaker and Langlands Types.
A representation π of GL n ( F )is said to be an induced representation of Whittaker type if it is the isobaric sum of π , . . . , π r and each π j is irreducible and essentially square-integrable. Such a representation is admissibleand smooth; moreover, if F is archimedean, then it is a Fr´echet representation of moderategrowth and of finite length. The contragredient of an induced representation of Whittaker type π = π (cid:1) · · · (cid:1) π r is (cid:101) π = (cid:102) π (cid:1) · · · (cid:1) (cid:101) π r , which is again an induced representation of Whittakertype. If each π j is additionally of the form σ j ⊗ | det | t j , where σ j is irreducible, unitary, andsquare-integrable, and (cid:60) ( t ) ≥ · · · ≥ (cid:60) ( t r ), then π is said to be an induced representation ofLanglands type.2.2.3. Whittaker Models.
A Whittaker functional Λ : V π → C of an admissible smooth represen-tation ( π, V π ) of GL n ( F ) is a continuous linear functional that satisfiesΛ( π ( u ) · v ) = ψ n ( u )Λ( v )for all v ∈ V π and u ∈ N n ( F ); here ψ n ( u ) := ψ n − (cid:88) j =1 u j,j +1 . If π is additionally irreducible, then the space of Whittaker functionals of π is at most one-dimensional. If this space is indeed one-dimensional, so that there exists a unique such functionalup to scalar multiplication, then π is said to be generic. Every induced representation ofLanglands type ( π, V π ) admits a nontrivial Whittaker functional Λ; moverover, π is isomorphicto its unique Whittaker model W ( π, ψ ), which is the image of V π under the map v (cid:55)→ Λ( π ( · ) · v ),so that W ( π, ψ ) consists of Whittaker functions on GL n ( F ) of the form W ( g ) := Λ( π ( g ) · v ).An induced representation of Whittaker type π also has a one-dimensional space of Whittakerfunctionals, but the map v (cid:55)→ Λ( π ( · ) · v ) need not be injective, so that the Whittaker modelmay only be a model of a quotient of π .2.2.4. Irreducible Representations.
For nonarchimedean F , every irreducible admissible smoothrepresentation π of GL n ( F ) is isomorphic to the unique irreducible quotient of some inducedrepresentation of Langlands type. If π is also generic, then it is isomorphic to some (necessarilyirreducible) induced representation of Langlands type [CS98].For archimedean F , we recall that a Casselman–Wallach representation of GL n ( F ) is an ad-missible smooth Fr´echet representation of moderate growth and of finite length [Wal92, BK14];in particular, induced representations of Whittaker type are Casselman–Wallach representations.Every irreducible Casselman–Wallach representation π of GL n ( F ) is isomorphic to the uniqueirreducible quotient of some induced representation of Langlands type. Again, if π is addition-ally generic, then it is isomorphic to some (necessarily irreducible) induced representation ofLanglands type. RCHIMEDEAN NEWFORM THEORY FOR GL n Spherical Representations.
An induced representation of Whittaker type π is said to bespherical if it has a K -fixed vector. Such a spherical representation π must then be a principalseries representation of the form | · | t (cid:1) · · · (cid:1) | · | t n ; furthermore, the subspace of K -fixed vectorsmust be one-dimensional. This K -fixed vector, which is unique up to scalar multiplication, iscalled the spherical vector of π . For a spherical representation of Langlands type π , the sphericalWhittaker function W ◦ in the Whittaker model W ( π, ψ ) is given by the Jacquet integral W ◦ ( g ) := (cid:90) N n ( F ) f ◦ ( w n ug ) ψ n ( u ) du, where w n := antidiag(1 , . . . ,
1) is the long Weyl element and f ◦ is the canonically normalisedspherical vector in the induced model: the unique smooth function f ◦ : GL n ( F ) → C satisfying f ◦ (1 n ) = n − (cid:89) j =1 n (cid:89) (cid:96) = j +1 ζ F (1 + t j − t (cid:96) ) , f ◦ ( uag ) = f ◦ ( g ) δ / n ( a ) n (cid:89) j =1 | a j | t j , f ◦ ( gk ) = f ◦ ( g )for all u ∈ N n ( F ), a = diag( a , . . . , a n ) ∈ A n ( F ), g ∈ GL n ( F ), and k ∈ K . The Jacquet integralof f ◦ converges absolutely when (cid:60) ( t ) > · · · > (cid:60) ( t n ) and extends holomorphically to all of C n (cid:51) ( t , . . . , t n ) (cf. Section 9.1). For nonarchimedean F , the normalisation of W ◦ is such that W ◦ (1 n ) = 1.2.3. Integral Representations of L -Functions. Rankin–Selberg Integrals.
We recall the definition of Rankin–Selberg integrals over a localfield F ; see [Cog04] for further details. Given induced representations of Whittaker type π of GL n ( F ) and π (cid:48) of GL m ( F ) with m ≤ n , we take Whittaker functions W ∈ W ( π, ψ ) and W (cid:48) ∈ W (cid:0) π (cid:48) , ψ (cid:1) and form the local GL n × GL m Rankin–Selberg integral defined byΨ( s, W, W (cid:48) ) := (cid:90) N m ( F ) \ GL m ( F ) W (cid:18) g
00 1 n − m (cid:19) W (cid:48) ( g ) | det g | s − n − m dg for m < n ,(2.1) Ψ( s, W, W (cid:48) , Φ) := (cid:90) N n ( F ) \ GL n ( F ) W ( g ) W (cid:48) ( g )Φ( e n g ) | det g | s dg for m = n ,(2.2)where Φ ∈ S (Mat × n ( F )) is a Schwartz–Bruhat function and e n := (0 , . . . , , ∈ Mat × n ( F ) = F n . These integrals converge absolutely for (cid:60) ( s ) sufficiently large and extend meromorphicallyto the entire complex plane via the local functional equation.2.3.2. Godement–Jacquet Zeta Integrals.
Following [GJ72, Jac79], we define Godement–Jacquetzeta integrals over a local field F . Given an induced representation of Whittaker type ( π, V π ) ofGL n ( F ), we take v ∈ V π , (cid:101) v ∈ V (cid:101) π , with associated matrix coefficient β ( g ) := (cid:104) π ( g ) · v , (cid:101) v (cid:105) of π ,and form the local Godement–Jacquet zeta integral defined by(2.3) Z ( s, β, Φ) := (cid:90) GL n ( F ) β ( g )Φ( g ) | det g | s + n − dg, where Φ ∈ S (Mat n × n ( F )) is a Schwartz–Bruhat function. This integral converges absolutelyfor (cid:60) ( s ) sufficiently large and extends meromorphically to the entire complex plane via the localfunctional equation.2.4. L -Functions and Epsilon Factors. Rankin–Selberg L -Functions and Standard L -Functions. For nonarchimedean F , theRankin–Selberg L -function L ( s, π × π (cid:48) ) is the generator of the C [ q s , q − s ]-fractional ideal of C ( q − s ) generated by the family of Rankin–Selberg integrals Ψ( s, W, W (cid:48) ) (or Ψ( s, W, W (cid:48) , Φ) if m = n ) with W ∈ W ( π, ψ ) and W (cid:48) ∈ W (cid:0) π (cid:48) , ψ (cid:1) (and Φ ∈ S (Mat × n ( F )) if m = n ) with L ( s, π × π (cid:48) ) normalised to be of the form P ( q − s ) − for some P ( q − s ) ∈ C [ q − s ] whose constantterm is 1. For archimedean F , the Rankin–Selberg L -function L ( s, π × π (cid:48) ) is defined via thelocal Langlands correspondence as explicated in [Kna94]. PETER HUMPHRIES
Similarly, for nonarchimedean F , the standard L -function L ( s, π ) is the generator of the C [ q s , q − s ]-fractional ideal of C ( q − s ) generated by the family of Godement–Jacquet zeta integrals Z ( s, β, Φ) with v ∈ V π , (cid:101) v ∈ V (cid:101) π , and Φ ∈ S (Mat n × n ( F )), with L ( s, π ) normalised to be of theform P ( q − s ) − for some P ( q − s ) ∈ C [ q − s ] whose constant term is 1. For archimedean F , thestandard L -function L ( s, π ) is defined via the local Langlands correspondence as explicated in[Kna94].In both settings, upon decomposing π and π (cid:48) as isobaric sums(2.4) π = r (cid:1) j =1 π j , π (cid:48) = r (cid:48) (cid:1) j (cid:48) =1 π (cid:48) j (cid:48) , we have the identities(2.5) L ( s, π ) = r (cid:89) j =1 L ( s, π j ) , L ( s, π × π (cid:48) ) = r (cid:89) j =1 r (cid:48) (cid:89) j (cid:48) =1 L ( s, π j × π (cid:48) j (cid:48) ) . Moreover, Rankin–Selberg L -functions involving twists by a character are related to standard L -functions via the identity(2.6) L ( s, π × | · | t ) = L ( s + t, π ) . L -Functions for Representations of GL n ( C ) . Essentially square-integrable representationsof GL n ( C ) exist only for n = 1, in which case the representation must be a character of the form π ( z ) = χ κ ( z ) | z | t C for some κ ∈ Z and t ∈ C , where z ∈ GL ( C ) = C × and χ is the canonicalcharacter χ ( z ) := e i arg( z ) = z | z | / C . The L -function of π is(2.7) L ( s, π ) = ζ C (cid:18) s + t + (cid:107) κ (cid:107) (cid:19) , where(2.8) ζ C ( s ) := 2(2 π ) − s Γ( s ) = (cid:90) C × | x | s C exp ( − πxx ) d × x. This integral representation of ζ C ( s ) converges absolutely for (cid:60) ( s ) > π is (cid:101) π = χ − κ | · | − t C , so that(2.9) L ( s, (cid:101) π ) = ζ C (cid:18) s − t + (cid:107) κ (cid:107) (cid:19) . L -Functions for Representations of GL n ( R ) . Essentially square-integrable representationsof GL n ( R ) exist only for n ∈ { , } . An essentially square-integrable representation of GL ( R )must be a character of the form π ( x ) = χ κ ( x ) | x | t R for some κ ∈ { , } and t ∈ C , where x ∈ GL ( R ) = R × and χ is the canonical character χ ( x ) := sgn( x ) = x | x | R . The L -function of π is(2.10) L ( s, π ) = ζ R ( s + t + κ ) , where(2.11) ζ R ( s ) := π − s Γ (cid:16) s (cid:17) = (cid:90) R × | x | s R exp (cid:0) − πx (cid:1) d × x. RCHIMEDEAN NEWFORM THEORY FOR GL n This integral representation of ζ R ( s ) converges absolutely for (cid:60) ( s ) > (cid:101) π = χ κ | · | − t R , so that(2.12) L ( s, (cid:101) π ) = ζ R ( s − t + κ ) . For n = 2, we note that | · | t R (cid:1) χ | · | t R ∼ = Ind GL ( R )GL ( C ) | · | t C , where GL ( C ) is viewed as a subgroup of GL ( R ) via the identification a + ib (cid:55)→ (cid:0) a b − b a (cid:1) . For κ (cid:54) = 0, the essentially discrete series representation of weight (cid:107) κ (cid:107) + 1, D (cid:107) κ (cid:107) +1 ⊗ | det | t R := Ind GL ( R )GL ( C ) χ κ | · | t C ∼ = Ind GL ( R )GL ( C ) χ − κ | · | t C , is essentially square-integrable, and every essentially square-integrable representation of GL ( R )is of the form π = D κ ⊗ | det | t R for some integer κ ≥ t ∈ C . The L -function of π is(2.13) L ( s, π ) = ζ C (cid:18) s + t + κ − (cid:19) = ζ R (cid:18) s + t + κ − (cid:19) ζ R (cid:18) s + t + κ + 12 (cid:19) . The contragredient of π is (cid:101) π = D κ ⊗ | det | − t R , so that(2.14) L ( s, (cid:101) π ) = ζ C (cid:18) s − t + κ − (cid:19) = ζ R (cid:18) s − t + κ − (cid:19) ζ R (cid:18) s − t + κ + 12 (cid:19) . Epsilon Factors.
To any induced representations of Whittaker type π of GL n ( F ) and π (cid:48) of GL m ( F ) and any nontrivial additive character ψ of F , one can associate the epsilon factor ε ( s, π, ψ ) of π , ε ( s, π (cid:48) , ψ ) of π (cid:48) , and ε ( s, π × π (cid:48) , ψ ) of π × π (cid:48) , which arise via the local functionalequations for the Godement–Jacquet zeta integral and GL n × GL m Rankin–Selberg integralrespectively. In particular, the local functional equation for the Godement–Jacquet zeta integralis(2.15) Z (1 − s, (cid:101) β, (cid:98) Φ) L (1 − s, (cid:101) π ) = ε ( s, π, ψ ) Z ( s, β, Φ) L ( s, π ) , where (cid:101) β ( g ) := β (cid:0) t g − (cid:1) with t g denoting the transpose of g , while the Fourier transform (cid:98) Φ is (cid:98) Φ( y ) := (cid:90) Mat n × n ( F ) Φ( x ) ψ (Tr( x t y )) dy. Note that this normalisation of the Fourier transform differs from that in [GJ72], and that theepsilon factor is dependent on this normalisation. For the local functional equation for theGL n × GL m Rankin–Selberg integral, see, for example, [Jac09, Theorem 2.1].Upon decomposing π and π (cid:48) as isobaric sums as in (2.4), we have the identities(2.16) ε ( s, π, ψ ) = r (cid:89) j =1 ε ( s, π j , ψ ) , ε ( s, π × π (cid:48) , ψ ) = r (cid:89) j =1 r (cid:48) (cid:89) j (cid:48) =1 ε ( s, π j × π (cid:48) j (cid:48) , ψ )by [JP-SS83, Theorem (3.1), Proposition (8.4), Proposition (9.4)] for nonarchimedean F and viathe local Langlands correspondence for archimedean F . When π (cid:48) is the trivial representationof GL ( F ), we have the equality ε ( s, π, ψ ) = ε ( s, π × π (cid:48) , ψ ) between the epsilon factor of π defined via the Godement–Jacquet zeta integral and the epsilon factor of π × π (cid:48) defined via theGL n × GL Rankin–Selberg integral.For nonarchimedean F , the epsilon factors ε ( s, π, ψ ) and ε ( s, π × π (cid:48) , ψ ) are units in C [ q s , q − s ]of the form(2.17) ε ( s, π, ψ ) = ε (cid:18) , π, ψ (cid:19) q − c ( π ) ( s − ) , ε ( s, π × π (cid:48) , ψ ) = ε (cid:18) , π × π (cid:48) , ψ (cid:19) q − c ( π × π (cid:48) ) ( s − )for some nonnegative integers c ( π ) and c ( π × π (cid:48) ). PETER HUMPHRIES
For archimedean F , we have that(2.18) ε ( s, π, ψ ) = i − κ if F = R and π = χ κ | · | t R for κ ∈ { , } , i − κ if F = R and π = D κ ⊗ | det | t R for κ ≥ i −(cid:107) κ (cid:107) if F = C and π = χ κ | · | t C for κ ∈ Z ,which, via (2.15), may be used to determine ε ( s, π, ψ ) and ε ( s, π × π (cid:48) , ψ ) for any induced repre-sentations of Whittaker type π and π (cid:48) , not just for essentially square-integrable representations.3. Nonarchimedean Newform Theory
We now detail the theory of the conductor exponent, newforms, and oldforms associated togeneric irreducible admissible smooth representations of GL n ( F ) with F nonarchimedean (ormore generally associated to induced representations of Whittaker or Langlands type; cf. Section2.2.4), as well as the relation between newforms and test vectors for GL n × GL m Rankin–Selbergintegrals and test vectors for Godement–Jacquet zeta integrals. The results herein are all well-known; we recall them as motivation for Section 4, in which we discuss analogous yet new resultsfor archimedean F .3.1. The Conductor Exponent and the Newform.
Let F be a nonarchimedean local fieldand let K = GL n ( O ) be the maximal compact subgroup of GL n ( F ), unique up to conjugation.For a nonnegative integer m , we define the following finite index subgroup of K : K ( p m ) := { k ∈ K : k n, , . . . , k n,n − , k n,n − ∈ p m } . (This is not to be confused with K := GL ( O ) = O × , the maximal compact subgroup ofGL ( F ) = F × .) Given an induced representation of Langlands type ( π, V π ) of GL n ( F ), wedefine the vector subspace V K ( p m ) π of V π consisting of K ( p m )-fixed vectors: V K ( p m ) π := { v ∈ V π : π ( k ) · v = v for all k ∈ K ( p m ) } . The following theorem is due to Casselman [Cas73, Theorem 1] for n = 2 and Jacquet,Piatetski-Shapiro, and Shalika for arbitrary n (though cf. Remark 3.12). Theorem 3.1 (Jacquet–Piatetski-Shapiro–Shalika [JP-SS81, Th´eor`eme (5)]) . Let ( π, V π ) be aninduced representation of Langlands type of GL n ( F ) . There exists a minimal nonnegative integer m for which V K ( p m ) π is nontrivial. For this minimal value of m , V K ( p m ) π is one-dimensional. Definition 3.2.
We define the conductor exponent of π to be this minimal nonnegative integer m and denote it by c ( π ); we then call the ideal p c ( π ) the conductor of π . The newform of π isdefined to be the nonzero vector v ◦ ∈ V K ( p c ( π ) ) π , unique up to scalar multiplication.The uniqueness of the newform may be thought of as being a multiplicity-one theorem fornewforms. The reason for naming this distinguished vector a newform is due to its relation to theclassical theory of modular forms: as shown by Casselman [Cas73, Section 3], an automorphicform on GL ( A Q ) whose associated Whittaker function is a pure tensor composed of newformsin the Whittaker model is the ad`elic lift of a classical newform in the sense of Atkin and Lehner[AL70]. Remark . There is no consensus on the name of this distinguished vector: Casselman [Cas73]leaves it unnamed, Jacquet, Piatetski-Shapiro, and Shalika [JP-SS81] name it the essentialvector, whereas Reeder [Ree91] calls it the new vector, and Schmidt [Sch02], regarding π asbeing the local component of an automorphic representation, refers to it as the local newform.When viewed in the Whittaker model, this vector is referred to by Popa [Pop08] as the Whittakernewform, whereas Matringe [Mat13] calls it the essential Whittaker function. Similarly, someauthors instead call c ( π ) the conductor of π , while others yet refer to q c ( π ) as the conductor of π . Perhaps a more apt name for q c ( π ) is the absolute conductor of π , being the absolute normof the ideal p c ( π ) . RCHIMEDEAN NEWFORM THEORY FOR GL n Remark . Under the local Langlands correspondence, which gives a bijection between irre-ducible admissible smooth representations π of GL n ( F ) and n -dimensional Frobenius semisimpleWeil–Deligne representations ρ of F , the conductor exponent c ( π ) is equal to the Artin exponent a ( ρ ).If c ( π ) = 0, so that K ( p c ( π ) ) = K , then π must be a spherical representation and we say that π is unramified. If c ( π ) >
0, then π is said to be ramified. In this sense, the conductor exponentis a measure of the extent of ramification of π : it quantifies how ramified π may be.3.2. Oldforms.
While Jacquet, Piatetski-Shapiro, and Shalika merely show that V K ( p c ( π ) ) π isone-dimensional, one can also calculate the dimension of V K ( p m ) π for all m ≥ c ( π ) in terms of abinomial coefficient; for n = 2, this is due to Casselman [Cas73, Corollary to the Proof], whileReeder has proven this result for arbitrary n . Theorem 3.5 (Reeder [Ree91, Theorem 1]) . Let ( π, V π ) be an induced representation of Lang-lands type of GL n ( F ) with n ≥ . We have that dim V K ( p m ) π = (cid:18) m − c ( π ) + n − n − (cid:19) if m ≥ c ( π ) , otherwise. Casselman and Reeder also give a basis for each of these spaces in terms of the action ofcertain Hecke operators on the newform. For m > c ( π ), we call V K ( p m ) π the space of oldformsof exponent m . Once again, the reason for naming these distinguished vectors oldforms is dueto their relation to the classical theory of modular forms: an automorphic form on GL ( A Q )whose associated Whittaker function is a pure tensor composed of Whittaker newforms at allbut finitely many places and of Whittaker oldforms at the remaining places corresponds to anoldform in the sense of Atkin and Lehner [AL70].3.3. Additivity and Inductivity of the Conductor Exponent.
Associated to any inducedrepresentation of Langlands type π of GL n ( F ) is an integer c ( π ) defined via the epsilon factoras in (2.17). Theorem 3.6 (Jacquet–Piateski-Shapiro–Shalika [JP-SS83, Section 5]) . Let π be an inducedrepresentation of Langlands type of GL n ( F ) . The integer c ( π ) appearing in the epsilon factor ε ( s, π, ψ ) as in (2.17) is equal to the conductor exponent of π . From the multiplicativity of epsilon factors (2.16), we have the following.
Theorem 3.7 (Jacquet–Piateski-Shapiro–Shalika [JP-SS83, Theorem (3.1)]) . For an inducedrepresentation of Langlands type π = π (cid:1) · · · (cid:1) π r of GL n ( F ) , we have that c ( π ) = r (cid:88) j =1 c ( π j ) . Thus the conductor exponent c ( π ) is additive with respect to isobaric sums; equivalently, theconductor p c ( π ) is multiplicative. Remark . In the classical setting of automorphic forms on the upper half-plane, Theorem3.7 manifests itself via the conductor of an Eisenstein newform: an Eisenstein newform, in thesense of [You19], is associated to a pair of primitive Dirichlet characters, and the conductor ofsuch a newform is the product of the conductors of the two Dirichlet characters.Finally, the epsilon factor is inductive in degree zero: if π and π (cid:48) are induced representationsof Langlands type of GL n ( E ), where E is a finite cyclic extension of F of degree m ≥ AI E/F π and AI E/F π (cid:48) denote the induced representations of Langlands type of GL mn ( F )obtained by induction [HH95], then ε (cid:0) s, AI E/F π, ψ (cid:1) ε (cid:0) s, AI E/F π (cid:48) , ψ (cid:1) = ε (cid:0) s, π, ψ ◦ Tr E/F (cid:1) ε (cid:0) s, π (cid:48) , ψ ◦ Tr E/F (cid:1) . Taking π (cid:48) to be the isobaric sum of n copies of the trivial representation, we deduce the following. Theorem 3.9.
For an induced representation of Langlands type π of GL n ( E ) , where E is afinite cyclic extension of F of degree m ≥ , we have that c (cid:0) AI E/F π (cid:1) = f E/F c ( π ) + d E/F n, where f E/F denotes the residual degree of
E/F and d E/F denotes the valuation of the discrimi-nant of
E/F .Remark . This result also has a classical manifestation. Let ψ be a Hecke Gr¨oßencharakterof a quadratic extension E of Q with conductor q ⊂ O E . By automorphic induction, one canassociate to ψ a classical newform f ψ : H → C of conductor N( q ) D E/ Q , where N( q ) := O E / q is the absolute norm of the integral ideal q and D E/ Q is the absolute discriminant of E/ Q .3.4. Test Vectors for Rankin–Selberg Integrals.
Next, we discuss the relation betweennewforms and test vectors for Rankin–Selberg integrals. We first recall the test vector problemfor GL n × GL m Rankin–Selberg integrals.
Test Vector Problem.
Given induced representations of Langlands type π of GL n ( F ) and π (cid:48) of GL m ( F ) with n ≥ m , determine the existence of right K n - and K m -finite Whittaker functions W ∈ W ( π, ψ ) and W (cid:48) ∈ W ( π (cid:48) , ψ ) , and additionally a bi- K n -finite Schwartz–Bruhat function Φ ∈ S (Mat × n ( F )) = S ( F n ) should m be equal to n , such that L ( s, π × π (cid:48) ) = (cid:40) Ψ( s, W, W (cid:48) ) if n > m , Ψ( s, W, W (cid:48) , Φ) if n = m . In full generality, this problem remains unresolved. For m = n − π (cid:48) a spherical representationof Langlands type, and W (cid:48) = W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) the spherical vector normalised as in Section 2.2.5,this has been solved by Jacquet, Piatetski-Shapiro, and Shalika. Theorem 3.11 (Jacquet–Piatetski-Shapiro–Shalika [JP-SS81, Th´eor`eme (4)], Jacquet [Jac12],Matringe [Mat13, Corollary 3.3]) . For n ≥ , let π be an induced representation of Langlandstype of GL n ( F ) . There exists a Whittaker function W ∈ W ( π, ψ ) such that for any sphericalrepresentation of Langlands type π (cid:48) of GL n − ( F ) with spherical Whittaker function W (cid:48)◦ ∈W ( π (cid:48) , ψ ) , Ψ( s, W, W (cid:48)◦ ) = L ( s, π × π (cid:48) ) for (cid:60) ( s ) sufficiently large.Moreover, there exists a unique such function W ◦ ∈ W ( π, ψ ) that additionally satisfies W ◦ (cid:18) g (cid:18) k
00 1 (cid:19)(cid:19) = W ◦ ( g ) for all k ∈ K n − . Up to multiplication by a scalar, this function is the newform v ◦ ∈ V K ( p c ( π ) ) π viewed in the Whittaker model. That is, W ◦ ∈ W ( π, ψ ) is a right K n − -invariant test vector for the GL n × GL n − Rankin–Selberg integral for all spherical representations π (cid:48) of GL n − ( F ). We call W ◦ the Whittakernewform of π . Remark . The proof of [JP-SS81, Th´eor`eme (4)] is in fact incomplete, as was observedby Matringe; correct proofs have subsequently independently been given by Jacquet [Jac12,Theorem 1] and Matringe [Mat13, Corollary 3.3].
Remark . The normalisation of the Whittaker newform is such that W ◦ (1 n ) = 1.The Whittaker newform is a test vector for more than just the GL n × GL n − Rankin–Selbergintegral for all spherical representations π (cid:48) of GL n − ( F ). RCHIMEDEAN NEWFORM THEORY FOR GL n Theorem 3.14 (Kim [Kim10, Theorem 2.2.1], Matringe [Mat13, Corollary 3.3]) . Let π be aninduced representation of Langlands type of GL n ( F ) with n ≥ . For m ∈ { , . . . , n − } and forevery spherical representation of Langlands type π (cid:48) of GL m ( F ) with spherical Whittaker function W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) , the Whittaker newform W ◦ ∈ W ( π, ψ ) of π satisfies Ψ( s, W ◦ , W (cid:48)◦ ) = L ( s, π × π (cid:48) ) for (cid:60) ( s ) sufficiently large. Miyauchi also gave a proof when m = 1 [Miu14, Theorem 5.1] by a different method thatgeneralises easily to prove the result for m ∈ { , . . . , n − } .A similar theory also holds for the case m = n . Theorem 3.15 (Kim [Kim10, Theorem 2.1.1]) . Let π be an induced representation of Langlandstype of GL n ( F ) . Then for every spherical representation of Langlands type π (cid:48) of GL n ( F ) withspherical Whittaker function W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) , the Whittaker newform W ◦ ∈ W ( π, ψ ) of π satisfies Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) = L ( s, π × π (cid:48) ) for (cid:60) ( s ) sufficiently large, where Φ ◦ ∈ S (Mat × n ( F )) is given by Φ ◦ ( x , . . . , x n ) := ω − π ( x n )vol( K ( p c ( π ) )) if x , . . . , x n − ∈ p c ( π ) and x n ∈ O × , otherwise,if c ( π ) > , where ω π denotes the central character of π and K ( p m ) := { k ∈ K : k n, , . . . , k n,n − ∈ p m } , while for c ( π ) = 0 , Φ ◦ ( x , . . . , x n ) := (cid:26) if x , . . . , x n ∈ O , otherwise. Again, this can also be proven via the method of Miyauchi [Miu14].
Remark . Little is known about test vectors for Rankin–Selberg integrals when π (cid:48) is ramified.Kim [Kim10, Proposition 2.2.2] has shown that Whittaker newforms are not test vectors when π (cid:48) is ramified: if π (cid:48) is a ramified representation of GL m ( F ) with m < n and W ◦ , W (cid:48)◦ are newformsof π and π (cid:48) respectively, then Ψ ( s, W ◦ , W (cid:48)◦ ) = 0 for all s ∈ C . For n = m = 2, Kim hasdetermined test vectors for certain pairs of representations π, π (cid:48) [Kim10]. Recently, Kurinczukand Matringe have explicitly determined test vectors for n = m arbitrary, π supercuspidal,and π (cid:48) = (cid:101) π ⊗ χ for some unramified character χ of F × [KM19]. Booker, Krishnamurthy, andLee resolve a weakened version of the test vector problem in [BKL20]: they construct vectors W ∈ W ( π, ψ ) and W (cid:48) ∈ W ( π (cid:48) , ψ ) for which Ψ( s, W, W (cid:48) ) is a multiple of L ( s, π × π (cid:48) ) by anonzero polynomial in q − s . Finally, Jacquet, Piatetski-Shapiro, and Shalika have shown that foreach pair of representations π, π (cid:48) of GL n ( F ) , GL m ( F ) with m < n , there exist finite collections { W j } ⊂ W ( π, ψ ) and { W (cid:48) j } ⊂ W ( π (cid:48) , ψ ) of K n - and K m -finite Whittaker functions for which (cid:80) j Ψ( s, W j , W (cid:48) j ) = L ( s, π × π (cid:48) ) [JP-SS83, Theorem (2.7)]; a similar result (additionally involvingSchwartz–Bruhat functions { Φ j } ⊂ S (Mat × n ( F ))) also holds for m = n .3.5. Test Vectors for Godement–Jacquet Zeta Integrals.
Finally, we mention the relationbetween newforms and test vectors for Godement–Jacquet zeta integrals. The test vector problemfor the Godement–Jacquet zeta integral is the following.
Test Vector Problem.
Given an induced representation of Langlands type ( π, V π ) of GL n ( F ) ,determine the existence of K -finite vectors v ∈ V π , (cid:101) v ∈ V (cid:101) π , and a bi- K -finite Schwartz–Bruhatfunction Φ ∈ S (Mat n × n ( F )) such that Z ( s, β, Φ) = L ( s, π ) . This has been solved for spherical representations by Godement and Jacquet and for non-spherical representations by the author.
Theorem 3.17 (Godement–Jacquet [GJ72, Lemma 6.10], Humphries [Hum20, Theorem 1.2]) . Let ( π, V π ) be an induced representation of Langlands type of GL n ( F ) . Let β ( g ) denote the matrixcoefficient (cid:104) π ( g ) · v ◦ , (cid:101) v ◦ (cid:105) , where v ◦ ∈ V π denotes the newform and (cid:101) v ◦ ∈ V (cid:101) π is the correspondingnewform normalised such that β (1 n ) = 1 . Then Z ( s, β, Φ) = L ( s, π ) for (cid:60) ( s ) sufficiently large, where Φ ∈ S (Mat n × n ( F )) is given by Φ( x ) := ω − π ( x n,n )vol( K ( p c ( π ) )) if x ∈ Mat n × n ( O ) with x n, , . . . , x n,n − ∈ p c ( π ) and x n,n ∈ O × , otherwiseif c ( π ) > , while for c ( π ) = 0 , Φ( x ) := (cid:26) if x ∈ Mat n × n ( O ) , otherwise. Archimedean Newform Theory
Analogues of Nonarchimedean Results.
For an induced representation of Whittakertype ( π, V π ) of GL n ( F ) with F archimedean, it is not so clear what the definition of the conductorexponent and the newform of π ought to be. When F is nonarchimedean, the conductor exponentmay be defined either as the least nonnegative integer m for which V K ( p m ) π is nonempty or asthe exponent appearing in the epsilon factor ε ( s, π, ψ ). Neither of these properties, however,can be easily imported to the archimedean setting. Indeed, there is no obvious analogue ofthe congruence subgroup K ( p m ) (though cf. [JN19]), while the epsilon factor ε ( s, π, ψ ) is anintegral power of i for all s ∈ C ; this integer is therefore only determined modulo 4, and socannot be directly used to define the conductor exponent.Our starting observation that leads us to define the conductor exponent and newform in thearchimedean setting is that the key property shared by the newform defined in terms of thecongruence subgroup K ( p c ( π ) ) in Theorem 3.1 and the Whittaker newform defined in terms of atest vector for the GL n × GL n − Rankin–Selberg integral in Theorem 3.11 is its invariance underthe action of the subgroup K n − (cid:51) k (cid:48) embedded in GL n ( F ) via the map k (cid:48) (cid:55)→ (cid:0) k (cid:48)
00 1 (cid:1) . Moreover,Theorem 3.1 essentially states that the newform is the “simplest” such vector for which thisis so, in the sense that V K ( p m ) π is trivial for m < c ( π ). The following lemma exemplifies thenecessity of K n − -invariance for the Whittaker newform. Lemma 4.1.
Let ( π, V π ) be an induced representation of Langlands type of GL n ( F ) with n ≥ ,and let W ∈ W ( π, ψ ) be a right K n -finite Whittaker function, so that the action of π (cid:0) k (cid:48)
00 1 (cid:1) on W for k (cid:48) ∈ K n − generates a finite-dimensional representation τ (cid:48) of K n − . If Hom K n − (1 , τ (cid:48) ) istrivial, then the GL n × GL n − Rankin–Selberg integral
Ψ ( s, W, W (cid:48)◦ ) is identically equal to zerofor every spherical representation of Langlands type π (cid:48) of GL n − ( F ) with spherical Whittakerfunction W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) .Proof. We may write W ( g ) = Λ( π ( g ) · v ) for some v ∈ V π . Via the Iwasawa decompositionwith respect to the standard Borel subgroup and the fact that W (cid:48)◦ is the spherical Whittakerfunction, the Rankin–Selberg integral Ψ ( s, W, W (cid:48)◦ ), defined in (2.1), is equal to (cid:90) A n − ( F ) W (cid:48)◦ ( a (cid:48) ) (cid:12)(cid:12) det a (cid:48) (cid:12)(cid:12) s − δ − n − ( a (cid:48) )Λ (cid:32) π (cid:18) a (cid:48)
00 1 (cid:19) · (cid:90) K n − π (cid:18) k (cid:48)
00 1 (cid:19) · v dk (cid:48) (cid:33) d × a (cid:48) , and the integral over K n − vanishes whenever Hom K n − (1 , τ (cid:48) ) is trivial. (cid:3) This leads us to search for vectors v ∈ V π that are invariant under the action of K n − and todefine the newform of π to be the “simplest” such vector. A natural way to interpret “simplest”is to search for vectors in K -types of π that are “small” in a sense that we make precise. Oncewe have located the newform of π , we show that, when viewed in the Whittaker model, it is aWhittaker newform in the sense of Theorem 3.11. In this way, we work in the reverse direction RCHIMEDEAN NEWFORM THEORY FOR GL n of the nonarchimedean setting, where Jacquet, Piatetski-Shapiro, and Shalika [JP-SS81] firstprove Theorem 3.11 and then use this to deduce Theorem 3.1.4.2. K -Types. The representation π being admissible means that Hom K ( τ, π | K ) is finite-dimensional for any τ in (cid:98) K , the set of equivalence classes of irreducible representations of K . We call τ a K -type of π should the vector space Hom K ( τ, π | K ) be nontrivial. To each τ ∈ (cid:98) K , one can associate a nonnegative integer deg τ called the Howe degree of τ [How89]. Inorder to define this, we first recall the theory of highest weights for the two groups U( n ) andO( n ), then explain how the Howe degree of an irreducible representation is defined in terms ofthe highest weight.4.2.1. Highest Weight Theory for U( n ) . The equivalence classes of finite-dimensional irreduciblerepresentations of the unitary groupU( n ) := (cid:110) k ∈ Mat n × n ( C ) : k t k = 1 n (cid:111) are parametrised by the set of highest weights, which we may identify with n -tuples of integersthat are nonincreasing: Λ n := { µ = ( µ , . . . , µ n ) ∈ Z n : µ ≥ · · · ≥ µ n } . We denote by τ µ an irreducible representation of U( n ) with highest weight µ ∈ Λ n . The Howedegree of τ µ is given by(4.2) deg τ µ = n (cid:88) j =1 (cid:107) µ j (cid:107) . Remark . There is another natural invariant that one associate to an irreducible representation τ µ , namely its Vogan norm (cid:107) τ µ (cid:107) V [Vog81], which is (cid:107) τ µ (cid:107) = n (cid:88) j =1 ( µ j + n + 1 − j ) . Highest Weight Theory for O( n ) . The equivalence classes of finite-dimensional irreduciblerepresentations τ µ of the orthogonal groupO( n ) := (cid:8) k ∈ Mat n × n ( R ) : k t k = 1 n (cid:9) are parametrised by an n -tuple of integers µ , which we again call highest weights (though unlikeU( n ), the compact Lie group O( n ) is not connected). These highest weights µ are preciselythose for which the highest weight vector of the irreducible representation of U( n ) with highestweight µ generates τ µ when restricted to O( n ), and are of the form µ = ( µ , . . . , µ m , η, . . . , η (cid:124) (cid:123)(cid:122) (cid:125) n − m times , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) m times ) ∈ N n , where m ∈ { , . . . , (cid:98) n (cid:99)} , µ ≥ . . . ≥ µ m ≥ η ∈ { , } , and N := N ∪ { } . We again let Λ n denote the set of highest weights; note in particular that Λ = { , } . We denote by τ µ anirreducible representation of O( n ) with highest weight µ ∈ Λ n . The Howe degree of τ µ is(4.4) deg τ µ = n (cid:88) j =1 µ j . Remark . The Vogan norm (cid:107) τ µ (cid:107) V of τ µ is (cid:107) τ µ (cid:107) = m (cid:88) j =1 ( µ j + n − j ) + (cid:98) n (cid:99) (cid:88) j = m +1 ( n − j ) . The Conductor Exponent and the Newform.
Let ( π, V π ) be an induced representationof Whittaker type of GL n ( F ). We define the projection map Π K n − : V π → V π given byΠ K n − ( v ) := (cid:90) K n − π (cid:18) k (cid:48)
00 1 (cid:19) · v dk (cid:48) , whose image is the subspace of K n − -invariant vectors; for the sake of consistency and complete-ness, we define Π K to be the identity when n = 1. We also define the projection mapΠ τ ( v ) := (cid:90) K n ξ τ ( k ) π ( k ) · v dk for each irreducible representation τ ∈ (cid:99) K n , where ξ τ ( k ) := (dim τ ) Tr τ ( k − )is the elementary idempotent associated to τ . The image of V π under Π τ is the τ -isotypicsubspace V τπ of V π , which is finite-dimensional since π is admissible and is trivial unless τ is a K -type of π . The composition of these two projections is the projection (cid:16) Π τ | Kn − (cid:17) ( v ) := (cid:0) Π K n − ◦ Π τ (cid:1) ( v ) = (cid:0) Π τ ◦ Π K n − (cid:1) ( v ) = (cid:90) K n ξ τ | Kn − ( k ) π ( k ) · v dk onto the subspace of K n − -invariant τ -isotypic vectors V τ | Kn − π := (cid:16) Π τ | Kn − (cid:17) ( V π ) = (cid:26) v ∈ V τπ : π (cid:18) k (cid:48)
00 1 (cid:19) · v = v for all k (cid:48) ∈ K n − (cid:27) . Here(4.6) ξ τ | Kn − ( k ) := (cid:90) K n − ξ τ (cid:18)(cid:18) k (cid:48)
00 1 (cid:19) k (cid:19) dk (cid:48) = (cid:90) K n − ξ τ (cid:18) k (cid:18) k (cid:48)
00 1 (cid:19)(cid:19) dk (cid:48) . Note that ξ τ | Kn − is identically equal to zero if and only if Hom K n − (1 , τ | K n − ) is trivial. Finally,for any nonnegative integer m , we set V K n − ( m ) π := (cid:77) τ ∈ (cid:100) K n deg τ = m V τ | Kn − π , the subspace of K n − -invariant vectors that are τ -isotypic for some τ ∈ (cid:99) K n of degree m . Weprove the following. Theorem 4.7 (Cf. Theorem 3.1) . Let ( π, V π ) be an induced representation of Whittaker type of GL n ( F ) . There exists a minimal nonnegative integer m for which V K n − ( m ) π is nontrivial. Forthis minimal value of m , V K n − ( m ) π is one-dimensional. That is, we show that among the K -types τ ∈ (cid:99) K n of π for which V τ | Kn − π is nontrivial, thereexists a unique such K -type τ ◦ minimising deg τ ; moreover, V τ ◦ | Kn − π is one-dimensional. Definition 4.8 (Cf. Definition 3.2) . We call deg τ ◦ the conductor exponent of π and denote itby c ( π ), and the nonzero vector v ◦ ∈ V K n − ( c ( π )) π = V τ ◦ | Kn − π , unique up to scalar multiplication,is called the newform of π . The K -type τ ◦ containing the newform is called the newform K -typeof π .Analogously to the nonarchimedean theory of the conductor exponent, we regard c ( π ) asquantifying the extent to which π is ramified; once again, the conductor exponent c ( π ) is zeroif and only if π is unramified, so that π is a spherical representation. Moreover, the fact that v ◦ is unique up to scalar multiplication may be thought of as a multiplicity-one theorem fornewforms. RCHIMEDEAN NEWFORM THEORY FOR GL n Remark . In general, the newform K -type τ ◦ is not equal to the minimal K -type (in thesense of Vogan [Vog81]) of π , and deg τ ◦ is not equal to the Vogan norm of the minimal K -type. On the other hand, we shall see that the Howe degree of the minimal K -type is equalto deg τ ◦ = c ( π ). This gives another way to define the conductor exponent, albeit with thedownside that the connection between the minimal K -type and the newform is unclear. Remark . The conductor exponent and newform K -type of an induced representation ofWhittaker type π = π (cid:1) · · · (cid:1) π r remain unchanged when π is replaced by π σ (1) (cid:1) · · · (cid:1) π σ ( r ) for any permutation σ ∈ S r .4.4. Oldforms.
We also define oldforms in the archimedean setting.
Definition 4.11.
The space of oldforms of exponent m > c ( π ) is V K n − ( m ) π . Theorem 4.12 (Cf. Theorem 3.5) . Let ( π, V π ) be an induced representation of Whittaker typeof GL n ( F ) with n ≥ . We have that dim V K n − ( m ) π = (cid:18) m − c ( π )2 + n − n − (cid:19) if m ≥ c ( π ) and m ≡ c ( π ) (mod 2) , otherwise. More precisely, we prove that when m ≡ c ( π ) (mod 2), Hom K n ( τ, π | K n ) is trivial for all butone K -type τ with deg τ = m , and for this particular K -type,dim Hom K n ( τ, π | K n ) = (cid:18) m − c ( π )2 + n − n − (cid:19) , dim Hom K n − (cid:0) , τ | K n − (cid:1) = 1 . The fact that V K n − ( m ) π is trivial when m ≡ c ( π ) + 1 (mod 2) follows more generally from aresult of Fan [Fan18], namely that for τ ∈ (cid:99) K n , V τπ is trivial whenever deg τ ≡ c ( π ) + 1 (mod 2).We may think of (cid:76) mj =0 V K n − ( j ) π as being the archimedean analogue of V K ( p m ) π . This spacehas dimension (cid:0) (cid:98) m − c ( π )2 (cid:99) + n − n − (cid:1) if m ≥ c ( π ) and dimension zero otherwise. This aligns closelywith the dimension of V K ( p m ) π as in Theorem 3.5, namely (cid:0) m − c ( π )+ n − n − (cid:1) if m ≥ c ( π ) and zerootherwise. Remark . For nonarchimedean F , Reeder [Ree91] explicitly describes the space of oldformsas the image of the space of newforms under the action of certain elements of the Hecke algebra.It would be of interest to generalise this to the archimedean setting. In particular, when viewedin the Whittaker model, it would be of interest to describe oldforms in terms of the action ofcertain differential operators on the newform. For GL ( R ), these are simply the Maaß raisingand lowering operators, while the theory for GL ( R ) is explored in [BM19].4.5. Additivity and Inductivity of the Conductor Exponent.
Associated to any inducedrepresentation of Whittaker type π of GL n ( F ) is the epsilon factor ε ( s, π, ψ ); with ψ is chosenas in Section 2.1.2, the epsilon factor is an integral power of i via (2.16) and (2.18). This integeris only determined modulo 4, in contrast to the nonarchimedean setting. In this regard, thefollowing theorem is not so instructive as Theorem 3.6. Theorem 4.14 (Cf. Theorem 3.6) . Let π be an induced representation of Whittaker type of GL n ( F ) . The epsilon factor ε ( s, π, ψ ) is equal to i − c ( π ) , where c ( π ) denotes the conductorexponent of π . Upon decomposing π as the isobaric sums π = π (cid:1) · · · (cid:1) π r , we have the identity i − c ( π ) = ε ( s, π, ψ ) = r (cid:89) j =1 ε ( s, π j , ψ ) = i − c ( π ) −···− c ( π r ) via (2.18). This is only enough to conclude that c ( π ) is congruent to c ( π ) + · · · + c ( π r ) modulo4, rather than an equality of conductor exponents. Nonetheless, we still prove by different meansthat the conductor exponent is additive with respect to isobaric sums. Theorem 4.15 (Cf. Theorem 3.7) . For an induced representation of Whittaker type π = π (cid:1) · · · (cid:1) π r of GL n ( F ) , we have that c ( π ) = r (cid:88) j =1 c ( π j ) . The epsilon factor is again inductive in degree zero, so that if π and π (cid:48) are induced representa-tions of Whittaker type of GL n ( C ) and AI C / R π and AI C / R π (cid:48) denote the induced representationof Whittaker type of GL n ( R ) obtained by induction [Hen10], then ε (cid:0) s, AI C / R π, ψ (cid:1) ε (cid:0) s, AI C / R π (cid:48) , ψ (cid:1) = ε (cid:0) s, π, ψ ◦ Tr C / R (cid:1) ε (cid:0) s, π (cid:48) , ψ ◦ Tr C / R (cid:1) . Taking π (cid:48) to be the isobaric sum of n copies of the trivial representation, we find that c (cid:0) AI C / R π (cid:1) ≡ c ( π ) + n (mod 4) . We prove by different means that this congruence can be replaced by an equality.
Theorem 4.16 (Cf. Theorem 3.9) . For an induced representation of Whittaker type π of GL n ( C ) , we have that c (cid:0) AI C / R π (cid:1) = c ( π ) + n. Test Vectors for Rankin–Selberg Integrals.
One can once again ask about the ex-istence of test vectors for GL n × GL m Rankin–Selberg integrals. This is known for sphericalrepresentations π and π (cid:48) with m ∈ { n − , n } , due to Stade [Sta01, Theorem 3.4], [Sta02, Theo-rem 1.1] (see also [GLO08, Lemma 4.2] and [IsSt13, Theorem 3.1]), while recent work of Ishii andMiyazaki deals with the case of pairs of principal series representations of certain specific forms[IM20, Theorem 2.5]. We prove the existence of a test vector for nonspherical representations π of GL n ( F ) provided that π (cid:48) is a spherical representation of Langlands type of GL n − ( F ) and W (cid:48) = W (cid:48)◦ ∈ W (cid:0) π (cid:48) , ψ (cid:1) is the spherical vector normalised as in Section 2.2.5. Theorem 4.17 (Cf. Theorem 3.11) . Let π be an induced representation of Langlands typeof GL n ( F ) with n ≥ . There exists a Whittaker function W ∈ W ( π, ψ ) such that for anyspherical representation of Langlands type π (cid:48) of GL n − ( F ) with spherical Whittaker function W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) , Ψ( s, W, W (cid:48)◦ ) = L ( s, π × π (cid:48) ) for (cid:60) ( s ) sufficiently large.Moreover, there exists a unique such function W ◦ ∈ W ( π, ψ ) that additionally satisfies W ◦ (cid:18) g (cid:18) k
00 1 (cid:19)(cid:19) = W ◦ ( g ) for all k ∈ K n − . Up to multiplication by a scalar, this function is the newform v ◦ ∈ V K n − ( c ( π )) π viewed in the Whittaker model. We again call W ◦ the Whittaker newform. The normalisation of the Whittaker newform isspecified in Definition 9.2.Stade [Sta02, Theorem 1.1] has shown that for all spherical representations π, π (cid:48) of GL n ( R ),Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) = L ( s, π × π (cid:48) ) , where Φ ◦ ∈ S (Mat × n ( R )) is given by Φ ◦ ( x ) := exp (cid:0) − πx t x (cid:1) ; see also [GLO08, Lemma 4.1]and [IsSt13, Theorem 3.1]. We extend this to nonspherical representations π . Theorem 4.18 (Cf. Theorem 3.15) . Let π be an induced representation of Langlands type of GL n ( F ) . For every spherical representation of Langlands type π (cid:48) of GL n ( F ) with sphericalWhittaker function W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) , the Whittaker newform W ◦ ∈ W ( π, ψ ) of π satisfies Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) = L ( s, π × π (cid:48) ) RCHIMEDEAN NEWFORM THEORY FOR GL n for (cid:60) ( s ) sufficiently large, where Φ ◦ ∈ S (Mat × n ( F )) is given by (4.19) (dim τ ◦ ) P ◦ ( x ) exp (cid:0) − d F πx t x (cid:1) , d F := [ F : R ] = (cid:26) if F = R , if F = C ,with P ◦ a homogeneous harmonic polynomial that depends only on the newform K -type τ ◦ of π . We discuss this distinguished polynomial P ◦ in further detail in Section 7; it is the uniquehomogeneous harmonic polynomial that is right K n − -invariant and satisfies P ◦ ( e n ) = 1 in thevector space of such polynomials that forms a model of τ ◦ .In the nonarchimedean setting, it is not only the case that the GL n × GL n − Rankin–Selbergintegral Ψ( s, W ◦ , W (cid:48)◦ ) is equal to L ( s, π × π (cid:48) ) for all spherical representations π (cid:48) of GL n − ( F ),but also for all spherical representations π (cid:48) of GL m ( F ) and all m ∈ { , . . . , n − } ; see Theorem3.14. In the archimedean setting, on the other hand, it is widely believed (see, for example,[Bum89, Section 2.6]) that Ψ( s, W, W (cid:48) ) is never equal to L ( s, π × π (cid:48) ) when π (cid:48) is an inducedrepresentation of Whittaker type of GL m ( F ) with m ∈ { , . . . , n − } and the Whittaker functions W ∈ W ( π, ψ ) and W (cid:48) ∈ W ( π (cid:48) , ψ ) are K n - and K m -finite respectively.Notably, Ishii and Stade [IsSt13, Theorem 3.2], furthering the work of Hoffstein and Murty[HM89], have shown that for all spherical representations π of GL n ( R ) and π (cid:48) of GL n − ( R ),(4.20) Ψ( s, W ◦ , W (cid:48)◦ ) = L ( s, π × π (cid:48) ) 12 πi (cid:90) σ + i ∞ σ − i ∞ L ( w, (cid:101) π )2 L ( s + w, π (cid:48) ) dw for some sufficiently large σ . Though we do not include a proof, our methods may also be usedto show the identity (4.20) when π is a ramified induced representation of Langlands type and W ◦ is the Whittaker newform.4.7. Test Vectors for Godement–Jacquet Zeta Integrals.
Test vectors for archimedeanGodement–Jacquet zeta integrals are known to exist; see [Lin18] for F = R and [Ish19] for thecase F = C . We give a new resolution of the test vector problem via newforms. Remark . We were unable to verify certain aspects of [Lin18]; cf. [Hum20, Footnote † ]. Theorem 4.22 (Cf. Theorem 3.17) . Let ( π, V π ) be an induced representation of Whittaker typeof GL n ( F ) . Let β ( g ) denote the matrix coefficient (cid:104) π ( g ) · v ◦ , (cid:101) v ◦ (cid:105) , where v ◦ ∈ V π denotes thenewform and (cid:101) v ◦ ∈ V (cid:101) π is the corresponding newform normalised such that β (1 n ) = 1 . Then Z ( s, β, Φ) = L ( s, π ) for (cid:60) ( s ) sufficiently large, where Φ ∈ S (Mat n × n ( F )) is given by (4.23) (dim τ ◦ ) P ◦ ( e n x ) exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) , with P ◦ a homogeneous harmonic polynomial that depends only on the newform K -type τ ◦ of π . The homogeneous harmonic polynomial P ◦ appearing in Theorem 4.22 is the same as thatappearing in Theorem 4.18. 5. Further Discussion
Previous Results.
Previous cases of test vectors for GL n × GL n and GL n × GL n − Rankin–Selberg integrals over archimedean fields with n arbitrary were only known when either bothrepresentations are unramified — see [Sta01, Theorem 3.4], [GLO08, Lemma 4.2], and [IsSt13,Theorem 3.2] — or when both representations are principal series representations of certainspecific forms, due to recent work of Ishii and Miyazaki [IM20, Theorems 2.5 and 2.9]. Jacquet[Jac09, Theorem 2.7] has shown that for each pair π, π (cid:48) of generic irreducible Casselman–Wallachrepresentations of GL n ( F ) , GL n − ( F ), there exists a finite collection { W j } ⊂ W ( π, ψ ) and { W (cid:48) j } ⊂ W ( π (cid:48) , ψ ) of K n - and K n − -finite Whittaker functions for which (cid:80) j Ψ( s, W j , W j ) = L ( s, π × π (cid:48) ); a similar result (additionally involving Schwartz functions { Φ j } ⊂ S (Mat × n ( F )))also holds for pairs of representations of GL n ( F ).Explicit descriptions of specific Whittaker functions for ramified principal series represen-tations of GL n ( R ) are given in [IO14] and of GL ( C ) in [HO09], while recursive formulæ for specific Whittaker functions for principal series representations of GL n ( R ) and GL n ( C ) aregiven in [IM20]. In all three cases, these are Whittaker functions in the minimal K -type; asobserved in [HIM12, Theorem 6.1], such a Whittaker function W is generally not a test vec-tor for Rankin–Selberg integrals Ψ( s, W, W (cid:48)◦ ) when W (cid:48)◦ is the spherical vector of a sphericalrepresentation.For n ≤ m ≤
2, the state of affairs of test vectors for GL n × GL m Rankin–Selbergintegrals is in much better shape. In particular, the existence of the Whittaker newform forGL ( F ) has previously been proven by Popa [Pop08], though in a slightly different formulationthat we now describe.Let π be a generic irreducible Casselman–Wallach representation of GL ( F ) with centralcharacter ω π , let T denote the diagonal torus embedded in GL ( F ), and let χ T be a characterof T whose restriction to the centre of GL ( F ) is ω − π . Given τ ∈ (cid:99) K , we define W τ,T to be thesubspace of the τ -isotypic subspace W τ of W ( π, ψ ) for which π | K ∩ T acts by χ − T . Theorem 5.1 (Popa [Pop08, Proposition 1, Theorem 1]) . The space W τ,T is at most one-dimensional. Furthermore, if χ T (diag( a , a )) = | a | s − / | a | / − s ω − π ( a ) and τ = τ ◦ is the K -type of lowest degree for which W τ,T is nontrivial, then there exists W ◦ ∈ W τ ◦ ,T for which Ψ( s, W ◦ ,
1) = L ( s, π ) . While this superficially appears to be different than the definition of the Whittaker newform,it is readily checked that W τ,T = (cid:26) W ∈ W τ : W (cid:18) g (cid:18) k
00 1 (cid:19)(cid:19) = W ( g ) for all g ∈ GL ( F ) and k ∈ K (cid:27) , and so the Whittaker function W ◦ is indeed the Whittaker newform and W τ ◦ ,T is equal to thespace V K ( c ( π )) π = V τ ◦ | K π when viewed in the Whittaker model.Jacquet [Jac72, Theorem 17.2] has previously determined, in several cases, explicit right K -finite test vectors W ∈ W ( π, ψ ) and W (cid:48) ∈ W ( π (cid:48) , ψ ) and Schwartz functions Φ ∈ S (Mat × ( F ))for the GL × GL Rankin–Selberg integral Ψ( s, W, W (cid:48) , Φ) when F = R ; moreover, this includescases involving generic irreducible Casselman–Wallach representations π, π (cid:48) of GL ( F ) for which π (cid:48) need not be spherical. This has been extended to F = C by Miyazaki [Miz18, Theorem 6.1];see also [HIM20, Appendix A].For n = 3, Hirano, Ishii, and Miyazaki [HIM16, HIM20] have announced an explicit descriptionof right K - and K -finite Whittaker functions W ∈ W ( π, ψ ) and W (cid:48) ∈ W ( π (cid:48) , ψ ) for whichΨ( s, W, W (cid:48) ) = L ( s, π × π (cid:48) ), where π and π (cid:48) are any generic irreducible Casselman–Wallachrepresentations of GL ( F ) and GL ( F ) respectively; in particular, π (cid:48) need not be spherical.5.2. Examples and Applications.
The Conductor Exponent for Essentially Square-Integrable Representations.
For π = χ κ | · | t , where κ ∈ { , } for F = R and κ ∈ Z for F = C , the conductor exponent is c ( π ) = (cid:107) κ (cid:107) .For π = D κ ⊗ | det | t , where κ ≥
2, the conductor exponent is c ( π ) = κ . Via Theorem 4.15, thisallows one to calculate the conductor exponent of any induced representation of Whittaker type.5.2.2. GL Examples.
Let us consider the classical setting of automorphic forms on the upperhalf-plane.
Example . Let f be a holomorphic cuspidal newform of weight κ ≥
2, level q ∈ N , andnebentypus ψ modulo q , where ψ ( −
1) = ( − κ . Then the underlying automorphic representationof GL ( A Q ), as described in [Cas73, Section 3], has as its archimedean component a discreteseries of weight κ , D κ . In particular, the conductor exponent of this archimedean component issimply the weight κ . Example . Let f be a Maaß cuspidal newform of weight κ ∈ { , } , level q ∈ N , and neben-typus ψ , where ψ ( −
1) = ( − κ . The archimedean component of the underlying automorphicrepresentation of GL ( A Q ) is one of three possible representations: RCHIMEDEAN NEWFORM THEORY FOR GL n • If κ = 1, then the archimedean component is a principal series representation of theform | · | it f (cid:1) χ | · | − it f or χ | · | it f (cid:1) | · | − it f , where t f ∈ R is the spectral parameter of f .The conductor exponent is 1. • If κ = 0 and f is even, so that f ( − z ) = f ( z ), then the archimedean component is of theform | · | it f (cid:1) | · | − it f , where t f ∈ R ∪ i ( − / , / • If κ = 0 and f is odd, so that f ( − z ) = − f ( z ), then the archimedean component is ofthe form χ | · | it f (cid:1) χ | · | − it f , where again t f ∈ R ∪ i ( − / , / Remark . The newform K -type of the archimedean component of the automorphic represen-tation associated to an odd Maaß form of weight zero is not its minimal K -type: the newform K -type is τ (2 , , whereas the minimal K -type is the determinant representation τ (1 , . Classically,this manifests itself via the fact that an odd Maaß form f of weight zero is not a test vector forthe Rankin–Selberg integral (cid:90) ∞ f ( iy ) y s − dyy . Instead, one must use Maaß raising and lowering operators on such a Maaß form to obtain atest vector.5.2.3.
A Non-Application: the Analytic Conductor and Analytic Newvector.
We have introducedin Section 4.3 the notion of the conductor exponent of a generic irreducible Casselman–Wallachrepresentation π as a measure of the extent of ramification. This can also be thought of asquantifying the (representation-theoretic) complexity of π .There is another well-known quantification of the complexity of π : the analytic conductor.First introduced by Iwaniec and Sarnak [IwSa00, (31)], this is defined on essentially square-integrable representations π by q ( π ) := (cid:107) t + κ (cid:107) if F = R and π = χ κ | · | t R , (cid:18) (cid:13)(cid:13)(cid:13)(cid:13) t + κ − (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) (cid:18) (cid:13)(cid:13)(cid:13)(cid:13) t + κ + 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) if F = R and π = D κ ⊗ | det | t R , (cid:18) (cid:13)(cid:13)(cid:13)(cid:13) t + (cid:107) κ (cid:107) (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) if F = C and π = χ κ | · | t C ,and extended to arbitrary induced representations of Whittaker type π = π (cid:1) · · · (cid:1) π r viamultiplicativity: q ( π ) := r (cid:89) j =1 q ( π j ) . One can relate this to the asymptotic behaviour of the local γ -factor γ ( s, π, ψ ) as s tends to1 /
2; see in particular [JN19, Lemma 3.1 (1)] and also [CFKRS05, Section 3.1] (the latter alsoalludes to how this similarly encompasses the conductor at the nonarchimedean places).These two quantifications are entirely distinct. This can be seen most clearly for sphericalrepresentations π = | · | it f (cid:1) | · | − it f of GL ( R ) occurring as the archimedean component of anautomorphic representation associated to an even Maaß form f of weight zero. As discussed inExample 5.3, the conductor exponent of such a representation π is c ( π ) = 0; on the other hand,the analytic conductor is q ( π ) = (1 + (cid:107) t f (cid:107) ) . This distinction boils down to the fact that theanalytic conductor q ( π ) is better thought of as being a measure of the size of the L -function L ( s, π ) rather than a measure of the complexity of the representation itself.Similarly, the newform is unrelated to the analytic newvectors introduced by Jana and Nelsonin [JN19]. Analytic newvectors are a different archimedean analogue of newforms; they are a family of vectors in a generic irreducible unitary Casselman–Wallach representation that are almost invariant under certain subsets (but not subgroups) of GL n ( F ) that are archimedeananalogues of the congruence subgroups K ( p m ). Notably, analytic newvectors are not necessarily K n -finite, though they are K n − -invariant; in particular, they lie in the subspace spanned bythe newform and all oldforms. Note that Jana and Nelson call the newform an “algebraicnewvector”. Global Eulerian Integrals.
We mention a global application of the resolution of the testvector problem for Rankin–Selberg integrals. In order to do so, we require some discussion ofglobal automorphic forms.Let ψ A Q denote the standard additive character of A Q that is unramified at every place of Q .Given a global number field F , define the additive character ψ A F := ψ A Q ◦ Tr A F / A Q of A F . Theconductor of ψ A F is the inverse different d − of F ; furthermore, there is a finite id`ele d ∈ A × F representing d such that ψ A F = (cid:78) v ψ d v v , where the additive character ψ d v v of F v of conductor d − v is defined by ψ d v v ( x ) := ψ v ( d v x ) with ψ v an unramified additive character of F v as in Section2.1.2.Let ( π, V π ) be a cuspidal automorphic representation of GL n ( A F ) with n ≥
2, where V π is aspace of automorphic forms on GL n ( A F ). Let W ϕ ∈ W ( π, ψ A F ) denote the Whittaker functionof ϕ ∈ V π , so that W ϕ ( g ) := (cid:90) N n ( F ) \ N n ( A F ) ϕ ( ug ) ψ A F ,n ( u ) du, where du denotes the Tamagawa measure. If W ϕ is a pure tensor, we may write W ϕ = (cid:81) v W ϕ,v with W ϕ,v ∈ W ( π v , ψ d v v ), where the generic irreducible admissible representations π v are the localcomponents of the automorphic representation π = (cid:78) v π v . We define the global newform ϕ ◦ ∈ V π to be such that at each place v of F , we have that W ϕ ◦ ,v ( g v ) := W v (diag( d n − v , . . . , d v , g v ) ∈W ( π v , ψ d v v ) with W v ∈ W ( π v , ψ v ) the (local) Whittaker newform as in Theorems 3.11 and 4.17. Proposition 5.5.
Let F be a global number field of absolute discriminant D F/ Q , and let ( π, V π ) be a cuspidal automorphic representation of GL n ( A F ) with n ≥ . Then the global newform ϕ ◦ ∈ V π is such that for all cuspidal automorphic representations ( π (cid:48) , V π (cid:48) ) of GL n − ( A F ) thatare everywhere unramified with global newforms ϕ (cid:48)◦ ∈ V π (cid:48) and for (cid:60) ( s ) sufficiently large, theglobal GL n × GL n − Rankin–Selberg integral I ( s, ϕ ◦ , ϕ (cid:48)◦ ) := (cid:90) GL n − ( F ) \ GL n − ( A F ) ϕ ◦ (cid:18) g
00 1 (cid:19) ϕ (cid:48)◦ ( g ) | det g | s − A F dg with dg the Tamagawa measure is equal, up to multiplication by a constant dependent only on F , to the product of ω − π (cid:48) ( d ) D n ( n − s F/ Q , where ω π (cid:48) denotes the central character of π (cid:48) , and of theglobal completed Rankin–Selberg L -function Λ( s, π × π (cid:48) ) := (cid:81) v L ( s, π v × π (cid:48) v ) .Remark . Via a regularisation process due to Ichino and Yamana [IY15, Theorem 1.1], asuitably modified version of this result still holds even if either π or π (cid:48) (or both) is not cuspidal. Proof.
By unfolding, I ( s, ϕ ◦ , ϕ (cid:48)◦ ) = (cid:90) N n − ( A F ) \ GL n − ( A F ) W ϕ ◦ (cid:18) g
00 1 (cid:19) W ϕ (cid:48)◦ ( g ) | det g | s − A F dg = c F/ Q (cid:89) v Ψ( s, W ϕ ◦ ,v , W ϕ (cid:48)◦ ,v ) , where the local Rankin–Selberg integrals Ψ( s, W ϕ ◦ ,v , W ϕ (cid:48)◦ ,v ) are as in (2.1), and c F/ Q > F that arises from the compatibility of the global Tamagawameasure on N n − ( A F ) \ GL n − ( A F ) compared to the local Haar measure on N n − ( F v ) \ GL n − ( F v )as normalised in Section 2.1. Upon making the change of variables g (cid:48) v (cid:55)→ diag( d − nv , . . . , d − v ) g (cid:48) v and using the fact that W (cid:48) v (diag( d − v , . . . , d − v ) g (cid:48) v ) = ω − π (cid:48) v ( d v ) W (cid:48) v ( g (cid:48) v ), we see thatΨ( s, W ϕ ◦ ,v , W ϕ (cid:48)◦ ,v ) = ω − π (cid:48) v ( d v ) | d v | − n ( n − s + n ( n − n − v Ψ( s, W v , W (cid:48) v ) . The result now follows from Theorems 3.11 and 4.17. (cid:3)
RCHIMEDEAN NEWFORM THEORY FOR GL n A similar result holds for global GL n × GL n Rankin–Selberg integrals via Theorems 3.15 and4.18. These involve an Eisenstein series E ( g, s ; Φ , η ) = | det g | s A F (cid:90) F × \ A × F Θ (cid:48) Φ ( a, g ) η ( a ) | a | ns A F d × a, where η : F × \ A × F → C × is a Hecke character, Φ ∈ S (Mat × n ( A F )) is a Schwartz–Bruhatfunction, and Θ Φ ( a, g ) := (cid:88) ξ ∈ Mat × n ( F ) Φ( aξg ) , Θ (cid:48) Φ ( a, g ) := Θ Φ ( a, g ) − Φ(0) . Proposition 5.7.
Let F be a number field of absolute discriminant D F/ Q , and let ( π, V π ) and ( π (cid:48) , V π (cid:48) ) be automorphic representations of GL n ( A F ) with central characters ω π and ω π (cid:48) . Supposethat at least one of π and π (cid:48) is cuspidal and that π and π (cid:48) have disjoint ramification. Thenthere exists a bi- K -finite Schwartz–Bruhat function Φ ∈ S (Mat × n ( A F )) such that for (cid:60) ( s ) sufficiently large, the global GL n × GL n Rankin–Selberg integral I ( s, ϕ ◦ , ϕ (cid:48)◦ , Φ) := (cid:90) Z( A F ) GL n ( F ) \ GL n ( A F ) ϕ ◦ ( g ) ϕ (cid:48)◦ ( g ) E ( g, s ; Φ , ω π ω π (cid:48) ) dg with dg the Tamagawa measure is equal, up to multiplication by a constant dependent only on F ,to the product of D n ( n − s F/ Q and of the global completed Rankin–Selberg L -function Λ( s, π × π (cid:48) ) := (cid:81) v L ( s, π v × π (cid:48) v ) .Remark . When both π and π (cid:48) are noncuspidal, it ought to be possible to prove a suitablymodified version of this result by extending to GL n a regularisation process due to Zagier [Zag82,Theorem] for GL .We also may show the existence of a test vector for the global Godement–Jacquet zeta integralvia Theorems 3.17 and 4.22. Proposition 5.9.
Let F be a number field, and let ( π, V π ) be an automorphic representationof GL n ( A F ) . Then there exists a matrix coefficient β ( g ) := (cid:104) π ( g ) · ϕ ◦ , (cid:101) ϕ ◦ (cid:105) and a bi- K -finiteSchwartz–Bruhat function Φ ∈ S (Mat n × n ( A F )) such that for (cid:60) ( s ) sufficiently large, the globalGodement–Jacquet zeta integral Z ( s, β, Φ) := (cid:90) GL n ( A F ) β ( g )Φ( g ) | det g | s + n − A F dg with dg the Tamagawa measure is equal, up to multiplication by a constant dependent only on F , to the global completed standard L -function Λ( s, π ) := (cid:81) v L ( s, π v ) . A similar result also holds for the Piatetski-Shapiro–Rallis integral [P-SR87], since this dou-bling integral is equal to the Godement–Jacquet zeta integral [P-SR87, Proposition 3.2].
Remark . In fact, we show something slightly stronger, namely that (cid:90) GL n ( A F ) ϕ ◦ ( hg )Φ( g ) | det g | s + n − A F dg is equal, up to multiplication by a constant dependent only on F , to Λ( s, π ) ϕ ◦ ( h ) for all h ∈ GL n ( A F ).5.3. Strategy of the Proofs.
Nonarchimedean Strategies.
As discussed in Section 3, there are several approaches to-wards developing nonarchimedean newform theory for GL n . We briefly examine the challengesin transporting each of these methods to the archimedean setting.Matringe [Mat13] uses the nonarchimedean theory of Bernstein–Zelevinsky derivatives toexplicitly construct the Whittaker newform. The archimedean theory of Bernstein–Zelevinskyderivatives is less well-developed (though see [AGS15] and [Cha15] for two different approaches),and it does not seem straightforward to transport Matringe’s proof to this setting. The method of Jacquet [Jac12] does not use Bernstein–Zelevinsky derivatives; the proof,however, is nonconstructive and only shows the existence of a K n − -invariant test vector forGL n × GL n − Rankin–Selberg integrals. In the archimedean setting, it seems difficult to describethe newform K -type when one only knows of the existence of such a test vector.Finally, the method of Miyauchi [Miu14] assumes the existence of the newform in V K ( p c ( π ) ) π and uses the action of certain Hecke operators to derive a recursive relation between certainvalues of the newform in the Whittaker model; using this, one can then show that the newformin the Whittaker model is a test vector for Rankin–Selberg integrals. The archimedean analogueof this is to assume the existence of the newform in the newform K -type and use the actionof certain differential operators (arising from elements of the centre of the universal envelopingalgebra) to derive systems of partial differential equations satisfied by the newform in theWhittaker model. This is essentially the approach in [HO09] and [IO14] (where the Whittakerfunction studied lies in the minimal K -type, not the newform K -type); already for GL ( C ),however, this leads to immense combinatorial difficulties in solving these systems of partialdifferential equations.5.3.2. The Archimedean Strategy.
We take a different path. To prove the existence of thenewform and newform K -type, we use Frobenius reciprocity to reduce the problem to branchingrules on the associated maximal compact subgroups; here we benefit from the fact that, unlikein the nonarchimedean setting, irreducible representations of K and explicit branching rulesare well-understood, and the induced representations of Whittaker type are particularly easy todescribe, since essentially square-integrable representations do not exist for GL n ( F ) with n ≥ K -type, namely a space of homogeneous harmonicpolynomials. Using this, we explicitly construct the newform in the induced model of π inSection 8. We give three different constructions of the newform in the induced model: via theIwasawa decomposition, via convolution sections, and via Godement sections. Each construc-tion has its advantages and disadvantages. The construction via the Iwasawa decomposition isstraightforward but lacks a direct relation to Whittaker functions. The construction via convo-lution sections, following work of Jacquet [Jac04], gives a recursive formula for the newform interms of a convolution of the newform itself with an explicit standard Schwartz function; thisgives an immediate resolution of the test vector problem for archimedean Godement–Jacquetintegrals. Finally, following Jacquet [Jac09], the newform is shown to be given as an element ofa Godement section, which gives a recursive formula for the newform in terms of an integral of adistinguished newform of a representation of GL n − ( F ) against a particular standard Schwartzfunction; this is limited to certain induced representations of Whittaker type, but is invaluableas an inductive step.With these formulæ in hand, we then express the newform in the Whittaker model via theJacquet integral in Section 9. The usage of convolution sections and Godement sections giveus recursive formulæ for the Whittaker newform. The latter in particular gives what we call apropagation formula: this is a recursive formula for GL n ( F ) Whittaker functions in terms ofGL n − ( F ) Whittaker functions.Our expression for the newform via convolution sections gives a quick resolution of the testvector problem for the Godement–Jacquet zeta integral. Our strategy for resolving the testvector problems for GL n × GL n and GL n × GL n − Rankin–Selberg integrals follows an approachpioneered by Jacquet [Jac09] (which is also followed in [IM20]). We employ a double inductionargument presented in Section 10. This type of argument is due to Jacquet [Jac09] (who, inturn, attributes this strategy to Shalika); it expresses the GL n × GL n Rankin–Selberg integralas the product of a GL n × GL n − Rankin–Selberg integral and a GL n Godement–Jacquet zetaintegral, and similarly expresses the GL n × GL n − Rankin–Selberg integral as a product of aGL n − × GL n − Rankin–Selberg integral and a GL n − Godement–Jacquet zeta integral. (Infact, we find a slightly more direct approach via convolution sections that masks the presenceof Godement–Jacquet zeta integrals.)
RCHIMEDEAN NEWFORM THEORY FOR GL n Additional Remarks on the Proofs.
We emphasise that the proofs of Theorems 4.17 and4.18, given in Section 10, are independent of the proofs in [GLO08, IsSt13, Sta01, Sta02] ofthe unramified cases. Although our proofs are somewhat involved when ramification is present,they are particularly simple for spherical representations. In particular, these gives proofs ofStade’s formulæ, namely the unramified cases of Theorems 4.17 and 4.18. These reproofs ofStade’s formulæ also follow from the proofs of [IM20, Theorems 2.5 and 2.9]; they are essentiallyimplicit in the work of Jacquet [Jac09].Notably, we do not explicitly make use of the action of the universal enveloping algebra ofthe complexified Lie algebra of GL n ( F ) as differential operators on Whittaker functions, nordo we require any calculations involving Mellin transforms. In this regard, our constructionof Whittaker functions is entirely distinct to that of much of previous work on archimedeanWhittaker functions [HIM12, HIM16, HO09, IO14, IsSt13, Pop08, Sta90, Sta95, Sta01, Sta02].In particular, our proofs of Theorems 4.17 and 4.18 demonstrate that the Jacquet integral is adequate for the direct computation of archimedean Rankin–Selberg integrals, contrary to anassertion of Ishii and Oda [IO14, p. 1288], provided one couples this with the usage of convolutionsections and Godement sections. 6. The Newform K -Type To study the newform K -type of an induced representation of Whittaker type π of GL n ( F ),as well as determine the dimension of spaces of oldforms, we must determine the dimension ofHom K n ( τ, π | K n ) for each τ ∈ (cid:99) K n for which Hom K n − (1 , τ | K n − ) is nontrivial. This is achievedvia branching laws.6.1. Branching from GL n ( C ) to U( n ) . Let π = π (cid:1) · · · (cid:1) π n be an induced representationof Whittaker type of GL n ( C ), so that for each j ∈ { , . . . , n } , π j = χ κ j | · | t j for some κ j ∈ Z and t j ∈ C . Lemma 6.1.
For τ ∈ (cid:91) U( n ) , we have that (6.2) Hom U( n ) (cid:0) τ, π | U( n ) (cid:1) ∼ = Hom U(1) n τ | U(1) n , n (cid:2) j =1 τ κ j . Here we view U(1) n as the subgroup of diagonal matrices in U( n ); it is the maximal compactsubgroup of the Levi subgroup M (1 ,..., ( C ) ∼ = GL ( C ) n of the standard parabolic subgroupP( C ) = P (1 ,..., ( C ) of GL n ( C ) from which π is induced. Proof.
Mackey’s restriction-induction formula implies that π | U( n ) ∼ = Ind U( n )U(1) n π | U(1) n , and so by the Frobenius reciprocity theorem,Hom U( n ) (cid:0) τ, π | U( n ) (cid:1) ∼ = Hom U(1) n (cid:0) τ | U(1) n , π | U(1) n (cid:1) . It remains to note that π | U(1) n ∼ = n (cid:2) j =1 τ κ j . (cid:3) The right-hand side of (6.2) is a branching from U( n ) to U(1) n . This can be understood viaiterating the following branching law from U( n ) to U( n − × U(1).
Lemma 6.3 ([Pro94, Proposition 10.1]) . For τ µ ∈ (cid:91) U( n ) of highest weight µ = ( µ , . . . , µ n ) ∈ Λ n , τ µ | U( n − × U(1) ∼ = (cid:77) λ ∈ Λ (cid:77) ν =( ν ,...,ν n − ) ∈ Λ n − µ ≥ ν ≥ µ ≥···≥ ν n − ≥ µ n (cid:80) n − j =1 ν j = (cid:80) nj =1 µ j − λ τ ν (cid:2) τ λ . In particular, τ µ | U( n − ∼ = (cid:77) ν =( ν ,...,ν n − ) ∈ Λ n − µ ≥ ν ≥ µ ≥···≥ ν n − ≥ µ n τ ν . Corollary 6.4.
The restriction to U( n − of the irreducible representation τ µ ∈ (cid:91) U( n ) ofhighest weight µ = ( µ , . . . , µ n ) ∈ Λ n contains the trivial representation if and only if µ ≥ , µ = · · · = µ n − = 0 , and µ n ≤ , in which case the trivial representation occurs with multiplicityone. We now iterate the branching law in Lemma 6.3 to determine the multiplicity of a represen-tation τ λ (cid:2) · · · (cid:2) τ λ n of U(1) n in a given representation τ µ of U( n ). Lemma 6.5.
For τ µ ∈ (cid:91) U( n ) of highest weight µ = ( µ , , . . . , , µ n ) ∈ Λ n and for any λ , . . . , λ n ∈ Λ , dim Hom U(1) n τ µ | U(1) n , n (cid:2) j =1 τ λ j = (cid:18) (cid:96) + n − n − (cid:19) if µ = n (cid:88) j =1 max { λ j , } + (cid:96) and µ n = n (cid:88) j =1 min { λ j , } − (cid:96) for some (cid:96) ∈ N , otherwise.Proof. We take µ = ( µ , , . . . , , µ n ) in Lemma 6.3 and then iterate this branching law in orderto find that τ µ | U(1) n ∼ = n − (cid:77) j =1 (cid:77) λ j ∈ Λ (cid:77) ( ν j, , ,..., ,ν j,n − j ) ∈ Λ n − j ν j, ≤ ν j − , , ν j,n − j ≥ ν j − ,n − j +1 ν j, + ν j,n − j = ν j − , + ν j − ,n − j +1 − λ j n (cid:2) j =1 τ λ j , where we define ν , := µ , ν ,n := µ n , and λ n := ν n − , . By induction, the condition ν j, + ν j,n − j = ν j − , + ν j − ,n − j +1 − λ j implies that µ + µ n − j (cid:88) k =1 λ k = ν j, + ν j,n − j for j ∈ { , . . . , n − } , ν n − , for j = n − j = n .The multiplicity of τ λ (cid:2) · · · (cid:2) τ λ n in τ µ is thereby equal to (cid:8) ( ν , , . . . , ν n − , ) ∈ N n − : ν j − , ≥ ν j, + max { λ j , } for all j ∈ { , . . . , n − } (cid:9) , namely the number of ( n − ν , , . . . , ν n − , ) for which the system of inequalities µ ≥ ν , + max { λ , } ≥ · · · ≥ ν n − , + n − (cid:88) j =1 max { λ j , } ≥ n (cid:88) j =1 max { λ j , } holds. This is zero unless there exists some (cid:96) ∈ N such that µ = n (cid:88) j =1 max { λ j , } + (cid:96), µ n = n (cid:88) j =1 min { λ j , } − (cid:96), in which case the multiplicity is precisely the number of ordered ( n − (cid:96) , which is (cid:18) (cid:96) + n − n − (cid:19) . (cid:3) With this in hand, we can now explicitly determine the right-hand side of (6.2).
RCHIMEDEAN NEWFORM THEORY FOR GL n Lemma 6.6.
Suppose that the restriction of τ µ ∈ (cid:91) U( n ) to U( n − contains the trivial repre-sentation. Then the highest weight of τ µ is of the form µ = ( µ , , . . . , , µ n ) ∈ Λ n , the trivialrepresentation occurs with multiplicity one, and dim Hom U(1) n τ µ | U(1) n , n (cid:2) j =1 τ κ j = (cid:18) (cid:96) + n − n − (cid:19) if µ = n (cid:88) j =1 max { κ j , } + (cid:96), , . . . , , n (cid:88) j =1 min { κ j , } − (cid:96) for some (cid:96) ∈ N , otherwise.Proof. This is a direct consequence of Lemma 6.5 together with Corollary 6.4. (cid:3)
Proofs of Theorems 4.7, 4.12, 4.14, and 4.15 for F = C . Lemmata 6.1 and 6.6 combine to com-plete the proofs of Theorems 4.7 and 4.12 for F = C , noting that for π = π (cid:1) · · · (cid:1) π n with π j = χ κ j | · | t j , the newform K -type τ ◦ = τ µ ◦ has highest weight µ ◦ = n (cid:88) j =1 max { κ j , } , , . . . , , n (cid:88) j =1 min { κ j , } , so that, recalling the definition (4.2) of deg τ ◦ , c ( π ) := deg τ ◦ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) j =1 max { κ j , } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) j =1 min { κ j , } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = n (cid:88) j =1 (cid:107) κ j (cid:107) . Theorem 4.14 then holds for F = C via the fact that ε ( s, π, ψ ) = n (cid:89) j =1 ε ( s, π j , ψ ) = n (cid:89) j =1 i −(cid:107) κ j (cid:107) = i −(cid:107) κ (cid:107)−···−(cid:107) κ n (cid:107) = i − c ( π ) , recalling (2.18), while the case n = 1 implies Theorem 4.15 for F = C . (cid:3) Remark . Note that deg τ µ ( (cid:96) ) = deg τ ◦ + 2 (cid:96) for µ ( (cid:96) ) := n (cid:88) j =1 max { κ j , } + (cid:96), , . . . , , n (cid:88) j =1 min { κ j , } − (cid:96) , and in particular that deg τ µ ( (cid:96) ) ≡ deg τ ◦ (mod 2). This congruence holds not just for a K -type τ of π for which Hom U( n − (1 , τ | U( n − ) is nontrivial, but for any K -type of π ; see [Fan18,Theorem 2.1]. Remark . The minimal K -type of π has highest weight µ = ( κ σ (1) , . . . , κ σ ( n ) ) , where σ is a permutation for which κ σ (1) ≥ · · · ≥ κ σ ( n ) . The corresponding Vogan norm is (cid:107) τ µ (cid:107) = n (cid:88) j =1 (cid:0) κ σ ( j ) + n + 1 − j (cid:1) . This minimal K -type is the newform K -type τ ◦ if and only if κ σ (2) = · · · = κ σ ( n − = 0.In general, the Vogan norm of the minimal K -type is not equal to c ( π ). On the other hand,the same cannot be said for the Howe degree: the Howe degree of the minimal K -type is n (cid:88) j =1 (cid:13)(cid:13) κ σ ( j ) (cid:13)(cid:13) = deg τ ◦ = c ( π ) . Branching from GL n ( R ) to O( n ) . Let π = π (cid:1) · · · (cid:1) π r be an induced representationof Whittaker type of GL n ( R ). This is induced from a standard parabolic subgroup P( R ) =P ( n ,...,n r ) ( R ) of GL n ( R ); when n j = 1, π j is of the form χ κ j | · | t j for some κ j ∈ { , } and t j ∈ C ,while when n j = 2, π j is of the form D κ j ⊗ | det | t j for some κ j ≥ t j ∈ C . Lemma 6.9.
For τ ∈ (cid:91) O( n ) , the vector space Hom O( n ) (cid:0) τ, π | O( n ) (cid:1) is isomorphic to r (cid:77) j =1 n j =2 ∞ (cid:77) (cid:96) j = κ j (cid:96) j ≡ κ j (mod 2) Hom O( n ) ×···× O( n r ) τ | O( n ) ×···× O( n r ) , r (cid:2) j =1 n j =1 τ κ j (cid:2) r (cid:2) j =1 n j =2 τ ( (cid:96) j , . Here we view O( n ) × · · · × O( n r ) as a subgroup of block-diagonal matrices in O( n ); it is themaximal compact subgroup of the Levi subgroup M P ( R ) ∼ = GL n ( R ) × · · · × GL n r ( R ) of theparabolic subgroup P( R ) = P ( n ,...,n r ) ( R ). Proof.
Mackey’s restriction-induction formula implies that π | O( n ) ∼ = Ind O( n )O( n ) ×···× O( n r ) π | O( n ) ×···× O( n r ) , and Hom O( n ) (cid:0) τ, π | O( n ) (cid:1) is isomorphic toHom O( n ) ×···× O( n r ) (cid:0) τ | O( n ) ×···× O( n r ) , π | O( n ) ×···× O( n r ) (cid:1) by the Frobenius reciprocity theorem. Note that if n j = 1, so that π j = χ κ j | · | t j , then π j | O(1) ∼ = τ κ j , while if n j = 2, so that π j = D κ j ⊗ | det | t j , then π j | O(2) ∼ = ∞ (cid:77) (cid:96) j = κ j (cid:96) j ≡ κ j (mod 2) τ ( (cid:96) j , , from which it follows that π | O( n ) ×···× O( n r ) ∼ = r (cid:77) j =1 n j =2 ∞ (cid:77) (cid:96) j = κ j (cid:96) j ≡ κ j (mod 2) r (cid:2) j =1 n j =1 τ κ j (cid:2) r (cid:2) j =1 n j =2 τ ( (cid:96) j , . (cid:3) By restricting in stages and using the fact that { j : n j = 1 } = 2 r − n and { j : n j = 2 } = n − r , we deduce the following. Corollary 6.10.
For τ ∈ (cid:91) O( n ) , dim Hom O( n ) (cid:0) τ, π | O( n ) (cid:1) is equal to (6.11) r (cid:88) j =1 n j =2 ∞ (cid:88) (cid:96) j = κ j (cid:96) j ≡ κ j (mod 2) (cid:88) ν n − r ) ∈ Λ n − r ) dim Hom O(2) n − r τ ν n − r ) | O(2) n − r , r (cid:2) j =1 n j =2 τ ( (cid:96) j , × dim Hom O(2( n − r )) × O(1) r − n τ | O(2( n − r )) × O(1) r − n , τ ν n − r ) (cid:2) r (cid:2) j =1 n j =1 τ κ j . To understand (6.11), which involves branching from O( n ) to various subgroups, we makeuse of the following branching law from O( n ) to O( n − × O(1).
Lemma 6.12 ([Pro94, Proposition 10.1]) . For τ µ ∈ (cid:91) O( n ) of highest weight µ = ( µ , . . . , µ n ) ∈ Λ n , τ µ | O( n − × O(1) ∼ = (cid:77) λ ∈ Λ (cid:77) ν =( ν ,...,ν n − ) ∈ Λ n − µ ≥ ν ≥ µ ≥···≥ ν n − ≥ µ n (cid:80) n − j =1 ν j ≡ (cid:80) nj =1 µ j − λ (mod 2) τ ν (cid:2) τ λ . RCHIMEDEAN NEWFORM THEORY FOR GL n In particular, τ µ | O( n − ∼ = (cid:77) ν =( ν ,...,ν n − ) ∈ Λ n − µ ≥ ν ≥ µ ≥···≥ ν n − ≥ µ n τ ν . Corollary 6.13.
The restriction to O( n − of the irreducible representation τ µ ∈ (cid:91) O( n ) ofhighest weight µ = ( µ , . . . , µ n ) ∈ Λ n contains the trivial representation if and only if µ ≥ and µ = · · · = µ n = 0 , in which case the trivial representation occurs with multiplicity one. Now we iterate the branching law in Lemma 6.12.
Lemma 6.14.
For τ µ ∈ (cid:91) O( n ) of highest weight µ = ( µ , , . . . , ∈ Λ n , and for λ , . . . , λ r − n ∈ Λ and ν n − r ) ∈ Λ n − r ) , dim Hom O(2( n − r )) × O(1) r − n τ µ | O(2( n − r )) × O(1) r − n , τ ν n − r ) (cid:2) r − n (cid:2) j =1 τ λ j is equal to (cid:26) if ν n − r ) = ( µ − λ − (cid:96), , . . . , for some (cid:96) ∈ N , otherwiseif { j : n j = 1 } = 1 , while if { j : n j = 1 } ≥ , this is equal to (cid:18) (cid:96) + n (cid:48) − n (cid:48) − (cid:19) if ν n − r ) = µ − r − n (cid:88) j =1 λ j − (cid:96), , . . . , for some (cid:96) ∈ N , otherwise.Proof. We take µ = ( µ , , . . . ,
0) in Lemma 6.12; the case { j : n j = 1 } = 1 is then immediate,while if { j : n j = 1 } ≥
2, we iterate this branching law in order to find that τ µ | O(2( n − r )) × O(1) r − n ∼ = r − n (cid:77) j =1 (cid:77) λ j ∈ Λ (cid:77) ( ν j, , ,..., ∈ Λ n − j ν j, ≤ ν j − , ν j, ≡ ν j − , − λ j (mod 2) τ ( ν n (cid:48) , , ,..., (cid:2) r − n (cid:2) j =1 τ λ j , where we set ν , := µ . It follows that for fixed µ = ( µ , , . . . , ∈ Λ n , λ , . . . , λ r − n ∈ Λ ,and ν n − r ) ∈ Λ n − r ) , the multiplicity of τ ν n − r ) (cid:2) τ λ (cid:2) · · · (cid:2) τ λ r − n in τ µ is zero unless ν n − r ) is of the form ( ν n (cid:48) , , , . . . ,
0) for some ν n (cid:48) , ∈ N , in which case it is equal to (cid:110) ( ν , , . . . , ν n (cid:48) − , ) ∈ N n (cid:48) − : ν j − , ≥ ν j, , ν j, ≡ ν j − , − λ j (mod 2) for all j ∈ { , . . . , n (cid:48) } (cid:111) , namely the number of ( n (cid:48) − ν , , . . . , ν n (cid:48) − , ) for which the system of inequalities µ ≥ ν , + λ ≥ · · · ≥ ν n (cid:48) , + r − n (cid:88) j =1 λ j holds with each quantity being of the same parity. This is zero unless µ = ν n (cid:48) , + r − n (cid:88) j =1 λ j + 2 (cid:96) for some (cid:96) ∈ N , in which case the multiplicity is precisely the number of ordered ( n (cid:48) − (cid:96) , which is (cid:18) (cid:96) + n (cid:48) − n (cid:48) − (cid:19) . (cid:3) We also require a special case of the branching law from O( n ) to O( n − × O(2).
Lemma 6.15 ([Pro94, Proposition 10.3]) . For n ≥ and τ µ ∈ (cid:91) O( n ) of highest weight µ =( µ , , . . . , ∈ Λ n , τ µ | O( n − × O(2) ∼ = (cid:77) λ =( λ , ∈ Λ λ ≤ µ (cid:77) ν =( ν , ,..., ∈ Λ n − ν ≤ µ − λ ν ≡ µ − λ (mod 2) τ ν (cid:2) τ λ . We again iterate this branching law.
Lemma 6.16.
For τ µ ∈ (cid:92) O(2( n − r )) of highest weight µ = ( µ , , . . . , ∈ Λ n − r ) and for ( λ j, , λ j, ) ∈ Λ with j ∈ { , . . . , n − r } , dim Hom O(2) n − r τ µ | O(2) n − r , n − r (cid:2) j =1 τ ( λ j, ,λ j, ) = if { j : n j = 2 } = 1 and ( λ , , λ , ) = ( µ , , (cid:18) (cid:96) + n − r − n − r − (cid:19) if { j : n j = 2 } ≥ , µ = n − r (cid:88) j =1 λ j, + 2 (cid:96) for some (cid:96) ∈ N ,and λ j, = 0 for all j ∈ { , . . . , n − r } , otherwise.Proof. The result follows from Schur’s lemma if { j : n j = 2 } = 1. If { j : n j = 2 } ≥ τ µ | O(2) n − r ∼ = n − r − (cid:77) j =1 (cid:77) ( λ j, , ∈ Λ λ j, ≤ ν j − , (cid:77) ( ν j, , ,..., ∈ Λ n − r − j ) ν j, ≤ ν j − , − λ j, ν j, ≡ ν j − , − λ j, (mod 2) n − r (cid:2) j =1 τ ( λ j, ,λ j, ) , where we again set ν , := µ . So the multiplicity of τ ( λ , ,λ , ) (cid:2) · · · (cid:2) τ ( λ n − r, ,λ n − r, ) in τ µ isequal to the number of ( n − r − ν , , . . . , ν n − r − ) for which the system of inequalities µ ≥ ν , + λ , ≥ ν , + λ , + λ , ≥ · · · ≥ ν n − r − , + n − r − (cid:88) j =1 λ j, ≥ n − r (cid:88) j =1 λ j, holds with each quantity being of the same parity. This is zero unless µ = n − r (cid:88) j =1 λ j, + 2 (cid:96) for some (cid:96) ∈ N , in which case the multiplicity is precisely the number of ordered ( n − r − (cid:96) , which is (cid:18) (cid:96) + n − r − n − r − (cid:19) . (cid:3) Shortly we shall require the following combinatorial identity involving binomial coefficients.
Lemma 6.17.
For k, m, n ∈ N , we have that k (cid:88) j =0 (cid:18) j + mm (cid:19)(cid:18) k − j + nn (cid:19) = (cid:18) k + m + n + 1 m + n + 1 (cid:19) . RCHIMEDEAN NEWFORM THEORY FOR GL n Proof.
Such an identity begs for a proof via generating series: ∞ (cid:88) k =0 (cid:18) k + m + n + 1 m + n + 1 (cid:19) x k = 1(1 − x ) m + n +2 = 1(1 − x ) m +1 − x ) n +1 = ∞ (cid:88) k =0 (cid:18) k + mm (cid:19) x k ∞ (cid:88) k =0 (cid:18) k + nn (cid:19) x k = ∞ (cid:88) k =0 k (cid:88) j =0 (cid:18) j + mm (cid:19)(cid:18) k − j + nn (cid:19) x k . (cid:3) With these results in hand, we can now explicitly determine (6.11).
Lemma 6.18.
Suppose that the restriction of τ µ ∈ (cid:91) O( n ) to O( n − contains the trivialrepresentation. Then the highest weight of τ µ is of the form µ = ( µ , , . . . , , the trivialrepresentation occurs with multiplicity one, and (6.19) r (cid:88) j =1 n j =2 ∞ (cid:88) (cid:96) j = κ j (cid:96) j ≡ κ j (mod 2) (cid:88) ν n − r ) ∈ Λ n − r ) dim Hom O(2) n − r τ ν n − r ) | O(2) n − r , r (cid:2) j =1 n j =2 τ ( (cid:96) j , × dim Hom O(2( n − r )) × O(1) r − n τ | O(2( n − r )) × O(1) r − n , τ ν n − r ) (cid:2) r (cid:2) j =1 n j =1 τ κ j = (cid:18) (cid:96) + n − n − (cid:19) if µ = r (cid:88) j =1 κ j + 2 (cid:96), , . . . , for some (cid:96) ∈ N , otherwise.Proof. The first claim two claims are Corollary 6.13. To prove the identity (6.19), we first notethat if { j : n j = 2 } = 0, Lemma 6.14 implies the result upon replacing 2 r − n with 2 r − n − { j : n j = 2 } ≥
1, we combine Lemmata 6.14 and 6.16 to see that the inner sum over ν n − r ) ∈ Λ n − r ) in the left-hand side of (6.19) is equal to zero unless µ = r (cid:88) j =1 n j =1 κ j + r (cid:88) j =1 n j =2 (cid:96) j + 2 (cid:96) for some (cid:96) ∈ N , so that ν n − r ) = r (cid:88) j =1 n j =2 (cid:96) j + 2( (cid:96) − (cid:96) (cid:48) ) , , . . . , for some (cid:96) (cid:48) ∈ { , . . . , (cid:96) } , in which case this inner sum is equal to (cid:96) (cid:88) (cid:96) (cid:48) =0 (cid:18) (cid:96) (cid:48) + n (cid:48) − n (cid:48) − (cid:19)(cid:18) (cid:96) − (cid:96) (cid:48) + n − r − n − r − (cid:19) . Coupled with Corollary 6.10, we find that the left-hand side of (6.19) is equal to zero unless µ = r (cid:88) j =1 κ j + 2 (cid:96) for some (cid:96) ∈ N , in which case it is equal to n − r (cid:88) i =1 ∞ (cid:88) α i =0 (cid:96) − (cid:80) n − ri =1 α i (cid:88) (cid:96) (cid:48) =0 (cid:18) (cid:96) (cid:48) + 2 r − n − r − n − (cid:19)(cid:18) (cid:96) − (cid:96) (cid:48) − (cid:80) n − ri =1 α i + n − r − n − r − (cid:19) upon defining α i such that there is an equality of the sets { α i } and { ( (cid:96) j − κ j ) / n j = 2 } .Interchanging the order of summation, this becomes(6.20) (cid:96) (cid:88) (cid:96) (cid:48) =0 (cid:18) (cid:96) (cid:48) + 2 r − n − r − n − (cid:19) n − r (cid:88) i =1 (cid:96) − (cid:96) (cid:48) − (cid:80) i − m =1 α m (cid:88) α i =0 (cid:18) (cid:96) − (cid:96) (cid:48) − (cid:80) n − ri =1 α i + n − r − n − r − (cid:19) . Using the fact that (cid:0) n − k − (cid:1) = (cid:0) nk (cid:1) − (cid:0) n − k (cid:1) , the sum over α n − r telescopes to (cid:18) (cid:96) − (cid:96) (cid:48) − (cid:80) n − r − i =1 α i + n − r − n − r − (cid:19) , and so upon iterating this process, we find that the inner double sum in (6.20) is equal to (cid:18) (cid:96) − (cid:96) (cid:48) + 2 n − r − n − r − (cid:19) , at which point Lemma 6.17 completes the proof. (cid:3) Proofs of Theorems 4.7, 4.12, 4.14, 4.15, and 4.16 for F = R . Lemmata 6.9 and 6.18 combineto complete the proofs of Theorems 4.7 and 4.12 for F = R : for π = π (cid:1) · · · (cid:1) π r with π j = χ κ j | · | t j when n j = 1 and π j = D κ j ⊗ | det | t j when n j = 2, the newform K -type τ ◦ = τ µ ◦ has highest weight µ ◦ = r (cid:88) j =1 κ j , , . . . , , so that, recalling the definition (4.4) of deg τ ◦ , c ( π ) := deg τ ◦ = r (cid:88) j =1 κ j . Theorem 4.14 then holds for F = R via the fact that ε ( s, π, ψ ) = r (cid:89) j =1 ε ( s, π j , ψ ) = r (cid:89) j =1 i − κ j = i − κ −···− κ r = i − c ( π ) , while the cases n = { j : n j = 1 } = 1, so that π = χ κ | · | t , and n = 2 { j : n j = 2 } = 2, sothat π = D κ ⊗ | det | t , imply Theorem 4.15 for F = R . Finally, for an induced representation ofWhittaker type of GL n ( C ) of the form π = χ κ | · | t C (cid:1) · · · (cid:1) χ κ n | · | t n C , the induced representation AI C / R π is isomorphic to the isobaric sum n (cid:1) j =1 Ind GL ( R )GL ( C ) χ κ j | · | t j C , and Ind GL ( R )GL ( C ) χ κ | · | t C ∼ = (cid:40) | · | t R (cid:1) χ | · | t R if κ = 0, D (cid:107) κ (cid:107) +1 ⊗ | det | t R if κ (cid:54) = 0,so that c (cid:0) AI C / R π (cid:1) = n (cid:88) j =1 ( (cid:107) κ j (cid:107) + 1) = c ( π ) + n, thereby proving Theorem 4.16. (cid:3) Remark . Just as was observed in Remark 6.7 for F = C , the Howe degree of a K -type of π is always congruent to deg τ ◦ modulo 2. RCHIMEDEAN NEWFORM THEORY FOR GL n Remark . From [Lin18, Proposition 4.3], the minimal K -type of π = π (cid:1) · · · (cid:1) π r has highestweight µ = κ σ (1) , . . . , κ σ ( r ) , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n − r times , where σ is a permutation for which κ σ (1) ≥ · · · ≥ κ σ ( r ) . This minimal K -type is τ ◦ if and onlyif r = n − κ j = 0 whenever n j = 1 or r = n and κ j (cid:54) = 0 for at most one j .Once again, the Howe degree of the minimal K -type is n (cid:88) j =1 κ σ ( j ) = deg τ ◦ = c ( π ) . Homogeneous Harmonic Polynomials
Having identified the newform K -type, we now study a particular model, a space of homoge-neous harmonic polynomials, of this representation of K . This allows us to explicitly describethe matrix coefficients of this representation, which are used to construct the explicit Schwartzfunctions Φ ◦ ∈ S (Mat × n ( F )) and Φ ∈ S (Mat n × n ( F )) given in (4.19) and (4.23).7.1. Homogeneous Harmonic Polynomials and Representations of U( n ) . For nonnega-tive integers p, q , let H p,q ( C n ) denote the vector space consisting of harmonic polynomials thatare homogeneous of bidegree ( p, q ), namely the set of polynomials P ( z ) = P ( z , . . . , z n , z , . . . , z n )in z ∈ Mat × n ( C ) = C n that are annihilated by the Laplacian∆ = 4 n (cid:88) j =1 ∂ ∂z j ∂z j and satisfy P ( λz ) = P (cid:0) λz , . . . , λz n , λz , . . . , λz n (cid:1) = λ p λ q P ( z , . . . , z n , z , . . . , z n )for all λ ∈ C . The dimension of H p,q ( C n ) is 1 for n = 1 and( p + q + n − p + n − q + n − p ! q !( n − n − p + q + n − n − (cid:18) p + n − n − (cid:19)(cid:18) q + n − n − (cid:19) for n ≥ τ be an irreducible representation of U( n ) of highest weight µ = ( p, , . . . , , − q ); notethat for n = 1, either p or q must be zero. Then H p,q ( C n ) is a model of τ , where the groupU( n ) (cid:51) k acts on H p,q ( C n ) (cid:51) P via right translation, namely( τ ( k ) · P ) ( z , . . . , z n , z , . . . , z n ) := P (cid:0)(cid:0) z · · · z n (cid:1) k, (cid:0) z · · · z n (cid:1) k (cid:1) ;that is, ( τ ( k ) · P )( z ) := P ( zk ). We define a U( n )-invariant inner product on H p,q ( C n ) (cid:51) P, Q by (cid:104) P, Q (cid:105) := (cid:90) U( n ) P ( e n k ) Q ( e n k ) dk. From the branching law in Lemma 6.3, there is a one-dimensional subspace of H p,q ( C n ) thatis invariant under the action of U( n − (cid:51) k (cid:48) embedded in U( n ) via k (cid:48) (cid:55)→ (cid:0) k (cid:48)
00 1 (cid:1) , which we candescribe explicitly.
Lemma 7.1 ([Rud08, Proposition 12.2.6]) . There exists a unique homogeneous harmonic poly-nomial P ◦ ∈ H p,q ( C n ) satisfying P ◦ ( e n ) = 1 and τ (cid:0) k (cid:48)
00 1 (cid:1) · P ◦ = P ◦ for all k (cid:48) ∈ U( n − ,namely (7.2) P ◦ ( z ) := min { p,q } (cid:88) ν =0 ( − ν (cid:0) pν (cid:1)(cid:0) qν (cid:1)(cid:0) ν + n − n − (cid:1) ( z z + · · · + z n − z n − ) ν z p − νn z nq − ν . In particular, for n = 1 , so that either p or q is equal to , (7.3) P ◦ ( z ) = (cid:40) z p for p ∈ N and q = 0 , z q for p = 0 and q ∈ N .Remark . When restricted to the unit sphere in C n , this polynomial is sometimes referred toas the zonal spherical harmonic of bidegree ( p, q ) in dimension n . When z z + · · · + z n − z n − =1 − z n z n , P ◦ ( z , . . . , z n , z , . . . , z n ) is a polynomial in z n , z n and can be expressed in terms of thegeneralised Zernike polynomial P n − q,p (also called a generalised disc polynomial) or the Jacobipolynomial P ( n − ,p − q ) q .We make crucial use of the fact that for all P ∈ H p,q ( C n ) and k ∈ U( n ), P ( e n k ) is equal to amatrix coefficient of τ . This can be thought of as an explicit form of Schur orthogonality. Lemma 7.5 ([Rud08, Theorem 12.2.5]) . The reproducing kernel for H p,q ( C n ) is (dim τ ) P ◦ , sothat for all P ∈ H p,q ( C n ) and k ∈ U( n ) , P ( e n k ) = (dim τ ) (cid:104) τ ( k ) · P, P ◦ (cid:105) . In particular, (cid:104) P ◦ , P ◦ (cid:105) = (dim τ ) − . Moreover, for all P ∈ H p,q ( C n ) and z ∈ C n , P ( z ) = dim τ (cid:90) U( n ) P ( e n k − ) P ◦ ( zk ) dk. Finally, we also require the following identity, which states that homogeneous harmonicpolynomials P ∈ H p,q ( C n ) are eigenfunctions of the Fourier transform. Lemma 7.6 (Hecke’s Identity; cf. [SW71, Chapter IV, Theorem 3.4]) . For any homogeneousharmonic polynomial P ∈ H p,q ( C n ) and w ∈ C n , we have that (cid:90) C n P ( z ) exp (cid:0) − πz t z (cid:1) ψ (cid:0) z t w (cid:1) dz = i − p − q P ( w ) exp (cid:0) − πw t w (cid:1) . Proof.
First we prove this for P ( z ) = z p z nq , the highest weight vector of τ . In this case, i − p − q P ( w ) exp (cid:0) − πw t w (cid:1) = ( − πi ) − p − q ∂ p + q ∂w p ∂w qn exp (cid:0) − πw t w (cid:1) = ( − πi ) − p − q ∂ p + q ∂w p ∂w qn (cid:90) C n exp (cid:0) − πz t z (cid:1) ψ (cid:0) z t w (cid:1) dz = (cid:90) C n P ( z ) exp (cid:0) − πz t z (cid:1) ψ (cid:0) z t w (cid:1) dz. For any other Q ∈ H p,q ( C n ), we may write Q ( z ) as a linear combination of elements of the form P ( zk ) with k ∈ U( n ), and so using the above calculation with w replaced by wk and makingthe change of variables z (cid:55)→ zk yields the result upon recalling that k t k = 1 n . (cid:3) Homogeneous Harmonic Polynomials and Representations of O( n ) . Similarly,for a nonnegative integer p , let H p ( R n ) denote the vector space consisting of homogeneousharmonic polynomials of degree p , namely the set of polynomials P ( x ) = P ( x , . . . , x n ) in x ∈ Mat × n ( R ) = R n that are annihilated by the Laplacian∆ = n (cid:88) j =1 ∂ ∂x j and satisfy P ( λx ) = λ p P ( x ) for all λ ∈ R . This space has dimension 1 for n = 1 and p ∈ { , } and has dimension (cid:18) p + n − n − (cid:19) + (cid:18) p + n − n − (cid:19) = (2 p + n − p + n − p !( n − p + n − p + n − (cid:18) p + n − n − (cid:19) for n ≥ p ∈ N .Let τ be an irreducible representation of O( n ) of highest weight µ = ( p, , . . . , p is anonnegative integer; note that p ∈ { , } for n = 1. Then H p ( R n ) is a model of τ , where the group RCHIMEDEAN NEWFORM THEORY FOR GL n O( n ) (cid:51) k acts on the space H p ( R n ) (cid:51) P via right translation, namely ( τ ( k ) · P )( x ) := P ( xk ).We define an O( n )-invariant inner product on H p ( R n ) (cid:51) P, Q by (cid:104) P, Q (cid:105) := (cid:90) O( n ) P ( e n k ) Q ( e n k ) dk. We record the following results, all of which are analogous to those for H p,q ( C n ) in Section7.1. Lemma 7.7 ([AH12, Section 2.1.2]) . There exists a unique homogeneous harmonic polynomial P ◦ ∈ H p ( R n ) satisfying P ◦ ( e n ) = 1 and τ (cid:0) k (cid:48)
00 1 (cid:1) · P ◦ = P ◦ for all k (cid:48) ∈ O( n − , namely (7.8) P ◦ ( x ) := p (cid:88) ν =0 ν ≡ i ν p !Γ (cid:0) n − (cid:1) ν (cid:0) ν (cid:1) !( p − ν )!Γ (cid:0) ν + n − (cid:1) (cid:0) x + · · · + x n − (cid:1) ν x p − νn . In particular, for n = 1 , so that p ∈ { , } , (7.9) P ◦ ( x ) = x p , while for n = 2 , so that p ∈ N , (7.10) P ◦ ( x , x ) = 12 ( x − ix ) p + 12 ( x + ix ) p = p (cid:88) ν =0 ν ≡ (cid:18) pν (cid:19) ( ix ) ν x p − ν . Remark . Atkinson and Han name P ◦ the Legendre polynomial of degree p in n dimensions[AH12, Section 2.1.2]; when restricted to the unit sphere in R n , this polynomial is also referredto as the zonal spherical harmonic. When x + · · · + x n − = 1 − x n , P ◦ ( x , . . . , x n ) is apolynomial in x n and can be expressed in terms of the Gegenbauer polynomial C n − p (also calledan ultraspherical polynomial) or the Jacobi polynomial P ( n − , n − ) p ; in particular, when n = 2,this is just the usual Legendre polynomial of degree p . Lemma 7.12 ([AH12, Section 2.2]) . The reproducing kernel for H p ( R n ) is (dim τ ) P ◦ , so thatfor all P ∈ H p ( R n ) and k ∈ O( n ) , P ( e n k ) = (dim τ ) (cid:104) τ ( k ) · P, P ◦ (cid:105) . In particular, (cid:104) P ◦ , P ◦ (cid:105) = (dim τ ) − . Moreover, for all P ∈ H p ( R n ) and x ∈ R n , P ( x ) = dim τ (cid:90) O( n ) P ( e n k − ) P ◦ ( xk ) dk. Lemma 7.13 (Hecke’s Identity [SW71, Chapter IV, Theorem 3.4]) . For any homogeneousharmonic polynomial P ∈ H p ( R n ) and ξ ∈ R n , we have that (cid:90) R n P ( x ) exp (cid:0) − πx t x (cid:1) ψ ( x t ξ ) dx = i − p P ( ξ ) exp (cid:0) − πξ t ξ (cid:1) . The Newform in the Induced Model
The Induced Model.
Let π = π (cid:1) · · · (cid:1) π r be an induced representation of Whittakertype. Let V π j be the space of π j ; the space V π of π may then be viewed as the space of smoothfunctions f : GL n ( F ) → V π ⊗ · · · ⊗ V π r that satisfy f ( umg ) = δ / ( m ) π ( m ) ⊗ · · · ⊗ π r ( m r ) f ( g ) , for any u ∈ N P ( F ), m = blockdiag( m , . . . , m r ) ∈ M P ( F ), and g ∈ GL n ( F ), where P( F ) =P ( n ,...,n r ) ( F ). The action of π on V π is via right translation, namely ( π ( h ) · f )( g ) := f ( gh ).To make this more explicit, we first describe the space V π j of the essentially square-integrablerepresentation π j of GL n j ( F ). The following result is well-known; see, for example, [GH11,Chapter 7]. Lemma 8.1. (1)
Let π = χ κ | · | t be a character of GL ( F ) = F × . The space V π of π is simply the one-dimensional vector space spanned by the function χ κ ( x ) | x | t . (2) Let π = D κ ⊗ | det | t be an essentially discrete series representation of GL ( R ) . Then π is asubrepresentation of the reducible principal series representation π (cid:93) := | · | t + κ − (cid:1) χ κ | · | t − κ − of GL ( R ) . Moreover, if V π (cid:93) denotes the induced model of π (cid:93) consisting of smooth functions f : GL ( R ) → C that satisfy f ( uag ) = | a | t + κ χ κ ( a ) | a | t − κ f ( g ) for all u ∈ N ( R ) , a = diag( a , a ) ∈ A ( R ) , and g ∈ GL ( R ) , where π (cid:93) acts on V π (cid:93) viaright translation, then the induced model of π is precisely the subspace V π := κ − (cid:92) µ =0 µ ≡ κ (mod 2) ker Π τ ( µ , of V π (cid:93) , where Π τ denotes the projection of V π (cid:93) onto the τ -isotypic component V τπ (cid:93) . Corollary 8.2.
Let π = π (cid:1) · · · (cid:1) π r be an induced representation of Whittaker type. Then theinduced model of π may be taken to be the space V π of smooth functions f : GL n ( F ) × M P ( F ) → C satisfying (8.3) f ( umg ; m (cid:48) ) = δ / ( m ) f ( g ; m (cid:48) m ) for all u ∈ N P ( F ) , m, m (cid:48) ∈ M P ( F ) , and g ∈ GL n ( F ) and such that for each g ∈ GL n ( F ) , f ( g ; · ) : M P ( F ) → C is an element of V π ⊗ · · · ⊗ V π r with V π j as in Lemma 8.1. For f ∈ V π , we write f ( g ) to denote f ( g ; 1 n ). Example . Suppose that π = χ κ | · | t (cid:1) · · · (cid:1) χ κ n | · | t n is a principal series representation, sothat κ j ∈ Z for F = C for F = R and κ j ∈ { , } . The induced model of π is the vector space V π of smooth functions f : GL n ( F ) → C that satisfy f ( uag ) = f ( g ) δ / n ( a ) n (cid:89) j =1 χ κ j ( a j ) | a j | t j for all u ∈ N n ( F ), a = diag( a , . . . , a n ) ∈ A n ( F ), and g ∈ GL n ( F ).Our goal now is to explicitly describe the newform f ◦ in the induced model V π of an inducedrepresentation of Whittaker type π . We give three different explicit constructions: via theIwasawa decomposition, via convolution sections, and via Godement sections. Initially, wedefine the newform in the induced model only up to multiplication by a nonzero constant;eventually in Definitions 8.10 and 9.2 we specify a normalisation that is particularly useful whenproceeding to study the newform in the Whittaker model.8.2. The Newform via the Iwasawa Decomposition.
Essentially Square-Integrable Representations.
We first describe the newform in the in-duced model of essentially square-integrable representations.
Lemma 8.5. (1)
For π = χ κ | · | t , the newform in the induced model is simply (8.6) f ◦ ( x ) = c ◦ χ κ ( x ) | x | t for any c ◦ ∈ C × . (2) For F = R and π = D κ ⊗ | det | t , the newform in the induced model of π is (8.7) f ◦ ( g ) = c ◦ | a | t + κ χ κ ( a ) | a | t − κ P ◦ ( e k − ) for any c ◦ ∈ C × and g ∈ GL ( R ) having the Iwasawa decomposition g = uak with u ∈ N ( R ) , a = diag( a , a ) ∈ A ( R ) , and k ∈ O(2) , where P ◦ is the homogeneous harmonic polynomialassociated to the newform K -type τ ◦ given by (7.10) . RCHIMEDEAN NEWFORM THEORY FOR GL n Remark . Strictly speaking, there is no need to write P ◦ instead of P ◦ , since this is real-valued; we do this simply to ensure notational consistency when later treating the cases F = R and F = C simultaneously, for the distinction is no longer moot in the latter case. Proof.
This is clear for π = χ κ | · | t . For π = D κ ⊗ | det | t , we must first check that f ◦ is well-defined, for the Iwasawa decomposition is not unique as A ( R ) and O(2) intersect nontrivially.If a (cid:48) = diag( a (cid:48) , a (cid:48) ) ∈ A ( R ) ∩ O(2), so that a (cid:48) , a (cid:48) ∈ { , − } , then on the one hand, f ◦ ( uaa (cid:48) k ) = c ◦ | a | t + κ χ κ ( a (cid:48) ) χ κ ( a ) | a | t − κ P ◦ ( e k − )since a (cid:48) ∈ A ( R ) with | a (cid:48) | = | a (cid:48) | = 1, while on the other hand, f ◦ ( uaa (cid:48) k ) = c ◦ | a | t + κ χ κ ( a ) | a | t − κ P ◦ ( e a (cid:48) k − )since a (cid:48) ∈ O(2), and these are equal since e a (cid:48) = a (cid:48) e , P ◦ is homogeneous of degree κ as τ ◦ = τ ( κ, , so that P ◦ ∈ H κ ( R ), and a (cid:48) κ = χ κ ( a (cid:48) ) as a (cid:48) ∈ { , − } .Next, Schur orthogonality shows that f ◦ ∈ ker Π τ ( µ , for 0 ≤ µ ≤ κ − µ ≡ κ (mod 2)since P ◦ ∈ H κ ( R ), and so f ◦ is indeed an element of the induced model V π of π as defined inLemma 8.1.Finally, to prove that f ◦ is the newform, we must show that (cid:90) O(2) ξ τ ◦ | O(1) ( k ) π ( k ) · f ◦ ( g ) dk = f ◦ ( g ) . From the definition (8.7) of f ◦ , it suffices to show that(8.9) (cid:90) O(2) ξ τ ◦ | O(1) ( k ) P ◦ (cid:0) e k − k (cid:48)− (cid:1) dk = P ◦ (cid:0) e k (cid:48)− (cid:1) for any k (cid:48) ∈ O(2). We note that ξ τ ◦ | O(1) ( k ) = (dim τ ◦ ) (cid:10) τ ◦ ( k − ) · P ◦ , P ◦ (cid:11) (cid:104) P ◦ , P ◦ (cid:105) = (dim τ ◦ ) P ◦ ( e k − ) , where the first equality follows from the definitions (4.6) of ξ τ ◦ | O(1) and (7.10) of P ◦ , while thesecond follows from Lemma 7.12. The equality (8.9) then follows from one more application ofLemma 7.12. (cid:3) For our applications, we require explicit choices of the constants c ◦ appearing in (8.6) and(8.7). Definition 8.10.
Let π be an essentially square-integrable representation of GL n ( F ), and let f ◦ denote the newform in the induced model, which is given by (8.6) if n = 1, so that π = χ κ | · | t ,and by (8.7) if n = 2, so that F = R and π = D κ ⊗ | det | t . We say that f ◦ is canonicallynormalised if(8.11) c ◦ = (cid:40) π = χ κ | · | t , i κ ζ R ( κ ) ζ R ( κ + 1) if π = D κ ⊗ | det | t .8.2.2. Induced Representations of Whittaker Type.
We now give an explicit construction of thenewform f ◦ : GL n ( F ) × M P ( F ) → C in the induced model of an induced representation ofWhittaker type π = π (cid:1) · · · (cid:1) π r of GL n ( F ) when g ∈ GL n ( F ) is written in terms of its Iwasawadecomposition.This description involve a distinguished homogeneous harmonic polynomial P ◦ ( n ,...,n r ) that isdefined in terms of polynomials P ◦ j in the following way. To each essentially square-integrablerepresentation π j of GL n j ( F ) with newform K -type τ ◦ j , we associate a distinguished homogeneousharmonic polynomial P ◦ j . • For F = R , n j = 1, π j = χ κ j | · | t j R , and τ ◦ j the one-dimensional representation of O(1)of highest weight κ j ∈ { , } , P ◦ j ∈ H p ( R ) is the homogeneous harmonic polynomialassociated to τ = τ ◦ j given by (7.9) with p = κ j . • For F = R , n j = 2, π j = D κ j ⊗ | det | t j R , and τ ◦ j the two-dimensional representation of O(2)of highest weight ( κ j ,
0) with κ j ≥ P ◦ j ∈ H p ( R ) is the homogeneousharmonic polynomial associated to τ = τ ◦ j given by (7.10) with p = κ j . • For F = C , so that n j = 1, π j = χ κ j | · | t j C , and τ ◦ j the one-dimensional representation ofU(1) of highest weight κ j ∈ Z , P ◦ j ∈ H p,q ( C ) is the homogeneous harmonic polynomialassociated to τ = τ ◦ j given by (7.3) with p = max { κ j , } and q = − min { κ j , } .We then define P ◦ ( n ,...,n r ) ( x ) := r (cid:89) j =1 (cid:40) P ◦ j ( x (cid:96) ) if (cid:96) = n + · · · + n j and π j = χ κ j | · | t j R , P ◦ j ( x (cid:96) , x (cid:96) +1 ) if (cid:96) = n + · · · + n j − π j = D κ j ⊗ | det | t j R (8.12)for F = R , while for F = C , we define P ◦ ( n ,...,n r ) ( z ) := r (cid:89) j =1 P ◦ j ( z j ) . (8.13)It is straightforward to see that the polynomials P ◦ ( n ,...,n r ) ( x ) and P ◦ ( n ,...,n r ) ( z ) are elements of H µ ◦ ( R n ) and H µ ◦ , − µ ◦ n ( C n ) respectively, where µ ◦ = ( µ ◦ , . . . , µ ◦ n ) is the highest weight of thenewform K -type τ ◦ = τ µ ◦ of π = π (cid:1) · · · (cid:1) π r .The description of the newform f ◦ in the induced model of π = π (cid:1) · · · (cid:1) π r also involves thecanonically normalised newforms f ◦ , . . . , f ◦ r of the essentially square-integrable representations π , . . . , π r . Proposition 8.14.
Let π = π (cid:1) · · · (cid:1) π r be an induced representation of Whittaker type with r ≥ . For g ∈ GL n ( F ) having the Iwasawa decomposition g = umk with respect to the parabolicsubgroup P( F ) = P ( n ,...,n r ) ( F ) , so that u ∈ N P ( F ) , m = blockdiag( m , . . . , m r ) ∈ M P ( F ) , and k ∈ K , and for m (cid:48) = blockdiag( m (cid:48) , . . . , m (cid:48) r ) ∈ M P ( F ) , the newform f ◦ : GL n ( F ) × M P ( F ) → C in the induced model V π of π is of the form (8.15) f ◦ ( g ; m (cid:48) ) := c ◦ dim τ ◦ · · · dim τ ◦ r δ / ( m ) (cid:90) K n · · · (cid:90) K nr f ◦ (cid:0) m (cid:48) m k (cid:1) · · · f ◦ r (cid:0) m (cid:48) r m r k r (cid:1) × P ◦ ( n ,...,n r ) (cid:0) e n k − blockdiag ( k , . . . , k r ) (cid:1) dk r · · · dk for some constant c ◦ ∈ C × , where each f ◦ j is the canonically normalised newform of π j and τ ◦ j is the newform K n j -type of π j .Proof. Since the Iwasawa decomposition is not unique as M P ( F ) and K intersect nontrivially,our first task is to show that f ◦ ( umm (cid:48)(cid:48) k ; m (cid:48) ) is well-defined for m (cid:48)(cid:48) = blockdiag( m (cid:48)(cid:48) , . . . , m (cid:48)(cid:48) r ) ∈ M P ( F ) ∩ K . On the one hand, this is c ◦ dim τ ◦ · · · dim τ ◦ r δ / ( m ) (cid:90) K n · · · (cid:90) K nr f ◦ (cid:0) m (cid:48) m m (cid:48)(cid:48) k (cid:1) · · · f ◦ r (cid:0) m (cid:48) r m r m (cid:48)(cid:48) r k r (cid:1) × P ◦ ( n ,...,n r ) (cid:0) e n k − blockdiag ( k , . . . , k r ) (cid:1) dk r · · · dk since m (cid:48)(cid:48) ∈ M P ( F ), noting that δ P ( m (cid:48)(cid:48) ) = 1 as m (cid:48)(cid:48) ∈ K . On the other hand, this is c ◦ dim τ ◦ · · · dim τ ◦ r δ / ( m ) (cid:90) K n · · · (cid:90) K nr f ◦ (cid:0) m (cid:48) m k (cid:1) · · · f ◦ r (cid:0) m (cid:48) r m r k r (cid:1) × P ◦ ( n ,...,n r ) (cid:16) e n k − m (cid:48)(cid:48)− blockdiag ( k , . . . , k r ) (cid:17) dk r · · · dk since m (cid:48)(cid:48) ∈ K ; as m (cid:48)(cid:48) j ∈ K n j , this is seen to be equal to the first expression upon making thechange of variables k j (cid:55)→ m (cid:48)(cid:48) j k j .Next, we confirm that this is an element of the induced model V π of π . It is clear that f ◦ is asmooth function from GL n ( F ) × M P ( F ) to C that satisfies (8.3). Moreover, f ◦ ( g ; · ) is indeed anelement of V π ⊗ · · · ⊗ V π r for each g ∈ GL n ( F ), since upon writing P ◦ ( n ,...,n r ) = (cid:81) rj =1 P ◦ j , theintegrals over K n j (cid:51) k j are either trivial if n j = 1, or lead to f ◦ j P ◦ j being replaced with the sum RCHIMEDEAN NEWFORM THEORY FOR GL n of two such products of elements of V π j and homogeneous harmonic polynomials, for we may useSchur orthogonality for the two-dimensional representation τ ◦ j = τ ( κ j , for π j = D κ j ⊗ | det | t j .Finally, we show that this is the newform, which requires confirming that (cid:90) K n ξ τ ◦ | Kn − ( k ) π ( k ) · f ◦ ( g ; m (cid:48) ) dk = f ◦ ( g ; m (cid:48) ) . From the definition (8.15) of f ◦ together with the Iwasawa decomposition, it suffices to showthat for each k (cid:48) ∈ K n ,(8.16) (cid:90) K n ξ τ ◦ | Kn − ( k ) P ◦ ( n ,...,n r ) (cid:0) e n k − k (cid:48)− (cid:1) dk = P ◦ ( n ,...,n r ) (cid:0) e n k (cid:48)− (cid:1) . We note that ξ τ ◦ | Kn − ( k ) = (dim τ ◦ ) (cid:10) τ ◦ ( k − ) · P ◦ , P ◦ (cid:11) (cid:104) P ◦ , P ◦ (cid:105) = (dim τ ◦ ) P ◦ ( e n k − ) , where the first equality follows from the definitions (4.6) of ξ τ ◦ | Kn − and (7.2) and (7.8) of P ◦ ,while the second follows from Lemmata 7.5 and 7.12. The equality (8.16) then follows from onemore application of Lemmata 7.5 and 7.12. (cid:3) Proposition 8.14 completely prescribes the behaviour of the newform f ◦ ( g ) := f ◦ ( g ; 1 n ) when g = uak is given by the Iwasawa decomposition with respect to the standard Borel subgroup. Corollary 8.17.
For u ∈ N n ( F ) , a = diag( a , . . . , a n ) ∈ A n ( F ) , and g ∈ GL n ( F ) , the newformin the induced model satisfies (8.18) f ◦ ( uag ) = f ◦ ( g ) δ / n ( a ) r (cid:89) j =1 χ κ j ( a (cid:96) ) | a (cid:96) | t j if (cid:96) = n + · · · + n j and π j = χ κ j | · | t j , χ κ j ( a (cid:96) +1 ) | a (cid:96) | t j + κj − | a (cid:96) +1 | t j − κj − if (cid:96) = n + · · · + n j − and π j = D κ j ⊗ | det | t j ,and for k ∈ K n , f ◦ ( k ) = c ◦ c ◦ · · · c ◦ r P ◦ ( n ,...,n r ) ( e n k − )= c ◦ r (cid:89) j =1 P ◦ j (cid:0) k (cid:96),n (cid:1) if (cid:96) = n + · · · + n j and π j = χ κ j | · | t j , i κ j ζ R ( κ j ) ζ R ( κ j + 1) P ◦ j (cid:0) k (cid:96),n , k (cid:96) +1 ,n (cid:1) if (cid:96) = n + · · · + n j − and π j = D κ j ⊗ | det | t j . (8.19) Proof.
Via the Iwasawa decomposition, it suffices to prove (8.18) for g = k ∈ K . We write u ∈ N n ( F ) as u (cid:48) u (cid:48)(cid:48) with u (cid:48) ∈ N P ( F ) and u (cid:48)(cid:48) = blockdiag( u (cid:48)(cid:48) , . . . , u (cid:48)(cid:48) r ) ∈ M P ( F ) with u (cid:48)(cid:48) j ∈ N n j ( F ),so that u (cid:48)(cid:48) a ∈ M P ( F ) whenever a ∈ A n ( F ). We then take g = uak = u (cid:48) u (cid:48)(cid:48) ak and m (cid:48) = 1 n in(8.15) and apply (8.6) and (8.7) to deduce (8.18). The identity (8.19) then follows upon taking u = a = 1 n , writing P ◦ ( n ,...,n r ) = (cid:81) rj =1 P ◦ j , and invoking Lemmata 7.5 and 7.12 to evaluate theintegrals over K n j (cid:51) k j . (cid:3) The Newform via Convolution Sections.
We now give a different description of thenewform in the induced model. This description is a recursive formula for f ◦ as an inte-gral over GL n ( F ) involving f ◦ itself and a distinguished standard Schwartz function Φ ∈ S (Mat n × n ( F )), where the space of standard Schwartz functions S (Mat n × n ( F )) consists offunctions Φ : Mat n × n ( F ) → C of the formΦ( x ) = P ( x ) exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) with P a polynomial in the entries of x and x and d F := [ F : R ] as in (4.19). When π is a sphericalrepresentation and f is the spherical vector, such a formula is known by the work of Gerasimov,Lebedev, and Oblezin [GLO08, Theorem 5.1] and Ishii and Stade [IsSt13, Proposition 2.6] (withthe latter expressed in terms of the Mellin transform of the Whittaker function); see also [IsSt13,Section 5]. Proposition 8.20.
Let π = π (cid:1) · · · (cid:1) π r be an induced representation of Whittaker type of GL n ( F ) with newform f ◦ in the induced model V π . Then for all h ∈ GL n ( F ) and for (cid:60) ( s ) sufficiently large, (8.21) (cid:90) GL n ( F ) f ◦ ( hg )Φ( g ) | det g | s + n − dg = L ( s, π ) f ◦ ( h ) , where Φ ∈ S (Mat n × n ( F )) is the standard Schwartz function (8.22) Φ( x ) := (dim τ ◦ ) P ◦ ( e n x ) exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) , with P ◦ the homogeneous harmonic polynomial associated to the newform K -type τ ◦ of π via (7.2) and (7.8) . In particular, the integral (8.21) converges absolutely if (cid:60) ( s ) > −(cid:60) ( t j ) for each j ∈ { , . . . , r } for which n j = 1, so that π j = χ κ j | · | t j , and (cid:60) ( s ) > −(cid:60) ( t j ) + ( κ j − / j ∈ { , . . . , r } for which n j = 2, so that F = R and π j = D κ j ⊗ | det | t j .We may think of the integral (8.21) as defining a convolution section of V π in the sense ofJacquet [Jac04], where the convolution is with respect to the function φ ( g ) := Φ( g ) | det g | s + n − .(Note that Jacquet deals only with functions φ : GL n ( F ) → C that are smooth and com-pactly supported.) Alternatively, the identity (8.21) may be thought of as a Pieri-type formula,generalising [Ish18, Theorem 3.8]. Proof.
Via the Iwasawa decomposition with respect to the standard Borel subgroup and (8.18),it suffices to show the identity (8.21) for h = k ∈ K n . We make the change of variables g (cid:55)→ k − g , then use the Iwasawa decomposition g = umk (cid:48) with respect to the parabolicsubgroup P( F ) = P ( n ,...,n r ) ( F ); the Haar measure becomes dg = δ − ( m ) du d × m dk (cid:48) . We seethat the left-hand side of (8.21) is(8.23) (cid:90) M P ( F ) | det m | s + n − δ − ( m ) (cid:90) N P ( F ) (cid:90) K n f ◦ ( umk (cid:48) )Φ( k − umk (cid:48) ) dk (cid:48) du d × m. The absolute convergence of this integral for (cid:60) ( s ) sufficiently large is not difficult; it followsdirectly from the definitions (8.15) of the newform in the induced model and (8.22) of thestandard Schwartz function Φ ∈ S (Mat n × n ( F )) together with the bounds from [Jac09, Lemma3.3 (ii)].We may evaluate the integral over K n (cid:51) k (cid:48) in (8.23) by inserting (8.15) and (8.22) and usingLemmata 7.5 and 7.12. We subsequently make the change of variables m j (cid:55)→ m j k − j , where m = blockdiag( m , . . . , m r ), to trivially evaluate the integrals over K n j (cid:51) k j , leading to(8.24) c ◦ dim τ ◦ · · · dim τ ◦ r (cid:90) M P ( F ) (cid:90) N P ( F ) | det m | s + n − δ − / ( m ) × f ◦ ( m ) · · · f ◦ r ( m r ) P ◦ ( n ,...,n r ) ( e n k − um ) exp (cid:0) − d F π Tr (cid:0) um t m t u (cid:1)(cid:1) du d × m, We evaluate the integrals over M P ( F ) (cid:51) m and N P ( F ) (cid:51) u in (8.24) by breaking these integralsup into parts, where this decomposition is dependent on the size of n j ∈ { , } for j ∈ { , . . . , r } .In doing so, we use the fact that k − = t k and recall the definitions (8.13) and (8.12) of thepolynomial P ◦ ( n ,...,n r ) in order to writedim τ ◦ · · · dim τ ◦ r P ◦ ( n ,...,n r ) ( e n g ) = r (cid:89) j =1 dim τ ◦ j P ◦ j ( g n,(cid:96) ) if (cid:96) = n + · · · + n j and π j = χ κ j | · | t j , P ◦ j ( g n,(cid:96) , g n,(cid:96) +1 ) if (cid:96) = n + · · · + n j − π j = D κ j ⊗ | det | t j .We first deal with the case of n j = 1, so that π j = χ κ j | · | t j ; in this case, we evaluate theintegrals over F × (cid:51) m j and F (cid:51) u i,(cid:96) with i ∈ { , . . . , (cid:96) − } for (cid:96) = n + · · · + n j . After makingthe change of variables u i,(cid:96) (cid:55)→ m − j u i,(cid:96) , recalling the definitions (8.6) of the newform in the RCHIMEDEAN NEWFORM THEORY FOR GL n induced model f ◦ j and (7.3) and (7.9) of the polynomial P ◦ j , and expanding this polynomial viathe multinomial theorem, we are left with evaluating c ◦ j (cid:88) ν + ··· + ν (cid:96) = (cid:107) κ j (cid:107) (cid:18) (cid:107) κ j (cid:107) ν , . . . , ν (cid:96) (cid:19) (cid:96) (cid:89) i =1 k max { sgn( κ j ) ν i , } i,n k i,n − min { sgn( κ j ) ν i , } × (cid:90) F × m j max { sgn( κ j ) ν (cid:96) , } m − min { sgn( κ j ) ν (cid:96) , } j χ κ j ( m j ) | m j | s + t j exp ( − d F πm j m j ) dm j × (cid:96) − (cid:89) i =1 (cid:90) F u i,(cid:96) max { sgn( κ j ) ν i , } u − min { sgn( κ j ) ν i , } i,(cid:96) exp ( − d F πu i,(cid:96) u i,(cid:96) ) du i,(cid:96) . Here (cid:18) κν , . . . , ν (cid:96) (cid:19) := κ ! ν ! · · · ν (cid:96) !denotes the multinomial coefficient for κ, ν , . . . , ν (cid:96) ∈ N with ν + · · · + ν (cid:96) = κ . The integral over F (cid:51) u i,(cid:96) vanishes unless ν i = 0, in which case it is 1, upon applying Hecke’s identity, Lemmata7.6 and 7.13. All that remains is the integral over F × (cid:51) m j , which is equal to c ◦ j k max { κ j , } (cid:96),n k (cid:96),n − min { κ j , } (cid:90) F × | m j | s + t j + (cid:107) κj (cid:107) dF exp( − d F πm j m j ) dm j = c ◦ j L ( s, π j ) P ◦ j (cid:0) k (cid:96),n (cid:1) having used the fact that(8.25) | x | (cid:107) κ (cid:107) dF = χ − κ ( x ) x max { κ, } x − min { κ, } = χ κ ( x ) x max { κ, } x − min { κ, } and recalling the definitions (7.3) and (7.9) of the polynomial P ◦ j , (2.7) and (2.10) of the L -function L ( s, π j ) in terms of ζ F ( s ), and (2.8) and (2.11) of the zeta function ζ F ( s ) as an integralover F × .Next, we deal with the case of n j = 2, so that F = R and π j = D κ j ⊗ | det | t j ; we evaluate theintegrals over GL ( R ) (cid:51) m j and R (cid:51) ( u i,(cid:96) , u i,(cid:96) +1 ) with i ∈ { , . . . , (cid:96) − } for (cid:96) = n + · · · + n j − m j = (cid:0) u (cid:96),(cid:96) +1 (cid:1)(cid:16) a (cid:96) a (cid:96) +1 (cid:17) k (cid:48) for u (cid:96),(cid:96) +1 ∈ R , a (cid:96) , a (cid:96) +1 ∈ R × , and k (cid:48) ∈ O(2); the Haarmeasure becomes dm j = | a (cid:96) | − | a (cid:96) +1 | du (cid:96),(cid:96) +1 d × a (cid:96) d × a (cid:96) +1 dk (cid:48) . We use Schur orthogonality toevaluate the integral over O(2) (cid:51) k (cid:48) , then make the change of variables u (cid:96),(cid:96) +1 (cid:55)→ a − (cid:96) +1 u (cid:96),(cid:96) +1 , u i,(cid:96) (cid:55)→ a − (cid:96) u i,(cid:96) , and u i,(cid:96) +1 (cid:55)→ a − (cid:96) +1 u i,(cid:96) +1 − u (cid:96),(cid:96) +1 u i,(cid:96) for i ∈ { , . . . , (cid:96) − } . Recalling the definitions(8.7) of f ◦ j and (7.10) of P ◦ j , and expanding this polynomial via the multinomial theorem, weare led to c ◦ j (cid:88) ± (cid:88) ν + ··· + ν (cid:96) +1 = κ j (cid:18) κ j ν , . . . , ν (cid:96) +1 (cid:19) (cid:96) +1 (cid:89) i =1 k ν i i,n × (cid:90) R × a ν (cid:96) +1 (cid:96) +1 χ κ j ( a (cid:96) +1 ) | a (cid:96) +1 | s + t j − κj − exp (cid:0) − πa (cid:96) +1 (cid:1) d × a (cid:96) +1 × (cid:90) R × | a (cid:96) | s + t j + κj − exp (cid:0) − πa (cid:96) (cid:1) (cid:90) R ( u (cid:96),(cid:96) +1 ∓ ia (cid:96) ) ν (cid:96) exp (cid:0) − πu (cid:96),(cid:96) +1 (cid:1) du (cid:96),(cid:96) +1 d × a (cid:96) × (cid:96) − (cid:89) i =1 (cid:90) R ( u i,(cid:96) +1 ∓ iu i,(cid:96) ) ν i exp (cid:0) − π (cid:0) u i,(cid:96) + u i,(cid:96) +1 (cid:1)(cid:1) du i,(cid:96) du i,(cid:96) +1 . We use Hecke’s identity, Lemma 7.13, to see that the integral over R (cid:51) ( u i,(cid:96) , u i,(cid:96) +1 ) vanishesunless ν i = 0, in which case it is 1; consequently, the only nonzero summands are those forwhich ν (cid:96) = κ j − ν (cid:96) +1 =: ν . For the integral over R (cid:51) u (cid:96),(cid:96) +1 , we make the change of variables u (cid:96),(cid:96) +1 (cid:55)→ u (cid:96),(cid:96) +1 ± ia (cid:96) and shift the contour of integration back to the line (cid:61) ( u (cid:96),(cid:96) +1 ) = 0 viaCauchy’s integral theorem, for the integrand extends holomorphically to an entire function of the complex variable u (cid:96),(cid:96) +1 . Since (cid:90) R u ν exp (cid:0) − πu (cid:1) ψ ( ua ) du = i ν (2 π ) − ν ∂ ν ∂a ν exp (cid:0) − πa (cid:1) , we arrive at c ◦ j κ j (cid:88) ν =0 ν ≡ (cid:18) κ j ν (cid:19) ( ik (cid:96),n ) ν k κ j − ν(cid:96) +1 ,n × (2 π ) − ν (cid:90) R × | a (cid:96) +1 | s + t j + κj +12 − ν exp (cid:0) − πa (cid:96) +1 (cid:1) d × a (cid:96) +1 (cid:90) R × | a (cid:96) | s + t j + κj − ∂ ν ∂a ν(cid:96) exp (cid:0) − πa (cid:96) (cid:1) d × a (cid:96) , having observed the vanishing of the integral over R × (cid:51) a (cid:96) +1 for odd ν . We integrate by parts ν times with respect to a (cid:96) , then integrate by parts ν/ a (cid:96) , differentiatingexp( − πa (cid:96) ), and ν/ a (cid:96) +1 , differentiating exp( − πa (cid:96) +1 ). We end up at c ◦ j κ j (cid:88) ν =0 ν ≡ (cid:18) κ j ν (cid:19) ( ik (cid:96),n ) ν k κ j − ν(cid:96) +1 ,n × (cid:90) R × | a (cid:96) +1 | s + t j + κj +12 exp (cid:0) − πa (cid:96) +1 (cid:1) d × a (cid:96) +1 (cid:90) R × | a (cid:96) | s + t j + κj − exp (cid:0) − πa (cid:96) (cid:1) d × a (cid:96) = c ◦ j L ( s, π j ) P ◦ j (cid:0) k (cid:96),n , k (cid:96) +1 ,n (cid:1) , again recalling the definitions (7.10) of P ◦ j , (2.13) of L ( s, π j ) in terms of products of ζ F ( s ), and(2.11) of ζ F ( s ) as an integral over F × .Combining these calculations, we find that (cid:90) GL n ( F ) f ◦ ( kg )Φ( g ) | det g | s + n − dg = c ◦ r (cid:89) j =1 c ◦ j L ( s, π j ) (cid:40) P ◦ j ( k (cid:96),n ) if (cid:96) = n + · · · + n j and π j = χ κ j | · | t j , P ◦ j ( k (cid:96),n , k (cid:96) +1 ,n ) if (cid:96) = n + · · · + n j − π j = D κ j ⊗ | det | t j ,which is precisely L ( s, π ) f ◦ ( k ) via the isobaric decomposition (2.5) of L ( s, π ) and the identity(8.19) for f ◦ ( k ).Finally, an inspection of the proof above shows that the integral (8.24) converges absolutelyif (cid:60) ( s ) > −(cid:60) ( t j ) for each j ∈ { , . . . , r } for which n j = 1 and (cid:60) ( s ) > −(cid:60) ( t j ) + ( κ j − / j ∈ { , . . . , r } for which n j = 2. (cid:3) We may use the identity (8.21) to prove Theorem 4.22, thereby resolving the test vectorproblem for archimedean Godement–Jacquet zeta integrals.
Proof of Theorem 4.22.
From the definition (2.3) of the Godement–Jacquet zeta integral andProposition 8.20, we have that Z ( s, β, Φ) = (cid:90) GL n ( F ) (cid:104) π ( g ) · v ◦ , (cid:101) v ◦ (cid:105) Φ( g ) | det g | s + n − dg = (cid:42)(cid:90) GL n ( F ) ( π ( g ) · v ◦ ) Φ( g ) | det g | s + n − dg, (cid:101) v ◦ (cid:43) = (cid:104) L ( s, π ) v ◦ , (cid:101) v ◦ (cid:105) = L ( s, π ) (cid:104) v ◦ , (cid:101) v ◦ (cid:105) = L ( s, π ) . (cid:3) A similar calculation to that of the proof of Proposition 8.20 yields the following.
RCHIMEDEAN NEWFORM THEORY FOR GL n Proposition 8.26.
Let π be an induced representation of Whittaker type of GL n ( F ) withnewform f ◦ in the induced model V π . Define (cid:101) f ◦ ( g ) := f ◦ ( t g − ) . Then for all h ∈ GL n ( F ) andfor (cid:60) ( s ) sufficiently large, (8.27) (cid:90) GL n ( F ) (cid:101) f ◦ ( hg ) (cid:101) Φ( g ) | det g | s + n − dg = L ( s, (cid:101) π ) (cid:101) f ◦ ( h ) , where (cid:101) Φ ∈ S (Mat n × n ( F )) is the standard Schwartz function (8.28) (cid:101) Φ( x ) := (dim τ ◦ ) P ◦ ( e n x ) exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) with P ◦ the homogeneous harmonic polynomial associated to the newform K -type τ ◦ of π via (7.2) and (7.8) . In particular, the integral (8.27) converges absolutely if (cid:60) ( s ) > (cid:60) ( t j ) for each j ∈ { , . . . , r } for which n j = 1, so that π j = χ κ j | · | t j , and (cid:60) ( s ) > (cid:60) ( t j ) + ( κ j − / j ∈ { , . . . , r } for which n j = 2, so that F = R and π j = D κ j ⊗ | det | t j . Remark . From (8.27), we see that Z ( s, (cid:101) β, (cid:101) Φ) = L ( s, (cid:101) π ), where (cid:101) β ( g ) := β ( t g − ). This is inperfect accordance with the local functional equation (2.15) upon noting that (cid:101) Φ = i c ( π ) (cid:98) Φ viaHecke’s identity, Lemmata 7.6 and 7.13.8.4.
The Newform via Godement Sections.
Our third description of the newform in theinduced model is via Godement sections. This is a recursive formula for the newform f ◦ of π = π (cid:1) π (cid:1) · · · (cid:1) π r in terms of an integral involving the newform f ◦ of π := π (cid:1) · · · (cid:1) π r and a distinguished standard Schwartz function. Unlike our earlier descriptions of f ◦ via theIwasawa decomposition and via convolution sections, this description via Godement sections isonly valid for certain induced representations of Whittaker type; we require the parameter t associated to π to have sufficiently large real part. When we proceed to studying the Whittakernewform, we remove this condition via analytic continuation.8.4.1. The Case π = χ κ | · | t . We first consider the case for which π = π (cid:1) · · · (cid:1) π r with n = 1, so that π = χ κ | · | t . We begin with a simple modification of Proposition 8.14. Lemma 8.30.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be inducedrepresentations of Whittaker type of GL n ( F ) and GL n − ( F ) with π = χ κ | · | t . Let f ◦ be thenewform of π in the induced model V π . Then for v ∈ Mat × ( n − ( F ) , x ∈ F × , h ∈ GL n − ( F ) ,and k ∈ K n , the newform f ◦ in the induced model V π satisfies (8.31) f ◦ (cid:18)(cid:18) v n − (cid:19) (cid:18) x h (cid:19) k (cid:19) = c ◦ c ◦ (dim τ ◦ ) | x | n − | det h | − × (cid:90) K (cid:90) K n − χ κ ( xk ) | xk | t f ◦ ( hk ) P ◦ (1 ,n − (cid:18) e n k − (cid:18) k k (cid:19)(cid:19) dk dk . Here the constants c ◦ and c ◦ are those associated to f ◦ and f ◦ via (8.15) , while P ◦ (1 ,n − ( x ) := P ◦ ( x ) P ◦ ( x , . . . , x n ) , where P ◦ and P ◦ are the homogeneous harmonic polynomials associated to the newform K -types τ ◦ and τ ◦ of π and π respectively via (7.2) and (7.8) .Proof. We show that (8.31) reproduces Corollary 8.17, which determines f ◦ completely. Writing P ◦ (1 ,n − = P ◦ P ◦ , we see that the integral over K (cid:51) k is trivial by the homogeneity of P ◦ and (8.25). We then use the Iwasawa decomposition h = u (cid:48) a (cid:48) k (cid:48) with respect to the standardBorel subgroup, so that u (cid:48) ∈ N n − ( F ), a (cid:48) = diag( a (cid:48) , . . . , a (cid:48) n − ) ∈ A n − ( F ), and k (cid:48) ∈ K n − , andapply Corollary 8.17 in order to rewrite f ◦ ( hk ). The integral over K n − (cid:51) k may then beevaluated via Lemmata 7.5 and 7.12. The resulting expression for f ◦ is then precisely thatgiven in Corollary 8.17 with u = (cid:0) v vu (cid:48) (cid:1) ∈ N n ( F ), a = diag( x, a (cid:48) , . . . , a (cid:48) n − ) ∈ A n ( F ), and (cid:0) k (cid:48) (cid:1) k ∈ K n in place of k . (cid:3) We now use the identity (8.31) in conjunction with the convolution section (8.27) in order toprove that f ◦ may be written as a Godement section. Proposition 8.32.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be inducedrepresentations of Whittaker type of GL n ( F ) and GL n − ( F ) with π = χ κ | · | t . Let f ◦ be thenewform of π in the induced model V π . Let Φ ∈ S (Mat ( n − × n ( F )) be the standard Schwartzfunction of the form Φ( x ) := P ◦ (cid:18) det (cid:18) x (cid:18) n − (cid:19)(cid:19)(cid:19) (dim τ ◦ ) P ◦ (cid:0) e n t x (cid:1) exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) , where P ◦ and P ◦ are the homogeneous harmonic polynomials associated to the newform K -types τ ◦ and τ ◦ of π and π respectively via (7.2) and (7.8) . Then if (cid:60) ( t ) is sufficiently large, thenewform f ◦ in the induced model V π satisfies the identity (8.33) f ◦ ( g ) = c ◦ ( − κ ( n − c ◦ L (cid:16) t + (cid:107) κ (cid:107) d F , (cid:102) π (cid:17) χ κ (det g ) | det g | t + n − × (cid:90) GL n − ( F ) f ◦ ( h )Φ (cid:0) h − (cid:0) n − (cid:1) g (cid:1) χ − κ (det h ) | det h | − t − n dh, where the constants c ◦ and c ◦ are those associated to f ◦ and f ◦ via (8.15) . In particular, the integral (8.33) converges absolutely if (cid:60) ( t ) > (cid:60) ( t j ) − − (cid:107) κ (cid:107) /d F for each j ∈ { , . . . , r } for which n j = 1, so that π j = χ κ j | · | t j , and (cid:60) ( t ) > (cid:60) ( t j ) + ( κ j − / − − κ for each j ∈ { , . . . , r } for which n j = 2, so that F = R and π j = D κ j ⊗ | det | t j . Proof.
We take s = 1 + t + (cid:107) κ (cid:107) /d F in the convolution section identity (8.27), so that f ◦ ( g ) = 1 L (cid:16) t + (cid:107) κ (cid:107) d F , (cid:101) π (cid:17) (cid:90) GL n ( F ) f ◦ (cid:0) g t g (cid:48)− (cid:1) (cid:101) Φ( g (cid:48) ) (cid:12)(cid:12) det g (cid:48) (cid:12)(cid:12) t + (cid:107) κ (cid:107) dF + n − dg (cid:48) , with (cid:101) Φ as in (8.28). We make the change of variables g (cid:48) (cid:55)→ t g t g (cid:48)− and use the Iwasawadecomposition g (cid:48) = umk with respect to the parabolic subgroup P( F ) = P (1 ,n − ( F ), where u = (cid:0) v n − (cid:1) ∈ N P ( F ) with v ∈ Mat × ( n − ( F ), m = (cid:0) x h (cid:1) ∈ M P ( F ) with x ∈ F × and h ∈ GL n − ( F ), and k ∈ K n ; the Haar measure is dg (cid:48) = | x | − n | det h | dv d × x dh dk . We may nowinsert the identity (8.31) for f ◦ . Next, we make the change of variables x (cid:55)→ xk − , h (cid:55)→ hk − ,and k (cid:55)→ (cid:16) k k (cid:17) k , so that the integrals over K (cid:51) k and K n − (cid:51) k are trivial; subsequently,the integral over K n (cid:51) k may be evaluated via Lemmata 7.5 and 7.12 after inserting thedefinition (8.28) of (cid:101) Φ. After making the change of variables v (cid:55)→ − xv and x (cid:55)→ x − , we write v (cid:48) := (cid:0) x v (cid:1) ∈ Mat × n ( F ), so that dv (cid:48) = ζ F (1) − | x | d × x dv , and make the change of variables v (cid:48) (cid:55)→ v (cid:48) g − . Finally, we use (8.25) in conjunction with the homogeneity of P ◦ and the fact that L (cid:18) t + (cid:107) κ (cid:107) d F , (cid:101) π (cid:19) = ζ F (cid:18) (cid:107) κ (cid:107) d F (cid:19) L (cid:18) t + (cid:107) κ (cid:107) d F , (cid:102) π (cid:19) via (2.5), (2.9), and (2.12). We arrive at the identity(8.34) f ◦ ( g ) = c ◦ c ◦ L (cid:16) t + (cid:107) κ (cid:107) d F , (cid:102) π (cid:17) χ κ (det g ) | det g | t + n − (cid:90) GL n − ( F ) f ◦ ( h ) χ − κ (det h ) | det h | − t − n × (dim τ ◦ ) P ◦ (cid:18) e n t g (cid:18) n − (cid:19) t h − (cid:19) exp (cid:18) − d F π Tr (cid:18) h − (cid:0) n − (cid:1) g t g (cid:18) n − (cid:19) t h − (cid:19)(cid:19) × ζ F (1) ζ F (cid:16) (cid:107) κ (cid:107) d F (cid:17) (cid:90) Mat × n ( F ) P ◦ (cid:16) (det h − ) v (cid:48) adj( g ) t e e n t v (cid:48) (cid:17) exp (cid:16) − d F πv (cid:48) t v (cid:48) (cid:17) dv (cid:48) dh, RCHIMEDEAN NEWFORM THEORY FOR GL n having recalled that the adjugate of g is adj( g ) := (det g ) g − . To evaluate the last line of (8.34),we expand P ◦ as a polynomial in v (cid:48) , . . . , v (cid:48) n via the multinomial theorem, yielding ζ F (1) ζ F (cid:16) (cid:107) κ (cid:107) d F (cid:17) (det h − ) max { κ , } (det h − ) − min { κ , } (cid:88) ν + ··· + ν n = (cid:107) κ (cid:107) (cid:18) (cid:107) κ (cid:107) ν , . . . , ν n (cid:19) × n (cid:89) i =1 (adj( g ) i, ) max { sgn( κ ) ν i , } (adj( g ) i, ) − min { sgn( κ ) ν i , } × n − (cid:89) i =1 (cid:90) F v (cid:48) i max { sgn( κ ) ν i , } v (cid:48) i − min { sgn( κ ) ν i , } exp (cid:16) − d F πv (cid:48) i v (cid:48) i (cid:17) dv (cid:48) i × (cid:90) F v (cid:48) n max { sgn( κ ) ν n , }− min { κ , } v (cid:48) n − min { sgn( κ ) ν n , } +max { κ , } exp (cid:0) − d F πv (cid:48) n v (cid:48) n (cid:1) dv (cid:48) n . Via Hecke’s identity, Lemmata 7.6 and 7.13, the integral over F (cid:51) v (cid:48) n vanishes unless ν = · · · = ν n − = 0 and ν n = (cid:107) κ (cid:107) , in which case the integral over F (cid:51) v (cid:48) i for i ∈ { , . . . , n − } is equal to1. Via (8.25) and the fact that ζ F (1) | v (cid:48) n | − dv (cid:48) n = d × v (cid:48) n , we are left with P ◦ (cid:0) (det h − ) adj( g ) n, (cid:1) ζ F (cid:16) (cid:107) κ (cid:107) d F (cid:17) (cid:90) F × | v (cid:48) n | (cid:107) κ (cid:107) dF exp (cid:0) − d F πv (cid:48) n v (cid:48) n (cid:1) d × v (cid:48) n = P ◦ (cid:0) (det h − ) adj( g ) n, (cid:1) by (2.8) and (2.11). Sinceadj( g ) n, = ( − n − det (cid:18)(cid:0) n − (cid:1) g (cid:18) n − (cid:19)(cid:19) , the result then follows from (8.34) and the homogeneity of P ◦ . (cid:3) The Case π = D κ ⊗ | det | t . Next, we give a description of the newform in the inducedmodel when F = R , n = 2, and π = D κ ⊗ | det | t is an essentially discrete series representation.We do this first when n = 2, so that π = π . Proposition 8.35.
Let π = D κ ⊗ | det | t be an essentially discrete series representation of GL ( R ) . Let Φ ∈ S (Mat × ( R )) be the standard Schwartz function of the form (8.36) Φ( x ) := (cid:90) R P ◦ (cid:0) v, e t x (cid:1) exp( − πv ) dv exp (cid:0) − πx t x (cid:1) , where P ◦ is the homogeneous harmonic polynomial associated to the newform K -type τ ◦ = τ ( κ, of π via (7.8) . Then the canonically normalised newform f ◦ in the induced model V π satisfiesthe identity (8.37) f ◦ ( g ) = i κ | det g | t + κ (cid:90) R × a − κ Φ (cid:0) a − (cid:0) (cid:1) g (cid:1) d × a . Remark . The integral over R (cid:51) v in (8.36) may be expressed in terms of Hermite polynomials,though we do not make direct use of this fact. Proof.
We take s = 1 + t + ( κ − / f ◦ ( g ) = 1 L (cid:0) t + κ − , (cid:101) π (cid:1) (cid:90) GL ( R ) f ◦ (cid:0) g t g (cid:48)− (cid:1) (cid:101) Φ( g (cid:48) ) (cid:12)(cid:12) det g (cid:48) (cid:12)(cid:12) t + κ dg (cid:48) , with (cid:101) Φ as in (8.28). We make the change of variables g (cid:48) (cid:55)→ t g t g (cid:48)− and use the Iwasawadecomposition g (cid:48) = uak with respect to the standard Borel subgroup, where u = ( v ) ∈ N ( R )with v ∈ R , a = diag( a , a ) ∈ A ( R ) with a , a ∈ R × , and k ∈ O(2); the Haar measure is dg (cid:48) = | a | − | a | dv d × a d × a dk . We may now insert the identity (8.7) for f ◦ , at which pointLemma 7.12 allows us to evaluate the integral over O(2) (cid:51) k . Next, we make the change of variables a (cid:55)→ a − , v (cid:55)→ − a − v , and (cid:0) a v (cid:1) (cid:55)→ (cid:0) a v (cid:1) g − , noting that d × a = | a | − da as ζ R (1) = 1. Finally, we use the fact that L (cid:18) t + κ − , (cid:101) π (cid:19) = ζ R ( κ ) ζ R ( κ + 1)via (2.14). In this way, we find that f ◦ ( g ) = c ◦ ζ R ( κ ) ζ R ( κ + 1) | det g | t + κ (cid:90) R × a − κ (cid:90) R exp( − πa ) × (cid:90) R P ◦ (cid:18) v, e t g (cid:18) (cid:19) a − (cid:19) exp( − πv ) exp (cid:18) − πa − (cid:0) (cid:1) g t g (cid:18) (cid:19) a − (cid:19) dv da d × a . It remains to note that the integral over R (cid:51) a is equal to 1 and to recall the definition (8.11)of the normalising constant c ◦ . (cid:3) Finally, we consider the more general case for which π = π (cid:1) · · · (cid:1) π r with n = 2 and π = D κ ⊗ | det | t . We first require a simple modification of Proposition 8.14 akin to Lemma8.30. Lemma 8.39.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be inducedrepresentations of Whittaker type of GL n ( R ) and GL n − ( R ) with π = D κ ⊗ | det | t . Let f ◦ be the newform of π in the induced model V π . Then for v , v ∈ Mat × ( n − ( R ) , v ∈ R , a , a ∈ R × , h ∈ GL n − ( R ) , and k ∈ O( n ) , the newform f ◦ in the induced model V π satisfies (8.40) f ◦ v v v n − a a
00 0 h k = c ◦ c ◦ c ◦ | a | t + κ − + n − χ κ ( a ) | a | t − κ − + n − | det h | − × dim τ ◦ (cid:90) O( n − f ◦ ( hk ) P ◦ (2 ,n − (cid:18) e n k − (cid:18) k (cid:19)(cid:19) dk . Here the constants c ◦ , c ◦ , and c ◦ are those associated to f ◦ , f ◦ , and f ◦ via (8.15) and (8.7) ,while P ◦ (2 ,n − ( x ) := P ◦ ( x , x ) P ◦ ( x , . . . , x n ) , where P ◦ and P ◦ are the homogeneous harmonic polynomials associated to the newform K -types τ ◦ and τ ◦ of π and π respectively via (7.8) .Proof. The proof is essentially identical to that of Lemma 8.30. (cid:3)
We now use the identity (8.40) in conjunction with the convolution section (8.27) in order toprove that f ◦ may be written as a Godement section. Proposition 8.41.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be inducedrepresentations of Whittaker type of GL n ( R ) and GL n − ( R ) with π = D κ ⊗ | det | t . Let f ◦ be the newform of π in the induced model V π . Let Φ ∈ S (Mat ( n − × n ( R )) be the standardSchwartz function of the form Φ( x ) := (cid:90) R P ◦ (cid:0) v, e n t x t e (cid:1) exp( − πv ) dv (dim τ ◦ ) P ◦ (cid:18) e n t x (cid:18) n − (cid:19)(cid:19) exp (cid:0) − π Tr (cid:0) x t x (cid:1)(cid:1) , where P ◦ and P ◦ are the homogeneous harmonic polynomials associated to the newform K -types τ ◦ and τ ◦ of π and π respectively via (7.8) . Then if (cid:60) ( t ) is sufficiently large, the newform f ◦ in the induced model V π satisfies the identity (8.42) f ◦ ( g ) = c ◦ i κ c ◦ L (cid:0) t + κ − , (cid:102) π (cid:1) | det g | t + κ − + n − (cid:90) GL n − ( R ) f ◦ ( h ) | det h | − t − κ − − n − × (cid:90) R × a − κ (cid:90) Mat × ( n − ( R ) Φ (cid:18)(cid:18) a − v h − (cid:19) (cid:0) n − (cid:1) g (cid:19) dv d × a dh, RCHIMEDEAN NEWFORM THEORY FOR GL n where the constants c ◦ and c ◦ are those associated to f ◦ and f ◦ via (8.15) . In particular, the integral (8.42) converges absolutely if (cid:60) ( t ) > (cid:60) ( t j ) − − ( κ − / j ∈ { , . . . , r } for which n j = 1, so that π j = χ κ j |·| t j , and (cid:60) ( t ) > (cid:60) ( t j )+( κ j − / − − ( κ − / j ∈ { , . . . , r } for which n j = 2, so that π j = D κ j ⊗ | det | t j . Proof.
We take s = 1 + t + ( κ − / f ◦ ( g ) = 1 L (cid:0) t + κ − , (cid:101) π (cid:1) (cid:90) GL n ( R ) f ◦ (cid:0) g t g (cid:48)− (cid:1) (cid:101) Φ( g (cid:48) ) (cid:12)(cid:12) det g (cid:48) (cid:12)(cid:12) t + κ − + n − dg (cid:48) , with (cid:101) Φ as in (8.28). We make the change of variables g (cid:48) (cid:55)→ t g t g (cid:48)− , then use the Iwa-sawa decomposition g (cid:48) = umk with respect to the parabolic subgroup P( R ) = P (1 , ,n − ( R ),where u = (cid:18) v v v n − (cid:19) ∈ N P ( R ) with v ∈ R , v , v ∈ Mat × ( n − ( R ), m = (cid:16) a a
00 0 h (cid:17) ∈ M P ( R ) with a , a ∈ R × and h ∈ GL n − ( R ), and k ∈ O( n ); the Haar measure is dg (cid:48) = | a | − n | a | − n | det h | dv dv dv d × a d × a dh dk . We may now insert the identity (8.40) for f ◦ .Next, we make the change of variables h (cid:55)→ hk − and k (cid:48) (cid:55)→ (cid:0) k (cid:1) k (cid:48) , so that the integral overO( n − (cid:51) k is trivial; subsequently, the integral over O( n ) (cid:51) k may be evaluated via Lemma7.12. Next, we make the change of variables v (cid:55)→ − a v , v (cid:55)→ − a v − v v , v (cid:55)→ − a v , a (cid:55)→ a − , and (cid:0) a v v (cid:1) (cid:55)→ (cid:0) a v v (cid:1) g − , noting that d × a = | a | − da as ζ R (1) = 1.Finally, we use the fact that L (cid:18) t + κ − , (cid:101) π (cid:19) = ζ R ( κ ) ζ R ( κ + 1) L (cid:18) t + κ − , (cid:102) π (cid:19) via (2.5) and (2.14). In this way, we find that f ◦ ( g ) = c ◦ c ◦ c ◦ ζ R ( κ ) ζ R ( κ + 1) L (cid:0) t + κ − , (cid:102) π (cid:1) | det g | t + κ − + n − × (cid:90) GL n − ( R ) f ◦ ( h ) | det h | − t − κ − − n − × (dim τ ◦ ) P ◦ (cid:18) e n t g (cid:18) n − (cid:19) t h − (cid:19) exp (cid:18) − π Tr (cid:18) h − (cid:0) n − (cid:1) g t g (cid:18) n − (cid:19) t h − (cid:19)(cid:19) × (cid:90) R × a − κ (cid:90) R exp( − πa ) (cid:90) Mat × ( n − ( R ) exp − π Tr (cid:0) a − v (cid:1) g t g a − t v × (cid:90) Mat × ( n − ( R ) P ◦ e n − v , e n t g a − t v exp (cid:0) − πv t v (cid:1) (cid:90) R exp( − πv ) dv dv dv d × a d × a dh. It remains to note that the integrals over R (cid:51) a , R (cid:51) v , and R (cid:51) v ,i for i ∈ { , . . . , n − } aretrivial and to recall the definition (8.11) of the normalising constant c ◦ . (cid:3) In order to prove a recursive formula for the newform in the Whittaker model, we require asimilar identity to (8.42) for a slightly modified induced representation of Whittaker type.
Lemma 8.43.
For n ≥ , let π ∗ = π ∗ (cid:1) π ∗ (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be inducedrepresentations of Whittaker type of GL n ( R ) and GL n − ( R ) with π ∗ = | · | t ∗ and π ∗ = | · | t ∗ . Let f ◦ be the newform of π in the induced model V π . Let Φ ∈ S (Mat ( n − × n ( R )) be the standardSchwartz function of the form Φ ∗ ( x ) := (dim τ ◦ ) P ◦ (cid:18) e n t x (cid:18) n − (cid:19)(cid:19) exp (cid:0) − π Tr (cid:0) x t x (cid:1)(cid:1) , where P ◦ is the homogeneous harmonic polynomial associated to the newform K -type τ ◦ of π via (7.8) . Then if (cid:60) ( t ∗ ) is sufficiently large, the newform f ∗◦ in the induced model V π ∗ satisfies the identity (8.44) f ∗◦ ( g ) = c ∗◦ c ◦ ζ R (1 + t ∗ − t ∗ ) L (1 + t ∗ , (cid:102) π ) | det g | t ∗ + n − (cid:90) GL n − ( R ) f ◦ ( h ) | det h | − t ∗ − n − × (cid:90) R × | a | − − t ∗ + t ∗ (cid:90) Mat × ( n − ( R ) Φ (cid:18)(cid:18) a − v h − (cid:19) (cid:0) n − (cid:1) g (cid:19) dv d × a dh, where the constants c ∗◦ and c ◦ are those associated to f ∗◦ and f ◦ via (8.15) . In particular, the integral (8.44) converges absolutely if (cid:60) ( t ∗ ) > (cid:60) ( t ∗ ) − (cid:60) ( t ∗ ) > (cid:60) ( t j ) − j ∈ { , . . . , r } for which n j = 1, so that π j = χ κ j | · | t j , and (cid:60) ( t ∗ ) > (cid:60) ( t j ) + ( κ j − / − j ∈ { , . . . , r } for which n j = 2, so that π j = D κ j ⊗ | det | t j . Proof.
This follows via the same method as the proof of Proposition 8.41. (cid:3) The Newform in the Whittaker Model
The Jacquet Integral.
Let π = π (cid:1) · · · (cid:1) π r be an induced representation of Whittakertype of GL n ( F ), so that each π j is of the form χ κ j | · | t j or D κ j ⊗ | det | t j . Given an element f ofthe induced model V π of π , we define the Jacquet integral(9.1) W ( g ) := (cid:90) N n ( F ) f ( w n ug ) ψ n ( u ) du. This integral converges absolutely if (cid:60) ( t ) > · · · > (cid:60) ( t r ) and defines a Whittaker function W = W f ∈ W ( π, ψ ); that is, as a function of f ∈ V π , Λ( f ) := W f (1 n ) defines a Whittakerfunctional, which is therefore unique up to scalar multiplication.Wallach [Wal92] has shown that the Jacquet integral gives a Whittaker functional for all induced representations of Whittaker type, and not just those for which (cid:60) ( t ) > · · · > (cid:60) ( t r ),via analytic continuation in the following way. Write π = π t ,...,t r for such a representation,and let V π = V π t ,...,tr denote its induced model. Fixing each χ κ j and D κ j but regarding t j asa complex variable, we may view the space V π t ,...,tr as a holomorphic fibre bundle. A section f t ,...,t r ( g ; m (cid:48) ) is a map from GL n ( F ) × M P ( F ) × C r to C such that f t ,...,t r ( · ; · ) is an element of V π t ,...,tr for each fixed ( t , . . . , t r ) ∈ C r ; a standard section (or flat section) is a section for which f t ,...,t r ( k ; 1 n ) is independent of ( t , . . . , t r ) ∈ C r for all k ∈ K . From [Wal92, Theorem 15.4.1],the Jacquet integral (9.1) evaluated on a standard section extends holomorphically as a functionof ( t , . . . , t r ) ∈ C r with (cid:60) ( t ) > · · · > (cid:60) ( t r ) to all of C r , and hence via analytic continuationdefines an equivariant map from V π t ,...,tr to W ( π t ,...,t r , ψ ).From this, we see that the newform f ◦ in the induced model V π defined via the Iwasawadecomposition (8.15) gives a standard section of newforms f ◦ t ,...,t r provided that we choose thenormalising constant c ◦ to be independent of ( t , . . . , t r ) ∈ C r . The corresponding Whittakerfunction is then given via the analytic continuation of the Jacquet integral (9.1). Furthermore,we may choose the normalising constant c ◦ to be dependent on ( t , . . . , t r ) ∈ C r and still obtainthe corresponding Whittaker function via the analytic continuation of the Jacquet integral solong as c ◦ is holomorphic as a function of ( t , . . . , t r ) ∈ C r .With this in mind, we may now define the canonically normalised newform in the inducedand Whittaker models. Definition 9.2.
Let π = π (cid:1) · · · (cid:1) π r be an induced representation of Langlands type ofGL n ( F ). The canonically normalised newform f ◦ in the induced model V π is defined via (8.6)and (8.7) with normalising constant (8.11) if r = 1, while for r ≥
2, it is defined via (8.15) withnormalising constant c ◦ := r − (cid:89) j =1 r (cid:89) (cid:96) = j +1 i c ( π (cid:96) ) L (cid:18) t j + (cid:107) κ j (cid:107) d F , (cid:101) π (cid:96) (cid:19) if π j = χ κ j | · | t j , L (cid:18) t j + κ j − , (cid:101) π (cid:96) (cid:19) L (cid:18) t j + κ j + 12 , (cid:101) π (cid:96) (cid:19) if π j = D κ j ⊗ | det | t j . RCHIMEDEAN NEWFORM THEORY FOR GL n The canonically normalised newform W ◦ in the Whittaker model W ( π, ψ ) is given by the analyticcontinuation of the Jacquet integral (9.1) of the canonically normalised newform f ◦ . We call W ◦ the Whittaker newform. Remark . When π is spherical, some authors refer to the canonically normalised Whittakerfunction as the completed Whittaker function; see, for example, [BHM20, Section 2.1]. Wefollow the nomenclature of [GMW20, Section 8].Recalling the identities (2.9), (2.12), and (2.14) relating L -functions to zeta functions, weobserve that the normalising constant c ◦ is well-defined since ζ F ( s ) is holomorphic for (cid:60) ( s ) > π being an induced representation of Langlands type means that (cid:60) ( t ) ≥ · · · ≥ (cid:60) ( t r ). Wealso note that if π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r , then the associated normalisingconstants satisfy the relation(9.4) c ◦ = i c ( π ) L (cid:18) t + (cid:107) κ (cid:107) d F , (cid:102) π (cid:19) c ◦ if π = χ κ | · | t , i c ( π ) L (cid:18) t + κ − , (cid:102) π (cid:19) L (cid:18) t + κ + 12 , (cid:102) π (cid:19) c ◦ if π = D κ ⊗ | det | t .This is a consequence of Definition 9.2, Theorem 4.15, and (2.5). Remark . While we do not prove this, our methods below can be extended to show that notonly is the Whittaker newform well-defined for induced representations of Langlands type, it isalso well-defined for induced representations of Whittaker type, including those for which c ◦ is not well-defined (note that in these cases, the Whittaker model is a model for a quotient of π ratherthan for π itself). Furthermore, one can show that the Whittaker newform of π = π (cid:1) · · · (cid:1) π r remains unchanged when π is replaced by π σ (1) (cid:1) · · · (cid:1) π σ ( r ) for any permutation σ ∈ S r (cf. Remark 4.10). When π is a spherical induced representation of Whittaker type, these claimsfollow from the work of Jacquet [Jac67, Th´eor`eme (8.6)].9.2. The Newform via Convolution Sections.
We now use the convolution section identity(8.21) for the newform in the induced model together with the Jacquet integral (9.1) in order togive a convolution section identity for the Whittaker newform. This is a recursive formula for W ◦ as an integral over GL n ( F ) involving W ◦ and the distinguished standard Schwartz functionΦ given by (8.22). Lemma 9.6.
Let π be an induced representation of Langlands type of GL n ( F ) with Whittakernewform W ◦ ∈ W ( π, ψ ) . Then for all h ∈ GL n ( F ) and for (cid:60) ( s ) sufficiently large, (9.7) (cid:90) GL n ( F ) W ◦ ( hg )Φ( g ) | det g | s + n − dg = L ( s, π ) W ◦ ( h ) , where Φ ∈ S (Mat n × n ( F )) is the standard Schwartz function given by (8.22) .Proof. We show this initially for (cid:60) ( t ) > · · · > (cid:60) ( t r ) with (cid:60) ( t ) sufficiently large; this identitythen extends via analytic continuation to all induced representations of Langlands type, for theleft-hand side is absolutely convergent for (cid:60) ( s ) sufficiently large by [Jac09, Lemma 3.2 (ii) andProposition 3.3].We replace h with w n uh in the convolution section identity (8.21) for f ◦ and insert thisidentity into the Jacquet integral (9.1). The result then follows upon interchanging the order ofintegration, which is justified by the absolute convergence of the Jacquet integral together withthe absolute convergence of the integral (8.21). (cid:3) The Newform via Godement Sections.
Next, we use Godement section identities forthe newform in the induced model together with the Jacquet integral (9.1) in order to givea Godement section identity for the Whittaker newform. This is a propagation formula: arecursive formula for W ◦ in terms of an integral over GL n − ( F ) involving a GL n − Whittakerfunction and a distinguished standard Schwartz function.
The Case π = χ κ |·| t . As in Section 8.4, we first treat the case for which π = π (cid:1) · · · (cid:1) π r with n = 1, so that π = χ κ | · | t . Lemma 9.8.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be induced representationsof Langlands type of GL n ( F ) and GL n − ( F ) with π = χ κ | · | t . Let W ◦ ∈ W ( π, ψ ) and W ◦ ∈ W ( π , ψ ) be the Whittaker newforms of π and π . Then for g ∈ GL n − ( F ) , (9.9) W ◦ (cid:18) g
00 1 (cid:19) = | det g | t + (cid:107) κ (cid:107) dF + n − (cid:90) GL n − ( F ) W ◦ ( h )Φ ( h − g )Φ ◦ ( e n − h ) | det h | − t − (cid:107) κ (cid:107) dF − n +1 dh, with the standard Schwartz functions Φ ∈ S (Mat ( n − × ( n − ( F )) and Φ ◦ ∈ S (Mat × ( n − ( F )) given by Φ ( x ) := exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) , (9.10) Φ ◦ ( x ) := (dim τ ◦ ) P ◦ ( x ) exp (cid:0) − d F πx t x (cid:1) , (9.11) where P ◦ is the homogeneous harmonic polynomial associated to the newform K n − -type τ ◦ of π via (7.2) and (7.8) .Remark . When n = 2, so that π = χ κ | · | t (cid:1) χ κ | · | t , the integral over GL ( F ) = F × (cid:51) h in (9.9) may be explicitly evaluated in order to show that W ◦ (cid:18) g
00 1 (cid:19) = | g | t t κ κ K t − t κ − κ (2 π | g | ) if F = R ,4 | g | t t + (cid:107) κ (cid:107) + (cid:107) κ (cid:107) K t − t + (cid:107) κ (cid:107)−(cid:107) κ (cid:107) (4 π | g | / ) if F = C ,where K ν ( z ) denotes the K -Bessel function. Proof.
We show this initially for (cid:60) ( t ) > · · · > (cid:60) ( t r ) with (cid:60) ( t ) sufficiently large; from [Jac09,Proposition 7.2], this identity then extends via analytic continuation to all induced representa-tions of Langlands type (note that Jacquet instead works with representations that are inducedfrom a lower parabolic subgroup rather than an upper parabolic subgroup). We may write W ◦ (cid:18) g
00 1 (cid:19) = (cid:90) Mat ( n − × ( F ) (cid:90) N n − ( F ) f ◦ (cid:18)(cid:18) w n − (cid:19) (cid:18) u v (cid:19) (cid:18) g
00 1 (cid:19)(cid:19) ψ n − ( u ) ψ ( e n − v ) du dv from the definition (9.1) of the Jacquet integral. We insert the Godement section identity(8.33) for f ◦ with g replaced by (cid:0) w n − (cid:1) ( u v ) (cid:0) g
00 1 (cid:1) into this expression, additionally insertingthe identity (9.4) for the normalising constant c ◦ . As explicated in [Jac09, Section 7.2], theensuing double integral is absolutely convergent, so that we can make the change of variables h (cid:55)→ w n − uh and v (cid:55)→ uhv ; we find that W ◦ (cid:0) g
00 1 (cid:1) is equal to i c ( π ) ( − κ ( n − χ κ (det w n ) χ − κ (det w n − ) χ κ (det g ) | det g | t + n − × (cid:90) GL n − ( F ) P ◦ (cid:0) det (cid:0) h − g (cid:1)(cid:1) exp (cid:0) − d F π Tr (cid:0) h − g t g t h − (cid:1)(cid:1) χ − κ (det h ) | det h | − t − n +1 × (dim τ ◦ ) (cid:90) Mat ( n − × ( F ) P ◦ (cid:0) t v (cid:1) exp (cid:0) − d F π t vv (cid:1) ψ ( e n − hv ) dv (cid:90) N n − ( F ) f ◦ ( w n − uh ) ψ n − ( u ) du dh. The integral over Mat ( n − × ( F ) (cid:51) v is equal to i − c ( π ) P ◦ ( e n − h ) exp( − πe n − h t h t e n − ) viaHecke’s identity, Lemmata 7.6 and 7.13, while the integral over N n − ( F ) (cid:51) u is equal to W ◦ ( h )via (9.1). It remains to use (8.25) in conjunction with the definition of P ◦ as well as to notethat det w n det w n − = ( − n − , so that χ κ (det w n ) χ − κ (det w n − ) = ( − κ ( n − . (cid:3) Remark . For spherical Whittaker functions, such a propagation formula (in a slightlymodified form) is due to Gerasimov, Lebedev, and Oblezin [GLO08, Proposition 4.1] and Ishiiand Stade [IsSt13, Proposition 2.1] (cf. [IM20, Appendix A]); iterating this propagation formulagives a recursive formula for GL n ( F ) Whittaker functions in terms of GL ( F ) and GL n − ( F )Whittaker functions known earlier by the work of Stade [Sta90, Theorem 2.1]. RCHIMEDEAN NEWFORM THEORY FOR GL n We also require the following propagation formula for W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) when π (cid:48) is spherical,which follows analogously to Lemma 9.8. Lemma 9.14.
For n ≥ , let π (cid:48) = | · | t (cid:48) (cid:1) | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n and π (cid:48) := | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n bespherical representations of Langlands type of GL n ( F ) and GL n − ( F ) . Let W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) and W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) be the spherical Whittaker functions of π (cid:48) and π (cid:48) . Then for g ∈ GL n ( F ) , (9.15) W (cid:48)◦ ( g ) = | det g | t (cid:48) + n − (cid:90) GL n − ( F ) W (cid:48)◦ ( h ) | det h | − t (cid:48) − n × (cid:90) Mat ( n − × ( F ) Φ (cid:48) (cid:0) h − (cid:0) n − v (cid:1) g (cid:1) ψ ( e n − v ) dv dh, with the standard Schwartz function Φ (cid:48) ∈ S (Mat ( n − × n ( F )) given by (9.16) Φ (cid:48) ( x ) := exp (cid:0) − d F π Tr (cid:0) x t x (cid:1)(cid:1) . The Case π = D κ ⊗ | det | t . We next treat the case for which π = π (cid:1) · · · (cid:1) π r with n = 2, so that F = R and π = D κ ⊗ | det | t . Lemma 9.17.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π ∗ := π ∗ (cid:1) π (cid:1) · · · (cid:1) π r be inducedrepresentations of Langlands type of GL n ( R ) and GL n − ( R ) with π = D κ ⊗ | det | t and π ∗ := | · | t + κ . Let W ◦ ∈ W ( π, ψ ) and W ∗◦ ∈ W ( π ∗ , ψ ) be the Whittaker newforms of π and π ∗ .Then for g ∈ GL n − ( R ) , (9.18) W ◦ (cid:18) g
00 1 (cid:19) = | det g | t + κ − + n − (cid:90) GL n − ( R ) W ∗◦ ( h )Φ ( h − g )Φ ◦ ( e n − h ) | det h | − t − κ − − n +1 dh, with the standard Schwartz functions Φ ∈ S (Mat ( n − × ( n − ( R )) as in (9.10) and Φ ∗◦ ∈ S (Mat × ( n − ( R )) given by (9.19) Φ ∗◦ ( x ) := (dim τ ∗◦ ) P ∗◦ ( x ) exp (cid:0) − πx t x (cid:1) , where P ∗◦ is the homogeneous harmonic polynomial associated to the newform O( n − -type τ ∗◦ of π ∗ via (7.8) .Remark . When n = 2, so that π = D κ ⊗ | det | t , the integral over GL ( R ) = R × (cid:51) h in(9.18) may be explicitly evaluated in order to show that W ◦ (cid:18) g
00 1 (cid:19) = | g | t + κ exp( − π | g | ) . Proof.
First we consider the case n = 2, so that π = D κ ⊗ | det | t and π ∗ = | · | t + κ +12 . Weinsert the Godement section identity (8.37) for f ◦ with g replaced by ( ) (cid:0) v (cid:48) (cid:1)(cid:0) g
00 1 (cid:1) into theJacquet integral (9.1), then interchange the order of integration and make the change of variables v (cid:48) (cid:55)→ a v (cid:48) , yielding W ◦ (cid:18) g
00 1 (cid:19) = i κ | g | t + κ (cid:90) R × | a | a − κ exp (cid:0) − π ( a − g ) (cid:1) × (cid:90) R (cid:90) R P ◦ ( v, v (cid:48) ) exp (cid:0) − π (cid:0) v + v (cid:48) (cid:1)(cid:1) ψ ( a v (cid:48) ) dv dv (cid:48) d × a . Via Hecke’s identity, Lemma 7.13, the integral over R (cid:51) ( v, v (cid:48) ) is i − κ P ◦ (0 , a ) exp( − πa ) = i − κ a κ exp( − πa ), and so we obtain the identity (9.18) upon relabelling a as h .Now we consider the case n ≥
3. We again show this initially for (cid:60) ( t ) > · · · > (cid:60) ( t r ) with (cid:60) ( t ) sufficiently large; from [Jac09, Proposition 7.2], this identity then extends via analyticcontinuation to all induced representations of Langlands type. The derivation of the identity(9.18) is somewhat indirect: we first determine an alternate expression for the right-hand sideof (9.18), and then show that the left-hand side is equal to this expression. To begin, let π ∗ := π ∗ (cid:1) π ∗ (cid:1) π (cid:1) · · · (cid:1) π r and π ∗ := π ∗ (cid:1) π (cid:1) · · · (cid:1) π r be inducedrepresentations of Langlands type of GL n ( R ) and GL n − ( R ) with π ∗ = | · | t ∗ and π ∗ = | · | t ∗ , andlet W ∗◦ ∈ W ( π ∗ , ψ ) and W ∗◦ ∈ W ( π ∗ , ψ ) be the Whittaker newforms of π ∗ and π ∗ .On the one hand, we have from (9.9) that W ∗◦ (cid:0) g
00 1 (cid:1) is equal to(9.21) | det g | t ∗ + n − (cid:90) GL n − ( R ) W ∗◦ ( h )Φ ( h − g )Φ ∗◦ ( e n − h ) | det h | − t ∗ − n +1 dh, with Φ as in (9.10) and Φ ∗◦ as in (9.19).On the other hand, for (cid:60) ( t ∗ ) sufficiently large, W ∗◦ (cid:0) g
00 1 (cid:1) is equal to (cid:90)
Mat ( n − × ( R ) (cid:90) Mat ( n − × ( R ) (cid:90) R (cid:90) N n − ( R ) f ∗◦ w n − u (cid:48) v (cid:48) v (cid:48) v (cid:48) (cid:18) g
00 1 (cid:19) × ψ n − ( u ) ψ ( e n − v (cid:48) ) ψ ( v (cid:48) ) du dv (cid:48) dv (cid:48) dv (cid:48) . We insert the Godement section identity (8.44) for f ∗◦ into this expression, additionally insertingthe identity (9.4) for the normalising constant c ∗◦ . By a straightforward extension of [Jac09,Proposition 7.2], the ensuing multiple integral is absolutely convergent, so that we can makethe change of variables h (cid:55)→ w n − u (cid:48) h , v (cid:48) (cid:55)→ u (cid:48) v (cid:48) , v (cid:48) (cid:55)→ u (cid:48) hv (cid:48) , v (cid:55)→ u (cid:48)− w n − , and v (cid:48) (cid:55)→ a ( v (cid:48) − v hv (cid:48) ). Using the definition of the Jacquet integral, (9.1), to evaluate the ensuingintegral over N n − ( R ) (cid:51) u (cid:48) and Hecke’s identity, Lemma 7.13, to evaluate the ensuing integralsover Mat ( n − × ( R ) (cid:51) v (cid:48) and R (cid:51) v (cid:48) , we find that(9.22) W ∗◦ (cid:18) g
00 1 (cid:19) = ( − c ( π ) L (1 + t ∗ , (cid:102) π ) | det g | t ∗ + n − (cid:90) GL n − ( R ) W ◦ ( h ) | det h | − t ∗ − n − × (cid:90) Mat ( n − × ( R ) exp (cid:18) − π Tr (cid:18) h − (cid:0) n − v (cid:48) (cid:1) g t g (cid:18) n − t v (cid:48) (cid:19) t h − (cid:19)(cid:19) ψ ( e n − v (cid:48) ) × (cid:90) R × | a | − t ∗ + t ∗ exp( − πa ) (cid:90) Mat × ( n − ( R ) (dim τ ◦ ) P ◦ ( a v h ) exp( − πa v h t h t v ) × exp (cid:18) − π Tr (cid:18)(cid:0) v v v (cid:48) + a − (cid:1) g t g (cid:18) t v v (cid:48) v + a − (cid:19)(cid:19)(cid:19) dv d × a dv (cid:48) dh. Here W ◦ is the Whittaker newform for π ◦ := π (cid:1) · · · (cid:1) π r and P ◦ is the homogeneous harmonicpolynomial associated to the newform O( n − τ ◦ of π ◦ via (7.8), and we have usedTheorem 4.15 to write c ( π ∗ ) = c ( π ∗ ) + c ( π ) = c ( π ).Next, we note that the identities (9.21) and (9.22) for W ∗◦ (cid:0) g
00 1 (cid:1) both extend holomorphicallyto t ∗ = t + ( κ − / t ∗ = t + ( κ + 1) /
2. From this, we see that the right-hand side of(9.18) is equal to(9.23)( − c ( π ) L (cid:18) t + κ + 12 , (cid:102) π (cid:19) | det g | t + κ − + n − (cid:90) GL n − ( R ) W ◦ ( h ) | det h | − t − κ − − n − × (cid:90) Mat ( n − × ( R ) exp (cid:18) − π Tr (cid:18) h − (cid:0) n − v (cid:48) (cid:1) g t g (cid:18) n − t v (cid:48) (cid:19) t h − (cid:19)(cid:19) ψ ( e n − v (cid:48) ) × (cid:90) R × | a | exp( − πa ) (cid:90) Mat × ( n − ( R ) (dim τ ◦ ) P ◦ ( a v h ) exp( − πa v h t h t v ) × exp (cid:18) − π Tr (cid:18)(cid:0) v v v (cid:48) + a − (cid:1) g t g (cid:18) t v v (cid:48) v + a − (cid:19)(cid:19)(cid:19) dv d × a dv (cid:48) dh. RCHIMEDEAN NEWFORM THEORY FOR GL n Now we show that W ◦ (cid:0) g
00 1 (cid:1) is equal to (9.23) when (cid:60) ( t ) is sufficiently large, from which theresult shall follow via analytic continuation. We begin by noting that it is equal to (cid:90) Mat ( n − × ( R ) (cid:90) Mat ( n − × ( R ) (cid:90) R (cid:90) N n − ( R ) f ◦ w n − u (cid:48) v (cid:48) v (cid:48) v (cid:48) (cid:18) g
00 1 (cid:19) × ψ n − ( u ) ψ ( e n − v (cid:48) ) ψ ( v (cid:48) ) du dv (cid:48) dv (cid:48) dv (cid:48) . We insert the Godement section identity (8.42) for f ◦ into this expression, additionally insertingthe identity (9.4) for the normalising constant c ◦ . The ensuing multiple integral is againabsolutely convergent, so that we can make the change of variables h (cid:55)→ w n − u (cid:48) h , v (cid:48) (cid:55)→ u (cid:48) v (cid:48) , v (cid:48) (cid:55)→ u (cid:48) hv (cid:48) , v (cid:55)→ u (cid:48)− w n − , and v (cid:48) (cid:55)→ a ( v (cid:48) − v hv (cid:48) ). We again evaluate the ensuing integralover N n − ( R ) (cid:51) u (cid:48) via the definition of the Jacquet integral and use Hecke’s identity, Lemma7.13, to evaluate the integrals over Mat ( n − × ( R ) (cid:51) v (cid:48) and R (cid:51) ( v , v (cid:48) ). The latter integralis equal to i − κ P ◦ (0 , a ) exp( − πa ) = i − κ a κ exp( − πa ). The resulting expression is precisely(9.23). (cid:3) Rankin–Selberg Integrals
It is time to put the propagation formulæ (9.9), (9.15), and (9.18) for W ◦ and W (cid:48)◦ to gooduse. Following the method of Jacquet [Jac09, Section 8], we use these formulæ to express theGL n × GL n Rankin–Selberg integral as the product of a GL n × GL n − Rankin–Selberg integraland a GL n × GL Rankin–Selberg L -function, and similarly express the GL n × GL n − Rankin–Selberg integral as a product of a GL n − × GL n − Rankin–Selberg integral and a GL × GL n − Rankin–Selberg L -function.10.1. GL n × GL n Rankin–Selberg Integrals.
We first consider the GL × GL Rankin–Selbergintegral defined by (2.2); this is simply the Tate zeta integral.
Proposition 10.1.
Let π = χ κ | · | t be a character of F × and let π (cid:48) = | · | t (cid:48) be a sphericalcharacter of F × . Let W ◦ ∈ W ( π, ψ ) be the Whittaker newform of π and let W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) bethe spherical Whittaker function of π (cid:48) . Then for (cid:60) ( s ) sufficiently large, the GL × GL Rankin–Selberg integral Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) is equal to L ( s, π × π (cid:48) ) with the standard Schwartz function Φ ◦ ∈ S (Mat × ( F )) given by Φ ◦ ( x ) := P ◦ ( x ) exp ( − d F πxx ) , where P ◦ is the homogeneous harmonic polynomial associated to the newform K -type τ ◦ of π via (7.2) and (7.8) .Proof. By definition, W ◦ ( g ) = χ κ ( g ) | g | t and W (cid:48)◦ ( g ) = | g | t (cid:48) . We then use (8.25) in conjunctionwith the definition of P ◦ in order to see thatΨ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) = (cid:90) F × | x | s + t + (cid:107) κ (cid:107) dF + t (cid:48) exp( − d F πxx ) d × x. By the identities (2.8) and (2.11) relating the integral to zeta functions, the identities (2.7) and(2.10) relating zeta functions to L -functions, and the identity (2.6) relating Rankin–Selberg L -functions involving twists by a character to standard L -functions, this is precisely L ( s, π × π (cid:48) ). (cid:3) Next, we prove a recursive formula for the GL n × GL n Rankin–Selberg integral for n ≥ Proposition 10.2.
For n ≥ , let π = π (cid:1) · · · (cid:1) π r be an induced representations of Langlandstype of GL n ( F ) , and let π (cid:48) = | · | t (cid:48) (cid:1) | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n and π (cid:48) := | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n bespherical representations of Langlands type of GL n ( F ) and GL n − ( F ) . Let W ◦ ∈ W ( π, ψ ) be theWhittaker newform of π and let W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) and W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) be the spherical Whittakerfunctions of π (cid:48) and π (cid:48) . Then for (cid:60) ( s ) sufficiently large, the GL n × GL n Rankin–Selberg integral Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) is equal to Ψ( s, W ◦ , W (cid:48)◦ ) L (cid:16) s, π × | · | t (cid:48) (cid:17) , with the standard Schwartz function Φ ◦ ∈ S (Mat × n ( F )) given by Φ ◦ ( x ) := (dim τ ◦ ) P ◦ ( x ) exp (cid:0) − d F πx t x (cid:1) , where P ◦ is the homogeneous harmonic polynomial associated to the newform K -type τ ◦ of π via (7.2) and (7.8) .Proof. Just as in [Jac09, Equation (8.1)], we insert the propagation formula (9.15) for W (cid:48)◦ ( g )into the definition (2.2) of Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ); the absolute convergence of the triple integralis shown in [Jac09, Section 8.2]. We replace h with uh , where now u ∈ N n − ( F ) and h ∈ N n − ( F ) \ GL n − ( F ), make the change of variables u (cid:55)→ u − and v (cid:55)→ u − v , then replace ( u v ) g with g , where now g ∈ GL n ( F ); in doing so, we use the fact that W (cid:48)◦ ( uh ) = ψ n − ( u ) W (cid:48)◦ ( h ) andthat ψ n − ( u ) ψ ( e n − v ) W ◦ ( g ) = W ◦ (( u v ) g ). We then make the change of variables g (cid:55)→ (cid:0) h
00 1 (cid:1) g .In this way, we find that Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) is equal to (cid:90) N n − ( F ) \ GL n − ( F ) W (cid:48)◦ ( h ) | det h | s − (cid:90) GL n ( F ) W ◦ (cid:18)(cid:18) h
00 1 (cid:19) g (cid:19) Φ( g ) | det g | s + t (cid:48) + n − dg dh, where we have defined Φ( g ) := Φ ◦ ( e n g )Φ (cid:48) (cid:0)(cid:0) n − (cid:1) g (cid:1) , where Φ (cid:48) ∈ S (Mat ( n − × n ( F )) is as in(9.16). Since the standard Schwartz function Φ ∈ S (Mat n × n ( F )) is as in (8.22), the integralover GL n ( F ) (cid:51) g is equal to L ( s + t (cid:48) , π ) W ◦ (cid:0) h
00 1 (cid:1) from (9.7). This yields the desired identity uponrecalling the definition (2.1) of Ψ( s, W ◦ , W (cid:48)◦ ) and the identity (2.6) relating Rankin–Selberg L -functions involving twists by a character to standard L -functions. (cid:3) n × GL n − Rankin–Selberg Integrals.
The Case π = χ κ |·| t . We now prove a recursive formula for the GL n × GL n − Rankin–Selberg integral. As in Sections 8.4 and 9.3, we first treat the case for which π = π (cid:1) · · · (cid:1) π r with n = 1, so that π = χ κ | · | t . Proposition 10.3.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r be inducedrepresentations of Langlands type of GL n ( F ) and GL n − ( F ) with π = χ κ | · | t a characterof F × , and let π (cid:48) = | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n − be a spherical representation of Langlands type of GL n − ( F ) . Let W ◦ ∈ W ( π, ψ ) and W ◦ ∈ W ( π , ψ ) be the Whittaker newforms of π and π andlet W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) be the spherical Whittaker function of π (cid:48) . Then for (cid:60) ( s ) sufficiently large,the GL n × GL n − Rankin–Selberg integral Ψ( s, W ◦ , W (cid:48)◦ ) is equal to Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ) L (cid:0) s, π × π (cid:48) (cid:1) , with the standard Schwartz function Φ ◦ ∈ S (Mat × ( n − ( F )) given by Φ ◦ ( x ) := (dim τ ◦ ) P ◦ ( x ) exp (cid:0) − d F πx t x (cid:1) , where P ◦ is the homogeneous harmonic polynomial associated to the newform K n − -type τ ◦ of π via (7.2) and (7.8) .Proof. Just as in [Jac09, Equation (8.3)], we insert the propagation formula (9.9) for W ◦ (cid:0) g
00 1 (cid:1) into the definition (2.1) of Ψ( s, W ◦ , W (cid:48)◦ ); the absolute convergence of the ensuing doubleintegral is justified in [Jac09, Section 8.3]. We replace h with uh , where now u ∈ N n − ( F )and h ∈ N n − ( F ) \ GL n − ( F ), make the change of variables u (cid:55)→ u − , then replace ug with g , where now g ∈ GL n − ( F ); in doing so, we use the fact that W ◦ ( uh ) = ψ n − ( u ) W ◦ ( h ) andthat ψ n − ( u ) W (cid:48)◦ ( g ) = W (cid:48)◦ ( ug ). We then make the change of variables g (cid:55)→ hg , leading to theidentityΨ( s, W ◦ , W (cid:48)◦ ) = (cid:90) N n − ( F ) \ GL n − ( F ) W ◦ ( h )Φ ◦ ( e n − h ) | det h | s × (cid:90) GL n − ( F ) W (cid:48)◦ ( hg )Φ ( g ) | det g | s + t + (cid:107) κ (cid:107) dF + n − dg dh, RCHIMEDEAN NEWFORM THEORY FOR GL n where the standard Schwartz functions Φ ∈ S (Mat ( n − × ( n − ( F )) and Φ ◦ ∈ S (Mat × ( n − ( F ))are as in (9.10) and (9.11). From (9.7), the integral over GL n − ( F ) (cid:51) g is equal to L (cid:18) s + t + (cid:107) κ (cid:107) d F , π (cid:48) (cid:19) W (cid:48)◦ ( h ) . From (2.5), (2.6), (2.7), and (2.10), we have that L (cid:18) s + t + (cid:107) κ (cid:107) d F , π (cid:48) (cid:19) = L ( s, π × π (cid:48) ) . This yields the desired identity upon recalling the definition (2.2) of Ψ( s, W ◦ , W (cid:48)◦ , Φ ◦ ). (cid:3) The Case π = D κ ⊗ | det | t . We next treat the case for which π = π (cid:1) · · · (cid:1) π r with n = 2, so that F = R and π = D κ ⊗ | det | t . Proposition 10.4.
For n ≥ , let π = π (cid:1) π (cid:1) · · · (cid:1) π r and π ∗ := π ∗ (cid:1) π (cid:1) · · · (cid:1) π r beinduced representations of Langlands type of GL n ( R ) and GL n − ( R ) with π = D κ ⊗ | det | t and π ∗ = | · | t + κ − , and let π (cid:48) = | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n − be a spherical representation of Langlands typeof GL n − ( R ) . Let W ◦ ∈ W ( π, ψ ) and W ∗◦ ∈ W ( π ∗ , ψ ) be the Whittaker newforms of π and π ∗ and let W (cid:48)◦ ∈ W ( π (cid:48) , ψ ) be the spherical Whittaker function of π (cid:48) . Then for (cid:60) ( s ) sufficientlylarge, the GL n × GL n − Rankin–Selberg integral Ψ( s, W ◦ , W (cid:48)◦ ) is equal to Ψ( s, W ∗◦ , W (cid:48)◦ , Φ ∗◦ ) L (cid:18) s + t + κ − , π (cid:48) (cid:19) , with the standard Schwartz function Φ ◦ ∈ S (Mat × ( n − ( R )) given by Φ ∗◦ ( x ) := (dim τ ∗◦ ) P ∗◦ ( x ) exp (cid:0) − πx t x (cid:1) , where P ∗◦ is the homogeneous harmonic polynomial associated to the newform O( n − -type τ ∗◦ of π ∗ via (7.8) .Proof. The proof is identical to that of Proposition 10.3 except that we use the Godementsection identity (9.18) for W ◦ (cid:0) g
00 1 (cid:1) in place of (9.9). (cid:3)
Proofs of Theorems 4.17 and 4.18.
We first record the following uniqueness principle.
Lemma 10.5.
Suppose that W is a smooth function on GL n − ( F ) of moderate growth thatsatisfies W ( ugk ) = ψ n − ( u ) W ( g ) for all u ∈ N n − ( F ) , g ∈ GL n − ( F ) , and k ∈ K n − . Then if (cid:90) N n − ( F ) \ GL n − ( F ) W ( g ) W (cid:48)◦ ( g ) | det g | s − dg = 0 for all s ∈ C and spherical representations π (cid:48) of GL n − ( F ) , we must have that W ( g ) = 0 for all g ∈ GL n − ( F ) .Proof. This is proved by Jacquet, Piatetski-Shapiro, and Shalika [JP-SS81, Lemme (3.5)] when F is nonarchimedean; the same proof holds for archimedean F with minimal modifications.Alternatively, one can show this via the Whittaker–Plancherel theorem [Wal92, Chapter 15]. (cid:3) With these results in hand, we may complete the proofs of Theorems 4.17 and 4.18.
Proofs of Theorems 4.17 and 4.18.
We prove these theorems by double induction. The basecase is the case n = 1 of Theorem 4.18, which is precisely Proposition 10.1.Suppose by induction that Theorem 4.18 holds with n − n . If π = π (cid:1) π (cid:1) · · · (cid:1) π r and π := π (cid:1) · · · (cid:1) π r with π = χ κ | · | t , then Proposition 10.3 and the induction hypothesisimply that Ψ( s, W ◦ , W (cid:48)◦ ) = L ( s, π × π (cid:48) ) L ( s, π × π (cid:48) ) . By (2.5), this is precisely L ( s, π × π (cid:48) ). Similarly, if π = π (cid:1) · · · (cid:1) π r and π ∗ := π ∗ (cid:1) π (cid:1) · · · (cid:1) π r with π = D κ ⊗ | det | t and π ∗ := | · | t + κ , then Proposition 10.4 and the induction hypothesisimply that Ψ( s, W ◦ , W (cid:48)◦ ) = L ( s, π ∗ × π (cid:48) ) L (cid:18) s + t + κ − , π (cid:48) (cid:19) . By (2.5) and (2.6), this is L (cid:18) s + t + κ − , π (cid:48) (cid:19) L (cid:18) s + t + κ + 12 , π (cid:48) (cid:19) r (cid:89) j =2 L ( s, π j × π (cid:48) ) , which by (2.10), (2.13), and (2.5) is again L ( s, π × π (cid:48) ).Next, suppose by induction that Theorem 4.17 holds. Then Proposition 10.2 and the inductionhypothesis imply that for π (cid:48) = | · | t (cid:48) (cid:1) | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n and π (cid:48) := | · | t (cid:48) (cid:1) · · · (cid:1) | · | t (cid:48) n ,Ψ( s, W ◦ , W (cid:48)◦ , Ψ ◦ ) = L ( s, π × π (cid:48) ) L ( s, π × | · | t (cid:48) ) . By (2.5), this is precisely L ( s, π × π (cid:48) ).Finally, Lemma 10.5 implies the uniqueness of W ◦ as a right K n − -invariant test function forthe GL n × GL n − Rankin–Selberg integral. (cid:3)
Acknowledgements.
I would like to thank Herv´e Jacquet and Akshay Venkatesh for theirencouragement as well as useful discussions. Thanks are also owed to Subhajit Jana for helpfulclarifications regarding the theory of analytic newvectors.
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