Rationality Theorems for Hecke Operators on GL n

Transcription

1 Rationality Theorems for Hecke Operators on GL n John A Rhodes and Thomas R Shemanske July 3, 2002 Abstract We define n families of Hecke operators Tk n(pl ) for GL n whose generating series T n k (p l )u l are rational functions of the form q k (u) where q k is a polynomial of degree ( n k), and whose form is that of the kth exterior product This work can be viewed as a refinement of work of Andrianov [], in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially k q k By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest We give explicit examples for GL 3 and GL 4 Introduction While Hecke theory for automorphic forms on GL 2 is fairly robust, far less is known for GL n when n > 2, especially as it relates to the arithmetic theory of modular forms It is well-known (see eg, Cartier [3], Theorem 4) that the Satake map shows that the p-part of the Hecke algebra associated to certain p-adic reductive groups is isomorphic to a polynomial ring invariant under an associated Weyl group In [2], Andrianov and Zhuravlev refer to this isomorphism as the spherical map, and give a description of it for the general linear and symplectic groups in terms of right cosets of the double cosets which generate the Hecke algebra An important aspect of this isomorphism is that it allows one to think about Hecke operators in the polynomial algebra, a setting in which the multiplication is more straightforward 2000 Mathematics Subject Classification Primary F25, 20E42; Secondary F60 Key Words and Phrases Hecke Operators, Rationality, Generating Series

2 2 John A Rhodes and Thomas R Shemanske than the usual product of double cosets Our goals in this paper are to define families of Hecke operators in the polynomial algebra which have generating series whose sum is an extremely nice rational function, and then to invert the Satake isomorphism, to produce classical Hecke operators having generating series of exactly the same form, and hence of number-theoretic interest In [], Andrianov considers classical Hecke operators for GL n, and studies their generating series He shows (Theorem 3) that the image of their sum under the spherical map is a rational function (in Q(x,, x n, u)) with nontrivial numerator, and whose denominator is a product essentially of the form: n k= i <i 2 < <i k n ( x i x i2 x ik u) Notice that the monomials which occur in the inner product, x i x i2 x ik, are kth exterior powers of the variables x,, x n In this paper, we take as a starting point the polynomial ring, and define families of symmetric polynomials, denoted t n k (pl ) (k =,, n) and which we will call polynomial Hecke operators, whose associated generating series are rational functions In particular (see Proposition 35) t n k (pl )u l = i <i 2 < <i k n ( x i x i2 x ik u) Thus the operators t k (p l ) give a significant refinement to the operators investigated by Andrianov In particular, we are producing a family of operators whose generating series are rational functions with trivial numerators, and whose denominators are the individual factors of Andrianov s product Moreover, we shall see that one family, the t n (p l ), are essentially formal analogs of the classical Hecke operators T (p l ) The more difficult task is to determine to what combination of double cosets these polynomial operators correspond To accomplish this one must invert the spherical map By studying the restriction of the spherical map to those double cosets of a fixed determinant, it is possible to (locally) invert the spherical map, the matrix of the restricted map being upper triangular At the heart of the computations to determine the matrix of the linear map is the question of determining the number of right cosets of a specified form which occur in the decomposition of a given double coset In general, this is quite onerous, however we give a number of ways in which the computation can be significantly simplified Intertwined with the general theory, we provide examples which explicitly compute families (Tk 4(pl )) of classical Hecke operators for GL 4 The interest in GL 4 is that n = 4 is the first case in which we associate Euler factors of degree greater than n with Hecke operators for GL n In particular, Proposition 35 shows that the denominator of the rational function associated to tn k (pl )u l is the product of ( n k) linear factors, thus producing Hecke operators for GL n with associated Euler factors of degree ( n k)

3 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 3 2 The Hecke Algebras The global Hecke algebra described by Shimura [6] is generated as a rational vector space by double cosets ΛξΛ with respect to the discrete group Λ = SL n (Z) and ξ M + n (Z) (integer matrices with positive determinant), while the algebra described by Andrianov and Zhuravlev [2] is generated by double cosets ΓξΓ with Γ = GL n (Z) and ξ GL n (Q) If we denote by H(K, S) the rational Hecke algebra generated by double cosets KξK where K is a subgroup of G, S K is a semigroup in the commensurator of K, and ξ S, then we have natural maps H(SL n (Z), M + n (Z)) H(SL n (Z), GL + n (Q)) H(GL n (Z), GL n (Q)), where the map on the left is an injection, and that on the right an isomorphism Moreover, a straightforward argument shows that Proposition 2 H(GL n (Z), GL n (Q)) is generated, as a ring, by H(SL n (Z), M n + (Z)) together with the elements GL n (Z)(p I n )GL n (Z) for all primes p (I n denotes the n n identity matrix) Thus for investigating Hecke theory, it makes little difference which algebra we choose We adopt the notation of [2], and set the global Hecke algebra H n = H(GL n (Z), GL n (Q)) By Theorems 23 and 28 of [2], H n is commutative, and generated by its local subrings, H n p one for each prime p, defined by H n p = H(GL n(z), GL n (Z[p ])) H n The integral subring H n p of H n p is H n p = ΓξΓ ξ M n(z), det(ξ) = ±p λ = Γ diag(p i,, p in )Γ λ i i n 0 where the angle brackets enclose generators as a Q-vector space As a ring, Hp n is generated by H n p together with the single element Γ(p I n )Γ Thus the study of the the global Hecke ring H n reduces to study of the local Hecke rings Hp n, which in turn can be understood through their integral subrings H n p Equivalently, we could focus on the p-adic Hecke ring Letting Γ p = GL n (Z p ), the p-adic Hecke ring is H n p = H(GL n(z p ), GL n (Q p )), and its integral subring H n p = Γ p(diag(p i,, p in ))Γ p i i n 0, which together with the single element Γ p (p I n )Γ p generates Hp n Since Hp n (resp H n p) is canonically isomorphic to Hp n (resp Hn p ), we could work in the p-adic setting (as in studying

4 4 John A Rhodes and Thomas R Shemanske p-adic reductive groups and buildings), but for our purposes here, we shall simply keep things local and not p-adic The integral Hecke algebra H p is generated by the n Hecke operators πk n (p) = Γ diag(p,, p,,, )Γ, }{{}}{{} k n k k =,, n and the local Hecke algebra H p is generated by the n elements above, plus the element πn n(p) (see [2]) The Satake map is an isomorphism between the (p-adically defined) local Hecke algebra and a ring of symmetric polynomials In [2], it is called the spherical map, and we will examine it carefully in 4 For now, simply note that if we let S n denote the symmetric group on n letters and Q[x ±,, x± n the ring of symmetric polynomials in the variables ]Sn x ±,, x ± n (ie those polynomials invariant under the natural action of the symmetric group on the variables), then the spherical map is denoted ω : H p Q[x ±,, x± n, and ]Sn takes H p isomorphically onto Q[x,, x n ] Sn (the usual ring of symmetric polynomials) In particular, it maps πk n (p) to a power of p times the kth elementary symmetric polynomial (see [2]) 3 Polynomial Hecke Operators As discussed in the introduction, the Satake isomorphism allows one to study the Hecke algebra in the context of a polynomial ring, with very familiar ring operations, instead of its defined setting where the multiplication of double cosets is cumbersome In this section, we shall define families of symmetric polynomials, denoted t n k (pl ) (k =,, n), which we will call polynomial Hecke operators, whose associated generating series are rational functions of an especially nice form (see Proposition 35 below) To begin our investigation, note that the symmetric group S n acts naturally on polynomials in n variables Thus for a polynomial p it makes sense to refer to the stabilizer Stab(p) in S n of p For a polynomial p in n variables, define the symmetrized polynomial associated to p, Sym n (p), by Sym n (p) = σ S n/ Stab(p) σ(p) We understand that if p is a constant, that Sym n (p) = p Example 3 In Q[z,, z n ], Sym n (z z k ) = i < <i k n z i z ik Let m and n be positive integers and denote by P n (m) the set of partitions of m into n pieces such that a given partition satisfies: m i i n 0 (and i k = m) Introduce the lexicographic ordering on P n (m), and let i = (i, i 2,, i n ) P n (m) For indeterminates z, z 2,, z n define

5 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 5 hp(0,, 0) =, hp(i) = hp(i, i 2,, i n ) = h n (l) = hp(l, 0,, 0) = }{{} n j i j P n(m) j P n(l) Sym n (z j zj 2 2 zjn n ), and Sym n (z j z j 2 2 z jn n ) = jk =l j k 0 z j z j 2 2 z jn n Note that the hp(i) form a basis for the Q-algebra of symmetric polynomials However, our current interest is in the polynomials h n (l) which have a generating series with a particularly simple rational expression Proposition 32 The generating series associated to the h r (l) satisfies h r (l)u l = [( uz ) ( uz r )] Proof This is essentially obvious: ( ) ( ) [( uz ) ( uz r )] = (uz ) a (uz r ) ar = a 0 u l [ ai =l a i 0 a r 0 ] z a zr ar It is clear from the definitions above that the coefficient of u l in the given expression is h r (l) Let x, x 2,, x n be indeterminates We now define our family of Hecke operators as symmetric polynomials in Q[x ±, x 2 ±,, x ± : For k n, define ]Sn n t n k(p l ) = h (n k) (l) z i σ i (x x 2 x k ) σ i S n/ Stab(x x 2 x k ) This definition is not as mysterious as it first appears We start with h r (l) where r = ( n k) is the size of the orbit, and substitute for the variables z i the elements in the orbit Thus there is one element σ i S n / Stab(x x 2 x k ) to correspond to each of the variables z,, z ( n k) The above definition is sufficiently complex to warrant an example, however we note before giving detail that t n k (p) is nothing more than the kth elementary symmetric polynomial in the variables x, x 2,, x n

6 6 John A Rhodes and Thomas R Shemanske Example 33 We begin by computing Then for example, h r () = hp(, 0,, 0) = Sym }{{} r (z ) = z + z z r r t n (p) = h n () z i σ i (x ) σ i S n/ Stab(x ) the first elementary symmetric polynomial, and in general t n k (p) = k) () h(n z i σ i (x x 2 x k ) = σ i S n/ Stab(x x 2 x k ) the k th elementary symmetric polynomial = x + x x n = s (x, x 2,, x n ), i < <i k n x i x ik = s k (x, x 2,, x n ), Remark 34 First recall that under the Satake isomorphism, the full local Hecke algebra H p is isomorphic to Q[x ±,, x ± n ] Sn = Q[x,, x n ] Sn [(x x n ) ], while the integral local Hecke algebra H p = Q[x,, x n ] Sn Then note that the polynomials t n k (pl ) are symmetric polynomials in the variables x, x 2,, x n, so our operators t n k (pl ) lie in the integral local Hecke algebra Turning to generating series, we find Proposition 35 For k =, 2,, n, the generating series for the Hecke operators t n k (pl ) is a rational function: [ t n k (pl )u l = σ S n/ Stab(x x 2 x k ) ] ( uσ(x x 2 x k )) Proof From Proposition 32, we know that But which completes the proof [ k) (n h k) (n (l)u l = ( uz i )] i= t n k (pl ) = h (n k) (l) z i σ i (x x 2 x k ) σ i S n/ Stab(x x 2 x k )

7 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 7 Example 36 Consider the case of n = 4 We have the four Hecke series: t 4 (p l )u l = [( ux )( ux 2 )( ux 3 )( ux 4 )] t 4 2 (pl )u l = [( ux x 2 )( ux x 3 )( ux x 4 )( ux 2 x 3 )( ux 2 x 4 )( ux 3 x 4 )] t 4 3(p l )u l = [( ux x 2 x 3 )( ux x 2 x 4 )( ux x 3 x 4 )( ux 2 x 3 x 4 )] t 4 4 (pl )u l = [( ux x 2 x 3 x 4 )] Remark 37 Later in this paper we establish a connection between these t n k and classical Hecke operators Accepting such a connection for now, note that the Euler factor corresponding to t n k has degree ( n k) and the monomials are the kth exterior powers of the xi s Then, observing that the Euler factor corresponding to t n (pl )u l seems most basic to GL n and that the Euler factor corresponding to t n k (pl )u l has degree ( n k) perhaps suggests a correspondence between forms on GL n and forms on GL ( n via the exterior power L-series k) 4 The Spherical Map Now that we have defined interesting Hecke operators in the polynomial ring, we need to invert the spherical map to characterize their classical counterparts Let Γ = GL n (Z) and G = GL n (Z[p ]) Andrianov [2] defines the spherical map ω : Γ\G Q[x ±,, x± n ], by observing that every right coset has a unique representative in upper-triangular form with powers of p on the diagonal, and entries in any given column taken modulo the corresponding diagonal entry (analogous to Hermite normal form) On such representatives he defines the map (extending linearly) by: p b ω Γ = p m x b 0 p bn x b 2 2 x bn where ib i = m and the b i are integers He later proves that the restriction of ω to H p (viewing the double coset as a sum of right cosets) gives an isomorphism onto the ring of symmetric polynomials Q[x ±,, x± n ]Sn n 4 Inverting the spherical map As we mentioned earlier, we may assume we are working with the integral Hecke ring H p, so we need to analyze the image of the spherical map on double cosets ΓξΓ with ξ GL n (Z[p ]) M n (Z), det(ξ) = p n Restricted to such double cosets, the spherical

8 8 John A Rhodes and Thomas R Shemanske map has image equal to the space spanned by symmetric polynomials of total degree n In fact, with respect to suitable bases, the restriction of the spherical map to the space of cosets spanned by double cosets ΓξΓ with det(ξ) = p n has a matrix which is upper triangular Given this fact, the spherical map is (at least in principle) easy to invert That the spherical map has this property could no doubt be deduced from the analogous result regarding representations of p-adic groups ([5],[3]), but our method of proof is also computationally valuable, so we include it here, especially since we need to find some explicit inverse images As noted above, for positive integers m and n, P n (m) is the set of partitions of m into n pieces with the lexicographic ordering Note that in particular, the elements of P n (m) are linearly ordered By standard elementary divisor theory, for any g GL n (Z[p ]) M n (Z), with det g = p m, ΓgΓ = Γ diag(p b, p b 2,, p bn )Γ for precisely one b = (b, b 2,, b n ) P n (m) For simplicity, write Γp b Γ for Γ diag(p b, p b 2,, p bn )Γ, and write Sym n (x b ) for Sym n (x b x bn n ), the symmetrized polynomial under the action of the symmetric group Now fix the integer m, and list the elements of P n (m) in order: P n (m) = {a < a 2 < < a t } Consider the double cosets {Γp a i Γ} and the symmetric polynomials {Sym n (x a i )} with an ordering induced from the one on P n (m) Each set is a linearly independent subset of the appropriate vector space Consider the restriction of Andrianov s spherical map ω : H p Q[x ±,, x ± n ], to the subspace generated by the {Γp a i Γ} The image of the spherical map restricted to this domain is spanned by the {Sym n (x a i )}, and we show below that the matrix of this linear map is upper triangular We begin with a proposition of use in its own right Proposition 4 Let n 2 and m be integers, and let b P n (m) Then Γp b Γ = p a Γ (disjoint) 0 p an where, in any given column, the entries are taken modulo the corresponding diagonal entry Moreover, for some σ S n, a = (a σ(),, a σ(n) ) P n (m) and a b Proof All but the last statement is established in Lemma 27 of [2] Since all diagonal entries are non-negative powers of p and any right coset representative must have the same determinant as p b, it follows that there is a σ S n with a = (a σ(),, a σ(n) ) P n (m) For any right coset occurring in the decomposition of the double coset Γp b Γ, we must have that the p a corresponding double cosets agree, that is Γ Γ = Γp b Γ From Theorem II0 0 p an p a of [4], this is true if and only if the two matrices, and diag(p b, p b 2,, p bn ), 0 p an have the same determinantal divisors Recall that the kth determinantal divisor is simply

9 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 9 the gcd of the determinants of all k k sub-matrices of the matrix, and that the quotients of successive determinantal divisors yield the elementary divisors By a k k sub-matrix we mean the matrix obtained by fixing k row and k column indices To proceed we need a small Lemma 42 Let a = (a,, a n ), b = (b,, b n ) P n (m) with b < a There is a smallest n n index k for which b k > a k For this k, b i > a i k Proof Since b < a there is an index i 0 such that b i = a i for i < i 0, and b i0 < a i0 Thus i 0 i 0 n n b i < a i But since a, b P n (m), a i = b i, so there must exist a smallest index k k k k (clearly > i 0 ) for which b k > a k It follows that b i < a i, so that required n b i > k n a i, as p a To complete the proof of the proposition, we have that Γ is a right coset 0 p an representative of Γp b Γ, and that a = (a σ(),, a σ(n) ) P n (m) for some σ S n Suppose to the contrary that a > b Recall we have m b b n 0 and m a σ() a σ(n) 0 Denote by d b k and da k the kth determinantal divisor of diag(pb,, p bn ) and p a respectively 0 p an The determinantal divisors of diag(p b,, p bn ) are clear: d b =, d b pbn 2 = pbn+b n,, d b n = pbn+ +b Since we are assuming that a > b, we have by the lemma a smallest index k for which b k > a σ(k), and hence for which d b n+ k = p n k b i > p n k a σ(i) However d a n+ k is the gcd of all possible determinants of (n + k) (n + k) sub-matrices, and choosing rows and columns σ(k),, σ(n) for one such matrix produces a determinant smaller than d b n+ k, thus showing that da n+ k < db n+ k, a contradiction k We now have a characterization of the spherical map Theorem 43 The matrix of the spherical map, restricted as above, is nonsingular and upper triangular In particular, if b P n (m), then ω(γp b Γ) = for constants c(a) with c(b) 0 a P n(m), a b c(a) Sym n (x a )

10 0 John A Rhodes and Thomas R Shemanske Proof It is enough to consider ω restricted to the integral subring H n p We compute the image of ω on the double coset Γ diag(p b,, p bn )Γ for b = (b,, b n ) P n (m) From p a σ() Proposition 4 above, the double coset is a union of right cosets Γ 0 p a σ(n) where a = (a,, a n ) P n (m), a b, and σ S n By definition of the spherical map [2], p a σ() ω Γ = p ia σ(i) x a σ() x a σ(n) n, 0 p a σ(n) so to understand the image of a double coset, we obviously need to count the number of such right cosets occur in Γp b Γ Denote this number by ψ σ a,b Then ω(γp b Γ) = a b, a P n(m) σ S n/ Stab(x a ) ψa,b σ p ia σ(i) x a σ() x a σ(n) n Note that the inner sum is over σ and not σ since if σ(i) = j, then x a σ(i) i = x a j σ (j) Moreover, the sum over σ is not generally the same as the sum over σ unless Stab(x a ) is a normal subgroup of S n On the other hand, ω(γp b Γ) is a symmetric polynomial, so that ψa,b σ p ia σ(i) = ψa,b τ p ia τ(i) for any two σ, τ Sn In particular, ψa,b σ = p i(a σ(i) a i ) ψa,b, where denotes the identity permutation, and consequently ω(γp b Γ) = a b, a P n(m) ψ a,b p ia i Sym n (x a ) (4) Thus c(a) = ψ a,b p ia i, and it is clear that c(b) 0 since one of the right coset representatives of the double coset is the diagonal matrix diag(p b,, p bn ) Remark 44 Of course the coefficients c(a) = ψa,b σ p ia σ(i) for any σ Sn, however ψ a,b p a is the easiest to compute since we are assuming the right coset has the form 0 p an with a a 2 a n, and where the entries in the ith column can be taken modulo p a i, and hence this case presents the fewest right cosets to analyze In any real computation, this simplification represents an enormous saving of time, and to some extent complexity since determining which right cosets belong to a given double coset involves computing determinantal divisors We note that the complexity of the problem depends not so much on n or m, but on the cardinality of P n (m) which grows extremely fast with m even for small n

11 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N Corollary 45 The matrix of the inverse of the spherical map, restricted as above, is upper triangular In particular, if b P n (m), then Sym(x b ) = d(a)ω(γp a Γ) for constants d(a) a P n(m), a b Remark 46 We point out three further observations which can be useful for simplifying such computations Let P n (m) = {λ,, λ r } and assume that λ < < λ r If λ i = (λ i,,, λ i,n ), let χ(λ i, λ j ) denote the number of right cosets of the form p λ i,σ() Γ 0 p λ i,σ(n) From the discussion above, we know χ(λ i, λ j ) = 0 whenever λ i > λ j which occur in the double coset Γp λ j Γ, as σ ranges over S n Certainly, we already know that for any a, b P n (m) (a b) χ(a, b) = ψa,b σ = p i(a σ(i) a i ) ψa,b, (42) σ S n/ Stab(x a ) but we also have the following relations: σ S n/ Stab(x a ) µ(λ j ) = χ(λ, λ j ) + + χ(λ j, λ j ) is the degree (ie, the total number of right cosets in the double coset) of the double coset Γp λ j Γ which is explicitly computed in Lemma 6 of [] 2 χ(λ j, λ j ) + + χ(λ j, λ r ) is the total number of possible right cosets with p λ j,k s (j fixed) on the diagonal This is trivial to compute since for any given upper triangular matrix, all the entries above are taken modulo the diagonal entry in that column 3 Refining the previous item, we have that ψ λ j,λ j + + ψ λ j,λ r = p l, l = r λ j,k (k ) k= The first two observations produce 2r linear relations among the r(r + )/2 variables χ(λ i, λ j ) These together with the third observation will be used in the example below to further reduce the computational load of inverting the spherical map 42 Some Examples In this section we provide two examples one trivial, one not, which will be used to define our new families of Hecke operators for GL 4, and which demonstrate the observations given in the previous section We shall compute the inverse image (under the spherical map) of t 4 (p2 ) and t 4 2 (p2 )

12 2 John A Rhodes and Thomas R Shemanske We consider first the almost trivial case of t 4 (p 2 ) which requires we analyze matrices of determinant p 2 P 4 (2) = {λ, λ 2 } where λ = (,, 0, 0) < λ 2 = (2, 0, 0, 0) To ease the notation, we write ψ ij for ψλ i,λ j in both examples below, and denote the kth determinantal divisor d λ i k as di k There is only one right coset with diagonal (p 2,,, ) since all the upper triangular entries are ( read modulo the diagonal entry, so it is clear that ψ 2,2 = The right cosets of the form p a 0 0 ) Γ p 0 0 belong either to the double coset Γ diag(p, p,, )Γ or to Γ diag(p 2,,, )Γ, and 0 the third determinantal divisor easily determines which for us, with d 3 = p while d2 3 = For a typical coset we see that the third determinantal divisor is d 3 = gcd(p, a) Now a runs mod p, so we immediately see that ψ = and ψ 2 = p Recall from equation 4, ω(γp b Γ) = a b, a P n(m) ψ a,b p ia i Sym(x a ) Let A = (a ij ) denote the (upper triangular) matrix of the (restricted) spherical map, so ω(γp λ j Γ) = i j a ij Sym(x λ i ) If λ i = (λ i,,, λ i,n ), then a ij = p ( kλ i,kψij, thus A = p 3 p 3 (p ) ) 0 p 2 ( ) It s inverse is A = p 3 p 2 (p ), that is 0 p 2 ω (Sym 4 (x λ )) = ω (Sym 4 (x x 2 )) = p 3 Γ diag(p, p,, )Γ and ω (Sym 4 (x λ 2 )) = ω (Sym 4 (x 2 )) = p2 Γ diag(p 2,,, )Γ p 2 (p )Γ diag(p, p,, )Γ Since t 4 (p 2 ) = Sym 4 (x 2 ) + Sym 4 (x x 2 ), it follows that ω (t 4 (p 2 )) = p 2 Γ diag(p 2,,, )Γ + p 2 Γ diag(p, p,, )Γ Next we consider the harder example of t 4 2 (p2 ) This requires that we study matrices of determinant p 4 P 4 (4) = {λ,, λ 5 }, where λ = (,,, ) < λ 2 = (2,,, 0) < λ 3 = (2, 2, 0, 0) < λ 4 = (3,, 0, 0) < λ 5 = (4, 0, 0, 0) We need to compute ψ ij for i j 5 For 2 i 5, the ψ ij are computed by means of analyzing determinantal divisors ) For example for i = 2, the task is to determine how many right cosets Γ (with a, b, c (mod p)) belong to the double cosets Γ diag(p 2, p, p, )Γ, ( p 2 a b 0 p c 0 p 0 Γ diag(p 2, p 2,, )Γ, Γ diag(p 3, p,, )Γ, and Γ diag(p 4,,, )Γ These numbers will be (respectively) ψ 22, ψ 23, ψ 24, ψ 25 If we put Ψ = (ψ ij ), then careful counting with determinantal divisors gives us all of Ψ except the first row: p 2p(p ) p(p ) 2 Ψ = p p(p ) p Note that as a check, the row sums are consistent with observation 3 in Remark 46 In principle, the first row could also be computed in this fashion although a quick inspection of

13 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 3 the problem reveals a myriad of cases which must be analyzed to determine the determinantal divisors So instead, we take advantage of the other observations we made in the previous section First we observe by equation 42, that χ,j = ψ j since the stabilizer of x λ = x x 2 x 3 x 4 is all of S 4 Then we note from Remark 46 that χ,j = µ(λ j ) χ 2,j χ 3,j χ j,j, where the µ(λ j ) is the degree of the double coset Γp λ j Γ which is explicitly computed in Lemma 6 of [] Unfortunately, we must also compute the χ i,j (i > ) from our knowledge of the ψ ij using equation 42 If we define ν(λ i ) by χ i,j = ν(λ i )ψ ij, we compute that µ(λ i ) ν(λ i ) λ 2 p(p 2 + p + )(p 2 + )(p + ) + 2p + 3p 2 + 3p 4 + 2p 5 + p 6 λ 3 p 4 (p 2 + p + )(p 2 + ) + p 2 + 2p 4 + p 6 + p 8 λ 4 p 5 (p 2 + p + )(p 2 + )(p + ) + p + 2p 2 + p 4 + 2p 5 + p 6 + 2p 8 + p 9 + p 0 λ 5 p 9 (p 2 + )(p + ) + p 4 + p 8 + p 2 Putting this together we conclude: ψ = χ = ψ 2 = χ 2 = 3p 3 p 2 p ψ 3 = χ 3 = 2p 4 3p 3 + p ψ 4 = χ 4 = 3p 5 5p 4 + p 3 + p 2 ψ 5 = χ 5 = p 6 3p 5 + 3p 4 p 3 Now if A = (a ij ) once again represents the matrix of the spherical map with respect to the canonical bases, then a ij = p kλ i,k ψij, and we have A = 3 p 3 p 2 p p 0 p p 4 3 p 3 + p 3 p 5 5 p 4 + p 3 + p 2 p 6 3 p p 4 p 3 p 0 p 0 p 0 p 2p(p ) p(p ) 2 p 7 p 7 p 7 p p p(p ) p 6 p 6 p 6 p p 5 p 5 p 4

14 4 John A Rhodes and Thomas R Shemanske It s inverse A is: p 0 p 7 (3p 3 p 2 p ) p 6 (p 4 p 3 p + ) p 5 (2p 5 p 4 2p 3 + ) p 4 (p 6 p 5 p 4 + p 2 + p ) 0 p 7 p 6 (p ) p 5 (p )(p + ) p 4 (p ) 2 (p + ) 0 0 p 6 p 5 (p ) p 4 (p ) p 5 p 4 (p ) p 4 We compute that t 4 2(p 2 ) = Sym 4 (x λ 3 ) + Sym 4 (x λ 2 ) + 3 Sym 4 (x λ ), so ω (t 4 2 (p2 )) = [(p 4 p 3 + p)p 6 Γp λ Γ p 6 (p )Γp λ 2 Γ + p 6 Γp λ 3 Γ] + [ p 7 (3p 3 p 2 p )Γp λ Γ + p 7 Γp λ 2 Γ] + 3[p 0 Γp λ Γ] = p 6 [ (p 4 + p 2 + )Γp λ Γ + Γp λ 2 Γ + Γp λ 3 Γ ] = p 6 [ (p 4 + p 2 + )Γ diag(p, p, p, p)γ + Γ diag(p 2, p, p, )Γ + Γ diag(p 2, p 2,, )Γ ] 5 New Families of Hecke Operators Recall that we have defined families of polynomial Hecke operators where h r (l) = j P r(l) t n k (pl ) = h (n k) (l) z i σ i (x x 2 x k ) σ i S n/ Stab(x x 2 x k ) Sym r (z j zj 2 2 zjr r ) = [ t n k (pl )u l = jk =l j k 0 σ S n/ Stab(x x 2 x k ) z j zj 2 2 zjr r, and which satisfy ( uσ(x x 2 x k ))], (5) and t n k (p) is the kth elementary symmetric polynomial in the variables x,, x n One could attempt simply to apply the inverse of the spherical map to the elements t n k (pl ), but this is neither practical, nor particularly illuminating Instead, we realize that even for GL 2 the Hecke operators are most clearly defined by defining some base cases and the recursion relations satisfied by the operators, so this is how we proceed here Equation 5 implicitly contains the recursion relations for the t n k (pl ) via [ ] t n k(p l )u l σ S n/ Stab(x x 2 x k ) ( uσ(x x 2 x k )) = (52) In the following sections, we completely characterize t n (p l ) and t n n(p l ) for any n Then as we saw hinted at in Example 36, we investigate a sort of duality between t n k and tn n k The

15 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 5 remaining operators are difficult to characterize in general, but in the last section, we work out the details for GL 4 This is the first really interesting case as it associates an Euler factor of degree > n to a Hecke operator for GL n 5 The operators t n (pl ) and t n n (pl ) The case k = recovers the classical Hecke operators (see 32 of [6]) In particular, we see that if s k = s k (x,, x n ) = t n k (p) is the kth elementary symmetric polynomial, then [ ] t n (pl )u l [ s u + s 2 u 2 s 3 u ( ) n s n u n] =, or (letting t n (p j ) = 0 for j < 0), t n (pl ) = t n (pl )s t (p l 2 )s ( ) n t n (pl n )s n = t n (pl )t n (p) t (p l 2 )t n 2 (p) + + ( )n t n (pl n )t n n (p) At the other extreme are the operators t n n which satisfy tn n(p l )u l = [ ux x n ] which is the sum of a geometric series, yielding t n n (pl ) = t n n (p)l = [s n (x,, x n )] l 52 Relations between t n k and tn n k We note that there is a sort of duality between t n k (pl ) and t n n k (pl ) Since ( ) ( n k = n see that t n n k(p l ) = h ( n n k) z i σ i (x x 2 x n k ) σ i S n/ Stab(x x 2 x n k ) h k) (n (l) z i σ i (x x 2 x k ) σ i S n/ Stab(x x 2 x k ) = h (n k) zi x x 2 x n/σ i (x x 2 x k ) σ i S n/ Stab(x x 2 x k ) n k), we while t n k (pl ) = The correspondence is most easily seen between t and t n An elementary computation shows s n (x,, x n ) s k (x,, x n ) = s n k(x,, x n ) and hence (letting s n = s n (x,, x n ) = x x n ), Thus, t n (pl )u l = s k (s n /x,, s n /x n ) = s k n (x,, x n ) s n k (x,, x n ) σ S n/ Stab(x ) ( uσ(x )) = [ us (x,, x n ) + u 2 s 2 (x,, x n ) + + ( ) n u n s n (x,, x n ) ]

16 6 John A Rhodes and Thomas R Shemanske while for t n n we have [ t n n (p l )u l = = σ S n/ Stab(x x 2 x n ) σ S n/ Stab(x ) [ n = ( us n /x j ) j= showing the nice symmetry ] ( uσ(x x 2 x n )) ( us n /σ(x )) ] = [ us (s n /x,, s n /x n ) + + ( ) n u n s n (s n /x,, s n /x n )] = [ us n (x,, x n ) + u 2 s n (x,, x n )s n 2 (x,, x n ) u 3 s 2 n (x,, x n )s n 3 (x,, x n ) + + ( ) n u n s n n (x,, x n )], 53 The recursion relations for Hecke operators on GL 2, GL 3, GL 4 Having done what we can for the operators for general n, here we work out all the recursion relations defining the Hecke operators on GL n for n = 2, 3, 4 For small n there is little choice in how these relations are written down, but as we shall see for n 4, it is possible to write down multiple recursion relations which appear quite distinct GL 4 is the first really interesting case, however for completeness we write down the recursion relations for n = 2, 3, 4 Throughout, we put t n k (pj ) = 0 for j < 0 (and t n k () = by definition), and recall t n n(p l ) = t n n(p) l The relations for the case n = 2 simply consists of a base case and one relation: t 2 k(p) = s k (x, x 2 ), k =, 2 t 2 (pl ) = t 2 (pl )t 2 (p) t2 (pl 2 )t 2 2 (p) For n = 3, we have a base case and two nontrivial relations: t 3 k(p) = s k (x, x 2, x 3 ), k =, 2, 3 t 3 (pl ) = t 3 (pl )t 3 (p) t3 (pl 2 )t 3 2 (p) + t3 (pl 3 )t 3 3 (p) t 3 2 (pl ) = t 3 2 (pl )t 3 2 (p) t3 2 (pl 2 )t 3 (p)t3 3 (p) + t3 2 (pl 3 )t 3 3 (p)2 For n = 4, we analogously have a base case and three nontrivial relations The relations for t 4 (pl ) and t 4 3 (pl ) are fairly easy to write down from what we have noted above, but the relation for t 4 2 (pl ) is somewhat more involved, and indeed is much more suggestive of the general case t n k (pl ) for n > 4 The relations for t 4 2(p l ) are deduced from the identity: [ t 4 2 (pl )u l ] [( ux x 2 )( ux x 3 )( ux x 4 )( ux 2 x 3 )( ux 2 x 4 )( ux 3 x 4 )] =

17 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 7 To write things in closed from, we adopt some ad hoc notation to simplify the exposition Let t(a, a 2, a 3, a 4 ) be the symmetric polynomial Sym 4 (x a x a 2 2 x a 3 3 x a 4 4 ) Then ( ux x 2 )( ux x 3 )( ux x 4 )( ux 2 x 3 )( ux 2 x 4 )( ux 3 x 4 ) = ut(,, 0, 0) + u 2 (t(2,,, 0) + 3t(,,, )) u 3 (t(3,,, ) + t(2, 2, 2, 0) + 2t(2, 2,, )) + u 4 (t(3, 2, 2, ) + 3t(2, 2, 2, 2)) u 5 t(3, 3, 2, 2) + u 6 t(3, 3, 3, 3) There are often many ways to represent the coefficients above in terms of the t 4 k (pl ) For example, the coefficient of u 3 is t(3,,, ) + t(2, 2, 2, 0) + 2t(2, 2,, ) = t 4 4 (p)t4 (p2 ) + 2t 4 3 (p2 ) t 4 3 (p)2 = t 4 4(p)(t 4 (p 2 ) + t 4 3(p)) and the shape of the recursion relations we write down depends upon the choice of the representation Here we opt for simplicity, and express ( ux x 2 )( ux x 3 )( ux x 4 )( ux 2 x 3 )( ux 2 x 4 )( ux 3 x 4 ) = ut 4 2(p) + u 2 (t 4 2(p) 2 t 4 2(p 2 )) u 3 t 4 4(p)(t (p 2 ) + t 4 3(p)) + u 4 t 4 4 (p)(t4 2 (p)2 t 4 2 (p2 )) u 5 t 4 4 (p)2 t 4 2 (p) + u6 t 4 4 (p)3 From this we deduce the recursion relations for n = 4: t 4 k (p) = s k(x, x 2, x 3, x 4 ), k =, 2, 3, 4 t 4 (p l ) = t 4 (p l )t 4 (p) t 4 (p l 2 )t 4 2(p) + t 4 (p l 3 )t 4 3(p) t 4 (p l 4 )t 4 4(p) t 4 2 (pl ) = t 4 2 (pl )t 4 2 (p) t4 2 (pl 2 )(t 4 2 (p)2 t 4 2 (p2 )) + t 4 2 (pl 3 )t 4 4 (p)(t4 (p2 ) + t 4 3 (p)) t 4 2 (pl 4 )t 4 4 (p)(t4 2 (p)2 t 4 2 (p2 )) + t 4 2 (pl 5 )t 4 2 (p)t4 4 (p)2 t 4 2 (pl 6 )t 4 4 (p)3 t 4 3(p l ) = t 4 3(p l )t 4 3(p) t 4 3(p l 2 )t 4 2(p)t 4 4(p) + t 4 3(p l 3 )t 4 (p)t 4 4(p) 2 t 4 3(p l 4 )t 4 (p) 3 6 The Hecke operators in the classical setting We finally reach the point where we can define the classical Hecke operators which correspond to our polynomial ones These are the operators defined in terms of double cosets which should be of number-theoretic interest because of their well structured generating series Recall our notation for the standard generators of H p : π n k (p) = Γ diag(p,, p,,, )Γ, }{{}}{{} k n k k =,, n We let T n k (pl ) be the inverse image under the spherical map of t n k (pl ) Since the spherical map ω is an isomorphism, the T n k (pl ) will satisfy exactly the same recursion relations as

18 8 John A Rhodes and Thomas R Shemanske the t n k (pl ) They will also have a generating series T n k (pl )u l which are rational functions of exactly the same form as the polynomial Hecke operators, in particular, T n k (pl )u l = [q n k (u)] where q n k is a polynomial of degree ( n k) Finally, we note that to define the operators T n k (pl ), we need only the base cases and the images of t n k (pl ) which occur in the defining recurrence relations So for n = 2 or 3, we need only the images of t n k (p), while for n = 4, we need the images of t4 k (p) as well as t4 k (p2 ) for k =, 2 which were the examples computed in a previous section By Lemma 22 of [2], we have that Tk n(p) = pk(k+)/2 πk n (p), and from our earlier examples we obtained: T 4 (p 2 ) = p 2 (Γ diag(p 2,,, )Γ + Γ diag(p, p,, )Γ T 4 2 (p 2 ) = p 6 [ (p 4 + p 2 + )Γ diag(p, p, p, p)γ + Γ diag(p 2, p, p, )Γ + Γ diag(p 2, p 2,, )Γ ] which, together with the recursion relations of the last section, characterizes the structure of these new Hecke operators for GL 4 Remark 6 It may occur to the reader that since the t n k (pl ) are symmetric polynomials, they are polynomials in the elementary symmetric polynomials (t n k (p)), and that the inverse image of the elementary symmetric polynomials is known, so why go through the trouble we have? The point is that by taking this later approach, even the generators (eg t 4 (p2 ) and t 4 2 (p2 )) of our classical Hecke algebra would only be defined as algebraic combinations of double cosets instead of linear combinations, and it was the cumbersome nature of computing products of double cosets which led us to the polynomial setting in the first place Of course, defining the algebra in terms of recursion formulas forces one to consider products, but perhaps it is slightly more esthetic to have at least the generators characterized simply

19 RATIONALITY THEOREMS FOR HECKE OPERATORS ON GL N 9 References [] A N Andrianov, Spherical functions for GL n over local fields and summation of hecke series, Math USSR Sbornik 2 (970), no 3 [2] A N Andrianov and V G Zhuravlëv, Modular forms and Hecke operators, American Mathematical Society, Providence, RI, 995, Translated from the 990 Russian original by Neal Koblitz MR 96d:045 [3] P Cartier, Representations of p-adic groups: a survey, Automorphic forms, representations and L-functions (Proc Sympos Pure Math, Oregon State Univ, Corvallis, Ore, 977), Part, Amer Math Soc, Providence, RI, 979, pp 55 MR 8e:22029 [4] Morris Newman, Integral matrices, Academic Press, New York, 972, Pure and Applied Mathematics, Vol 45 MR 49 #5038 [5] I Satake, Theory of spherical functions on reductive algebraic groups over P-adic fields, Inst Hautes Etudes Sci Publ Math 8 (963), 5 69 [6] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No Iwanami Shoten, Publishers, Tokyo, 97, Kanô Memorial Lectures, No MR 47 #338 Department of Mathematics, Bates College, Lewiston, Maine address: jrhodes@batesedu Department of Mathematics, 688 Bradley Hall, Dartmouth College, Hanover, NH Fax: (603) address: thomasrshemanske@dartmouthedu URL: