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Eisenstein Series, Crystals, and Ice Benjamin Brubaker, Daniel Bump, and Solomon Friedberg

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Eisenstein Series, Crystals, and Ice Benjamin Brubaker, Daniel Bump, and Solomon Friedberg Eisenstein Series, Crystals, and Ice Benjamin Brubaker, Daniel Bump, and Solomon Friedberg utomorphic forms are generalizations Schur Polynomials of periodic functions; they are func- The symmetric group on n letters, Sn, acts on tions on a group that are invariant the ring of polynomials Z[x1, . , xn] by permuting under a discrete subgroup. A natural the variables. A polynomial is symmetric if it is Away to arrange this invariance is by invariant under this action. Classical examples are averaging. Eisenstein series are an important class the familiar elementary symmetric functions of functions obtained in this way. It is possible to e = x ··· x . give explicit formulas for their Fourier coefficients. j i1 ij 1≤i1<···<ij ≤n Such formulas can provide clues to deep connec- tions with other fields. As an example, Langlands’s Since the property of being symmetric is preserved study of Eisenstein series inspired his far-reaching by sums and products, the symmetric polynomials conjectures that dictate the role of automorphic make up a subring n of Z[x1, . , xn]. The ej , forms in modern number theory. 1 ≤ j ≤ n, generate this subring. In this article, we present two new explicit Since n is also anΛ abelian group under poly- formulas for the Fourier coefficients of (certain) nomial addition, it is natural to seek a set that Eisenstein series, each given in terms of a com- generatesΛ n as an abelian group. One such set is given by the Schur polynomials (first considered binatorial model: crystal graphs and square ice. by Jacobi),Λ which are indexed by partitions. A Crystal graphs encode important data associated partition of a positive integer k is a nonincreasing to Lie group representations, whereas ice models sequence of nonnegative integers λ = (λ , λ , . .) arise in the study of statistical mechanics. Both will 1 2 such that k = λ ; necessarily only a finite number be described from scratch in subsequent sections. i of terms in the sequence are nonzero. Partitions We were led to these surprising combinatorial are added componentwise. If λ = (λ ) is a partition connections by studying Eisenstein series not just i with λi = 0 for i > n, let ρ = (n−1, n−2,..., 0, . .), on a group but more generally on a family of and let covers of the group. We will present formulas λ +n−j a = det(x j ) . for their Fourier coefficients that hold even in λ+ρ i 1≤i,j≤n this generality. In the simplest case, the Fourier Then aρ divides aλ+ρ, and the quotient sλ := coefficients of Eisenstein series are described in aλ+ρ/aρ is the Schur polynomial. It is a homo- terms of symmetric functions known as Schur geneous, symmetric polynomial of degree k. For polynomials, so that is where our story begins. example, we have k k−1 (1) s(k,0)(x1, x2) = x1 + x1 x2 + · · · k−1 k Benjamin Brubaker is associate professor of mathemat- + x1x2 + x2 ics at the Massachusetts Institute of Technology. His email = 2 + 2 + 2 address is [email protected]. (2) s(2,1,0)(x1, x2, x3) x1x2 x1x3 x1x2 2 Daniel Bump is professor of mathematics at Stanford + 2x1x2x3 + x1x3 University. His email address is [email protected]. + x2x + x x2. edu. 2 3 2 3 Solomon Friedberg is professor of mathematics at Boston The sλ, running over all partitions λ with λi = 0 College. His email address is [email protected]. for i > n, form a basis for n. Schur showed that This work was supported by NSF grants these polynomials describe the characters of rep- DMS-0844185, DMS-1001079, and DMS-1001326, and resentations of the symmetricΛ and general linear NSA grant H98230-10-1-0183. groups. (See Macdonald [17] for more details.) As December 2011 Notices of the AMS 1563 we will see in subsequent sections, these charac- E∗(z, s) = E∗(z, 1−s). Thismaybeprovedbyspec- ters are connected to the Fourier coefficients of tral methods, as E(z, s) is an eigenfunction of the Eisenstein series. Laplace-Beltrami operator on H . This fact has far-reaching consequences for the Eisenstein Series on SL(2) theory of automorphic forms. As an illustration Let H = {z = x + iy ∈ C | y > 0} denote the in our present case, observe that the invariance 1 1 ∗ ∗ complex upper half plane. The group SL2(R) acts under γ = 0 1 implies that E (z+1, s) = E (z, s). on H by linear fractional transformation: Hence the Eisenstein series admits a Fourier series with respect to the real variable x as follows: az + b a b γ(z)= , where γ = ∈SL (R). ∞ cz + d c d 2 E∗(z, s) = a(r, y, s)e2πinx, It is of interest to find functions that are auto- r=−∞ morphic—invariant under the action of a discrete where subgroupof SL2(R). The modular group = SL2(Z) 1 is of particular importance. One may create a fam- a(r, y, s) = E∗(x + iy, s)e−2πirx dx. ily of automorphic functions on byΓ averaging. 0 To this end, for each s ∈ C with Re(s) > 1, define In the special case r = 0, one can show that the unnormalized Eisenstein seriesΓ a(0, y, s) = y s ξ(2s) + y 1−s ξ(2 − 2s), E(z, s) = Im(γ(z))s , where ξ(2s) = π −s (s)ζ(2s). Because a(0, y, s) γ∈ ∞\ inherits the analytic properties of the Fourier where Γ Γ series, the analyticΓ continuation and functional 1 n 1 ∞ = n ∈ Z . equation of the Riemann zeta function follow. 0 1 What about the remaining Fourier coefficients? Note thatΓ we must quotient out by the subgroup A calculation (see, for example [7], Section 1.6) shows that if r ≠ 0, then ∞ since this is an infinite group that stabilizes s−1/2 1/2 the imaginary part of z. The definition makes a(r, y, s) = 2|r| σ1−2s (|r|)y Ks−1/2(2π|r|y) Γclear that the Eisenstein series is automorphic— where E(γ(z), s) = E(z, s) for all γ ∈ . Using the identity 1−2s Im(γz) = y/|cz + d|2, we may reparameterize the σ1−2s (r) = m sum in terms of integer pairs Γ(c, d). Indeed, each m|r pair of relatively prime integers (c, d) is the bottom and K denotes a K-Bessel function. + 1 rowofamatrixin , and two matrices γ1 and γ2 ∈ Let us shift s to s 2 and examine the “arithmetic −1 havethesamebottomrowifandonlyif γ1γ2 ∈ ∞. parts” of the nonconstant Fourier coefficients of ∗ + 1 Thus the EisensteinΓ series may be expressed inΓ E (z, s 2 ): the form Γ def s y s a(r) = |r| σ−2s (|r|). (3) E(z, s) = , |cz + d|2s They are multiplicative. That is, if gcd(r , r ) = 1, (c,d)∈Z2 1 2 gcd(c,d)=1 then a(r1r2) = a(r1)a(r2). Thus they are com- from which one may deduce that the series pletely determined by their values at prime powers k converges absolutely for Re(s) > 1. r = p . Moreover, The series E(z, s) has many spectacular an- a(pk) = pks + p(k−2)s + · · · + p−ks . alytic properties. To describe them, define the normalized Eisenstein series, A fundamental theme of automorphic forms iden- tifies these coefficients with values of characters (4) E∗(z, s) = 1 π −s (s)ζ(2s)E(z, s), 2 of a representation. Let V denote the standard k where ζ(s) is the Riemann zeta function and (s) representation of SL2(C) and let ∨ V denote Γ ∗ is the gamma function. One can show that E (z, s) the kth symmetric power. Thus if A ∈ SL2(C) has analytic continuation to a meromorphic func-Γ has eigenvalues α, β, then ∨kA has eigenvalues k k−1 k−1 k tion for s ∈ C and satisfies the functional equation α , α β, . , αβ , β . The character χk of the representation ∨kV is given by 1In the first line of this article, we described auto- k k1 k2 morphic forms as functions on groups, but here χk(A) = tr(∨ (A)) = α β . we’ve defined E(z, s) as a function on the upper k1+k2=k half plane H . The resolution of this apparent dis- Comparing this with our earlier expression for the crepancy is that H ≃ SL2(R)/SO2(R) where arithmetic piece a(pk), we find R = cos θ sin θ ∈ R SO2( ) − sin θ cos θ θ [0, 2π) . Indeed, SL2( ) s p acts transitively on the point i by linear fractional k = (5) a(p ) χk −s . transformation with stabilizer SO2(R). p 1564 Notices of the AMS Volume 58, Number 11 Notice that a(pk) is thus the Schur polynomial in group law requires moving between the sheets, s −s (1) evaluated at (x1, x2) = (p , p ): and the nth root of unity that arises in taking the k s −s product of two group elements is computed using (6) a(p ) = s(k,0)(p , p ). the arithmetic of the field L.3 In fact, the existence This identity has substantial generalizations. of this group is closely related to the nth power Indeed, one can define Eisenstein series analogous reciprocity law. to E(z, s) for any reductive group G. In this For these groups, one may define an Eisenstein generality, the notion of Fourier coefficient is series En(z, s) as an average, similar to (3). The replaced by that of Whittaker coefficient. The definition is modified by adding an extra factor in Casselman-Shalika formula [8], first proved for the average that keeps track of the sheets of the GL(n) by Shintani [18], states that the values cover.
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