##
**Fourier coefficients of modular forms on \(G_2\).**
*(English)*
Zbl 1165.11315

From the introduction: One of the most surprising aspects of the classical theory of modular forms \(f\) on the group \(\text{SL}_2(\mathbb Z)\) is the wealth of information carried by the Fourier coefficients \(a_n(f)\) for \(n\geq 0\). The Fourier coefficients of Eisenstein series were calculated by E. Hecke and C. Siegel and are instrumental in the study of zeta functions at negative integers. The Fourier coefficients of theta series have been studied since the work of C. Jacobi; they give many deep results on Euclidean lattices, such as the unicity of the Leech lattice. Finally, the action of Hecke operators on Fourier coefficients goes back to L. Mordell and allows one to show that the Mellin transform of an eigenform is an \(L\)-function with Euler product.

Siegel developed a theory of Fourier coefficients \(c_N(f)\) for holomorphic forms \(f\) on the symplectic group \(\text{Sp}_{2g}(\mathbb Z)\). Here the coefficients, for forms of even weight, are indexed by positive semidefinite, integral even quadratic spaces \(N\) of rank \(g\). There is an analogous theory for holomorphic forms on tube domains, where the Fourier coefficients are indexed by orbits on integral elements in the corresponding homogeneous cone.

On the other hand, one has a less refined notion of Fourier coefficients for a general automorphic form \(f\) on a general reductive group \(G\). Given any parabolic subgroup \(P=M\cdot N\) of \(G\) and a unitary character \(\chi\) of \(N(\mathbb A)\) trivial on \(N(\mathbb Q)\), the \(\chi\)th Fourier coefficient of \(f\) is the function on \(G(\mathbb A)\) given by

\[ f_\chi(g)= \int_{N(\mathbb Q)\setminus N(\mathbb A)}f(ng)\cdot \overline{\chi(n)}\,dn. \]

This notion of Fourier coefficients is useful for many purposes, such as the definition of cusp forms, but since \(f_\chi\) are functions rather than numbers, it is often difficult to extract arithmetic information from them. For arithmetic applications it is thus desirable to have a refined theory of Fourier coefficients analogous to that for the holomorphic forms discussed above.

In this paper we develop such a theory of Fourier coefficients for certain modular forms on the exceptional Chevalley group \(G_2(\mathbb Z)\). Here the symmetric space \(X=G_2(\mathbb R)/\text{SO}_4\) does not have an invariant complex structure; there are thus no holomorphic modular forms. The real components of the automorphic representations we consider are in the quaternionic discrete series. For forms of even weight, we show that the Fourier coefficients \(c_A(f)\) are indexed by totally real cubic rings \(A\): commutative rings with unit, which are free of rank 3 over \(\mathbb Z\) and such that the \(\mathbb R\)-algebra \(A\otimes\mathbb R\) is isomorphic to \(\mathbb R^3\).

As an illustration of the theory, we calculate the Fourier coefficients for Eisenstein series and analogs of theta series. There is a natural family of Eisenstein series \(E_{2k}\) (of even weight \(2k\)) which was first investigated by D. Jiang and S. Rallis. Assuming an extension of their local results, we show that for a maximal cubic ring \(A\), the \(A\)th Fourier coefficient of the Eisenstein series \(E_{ik}\) is the nonzero rational number \(\zeta_A(1-2k)\). The analogs of theta series are constructed via the dual pair correspondence arising from the restriction of the minimal representation of the quaternionic form of the exceptional group \(E_8\). The \(A\)th Fourier coefficients of the analogs of theta series count embeddings of the ring \(A\) into integral exceptional Jordan algebras, just as the coefficients of Siegel theta series count embeddings of quadratic spaces over \(\mathbb Z\).

The rest of the paper studies the action of spherical Hecke operators on Fourier coefficients. We give some background on the general theory and then work out the relative Satake transform when \(G=G_2\) and \(L=\text{GL}_2\) is the Levi factor of the Heisenberg parabolic subgroup \(P\). Using this transform, we determine the action of the two generators of the spherical Hecke algebra at \(p\) on the Fourier coefficients. This involves the determination of single coset representatives for the double cosets corresponding to the two generators, and the computations are carried out. The resulting formulas in Section 15 are analogs of the well-known formula

\[ a_n(T_p|f)= a_{np}(f)+ p^{2k-1}a_{n/p}(f) \]

for the action of the Hecke operator \(T_p\) on the Fourier coefficients of a holomorphic modular form \(f\) of weight \(2k\) on \(\text{SL}_2(\mathbb Z)\). Finally, we show in the last section that if \(f\) is a Hecke eigenform, then the primitive coefficients (i.e., those at Gorenstein cubic rings) and the Hecke eigenvalues determine the rest of the coefficients and hence \(f\) (if \(f\) is a cusp form). This is the analog of the classical result that if \(f\) is a holomorphic cuspidal Hecke eigenform on \(\text{SL}_2(\mathbb Z\)), then \(f\) is determined by \(a_1(f)\) and its Hecke eigenvalues.

Siegel developed a theory of Fourier coefficients \(c_N(f)\) for holomorphic forms \(f\) on the symplectic group \(\text{Sp}_{2g}(\mathbb Z)\). Here the coefficients, for forms of even weight, are indexed by positive semidefinite, integral even quadratic spaces \(N\) of rank \(g\). There is an analogous theory for holomorphic forms on tube domains, where the Fourier coefficients are indexed by orbits on integral elements in the corresponding homogeneous cone.

On the other hand, one has a less refined notion of Fourier coefficients for a general automorphic form \(f\) on a general reductive group \(G\). Given any parabolic subgroup \(P=M\cdot N\) of \(G\) and a unitary character \(\chi\) of \(N(\mathbb A)\) trivial on \(N(\mathbb Q)\), the \(\chi\)th Fourier coefficient of \(f\) is the function on \(G(\mathbb A)\) given by

\[ f_\chi(g)= \int_{N(\mathbb Q)\setminus N(\mathbb A)}f(ng)\cdot \overline{\chi(n)}\,dn. \]

This notion of Fourier coefficients is useful for many purposes, such as the definition of cusp forms, but since \(f_\chi\) are functions rather than numbers, it is often difficult to extract arithmetic information from them. For arithmetic applications it is thus desirable to have a refined theory of Fourier coefficients analogous to that for the holomorphic forms discussed above.

In this paper we develop such a theory of Fourier coefficients for certain modular forms on the exceptional Chevalley group \(G_2(\mathbb Z)\). Here the symmetric space \(X=G_2(\mathbb R)/\text{SO}_4\) does not have an invariant complex structure; there are thus no holomorphic modular forms. The real components of the automorphic representations we consider are in the quaternionic discrete series. For forms of even weight, we show that the Fourier coefficients \(c_A(f)\) are indexed by totally real cubic rings \(A\): commutative rings with unit, which are free of rank 3 over \(\mathbb Z\) and such that the \(\mathbb R\)-algebra \(A\otimes\mathbb R\) is isomorphic to \(\mathbb R^3\).

As an illustration of the theory, we calculate the Fourier coefficients for Eisenstein series and analogs of theta series. There is a natural family of Eisenstein series \(E_{2k}\) (of even weight \(2k\)) which was first investigated by D. Jiang and S. Rallis. Assuming an extension of their local results, we show that for a maximal cubic ring \(A\), the \(A\)th Fourier coefficient of the Eisenstein series \(E_{ik}\) is the nonzero rational number \(\zeta_A(1-2k)\). The analogs of theta series are constructed via the dual pair correspondence arising from the restriction of the minimal representation of the quaternionic form of the exceptional group \(E_8\). The \(A\)th Fourier coefficients of the analogs of theta series count embeddings of the ring \(A\) into integral exceptional Jordan algebras, just as the coefficients of Siegel theta series count embeddings of quadratic spaces over \(\mathbb Z\).

The rest of the paper studies the action of spherical Hecke operators on Fourier coefficients. We give some background on the general theory and then work out the relative Satake transform when \(G=G_2\) and \(L=\text{GL}_2\) is the Levi factor of the Heisenberg parabolic subgroup \(P\). Using this transform, we determine the action of the two generators of the spherical Hecke algebra at \(p\) on the Fourier coefficients. This involves the determination of single coset representatives for the double cosets corresponding to the two generators, and the computations are carried out. The resulting formulas in Section 15 are analogs of the well-known formula

\[ a_n(T_p|f)= a_{np}(f)+ p^{2k-1}a_{n/p}(f) \]

for the action of the Hecke operator \(T_p\) on the Fourier coefficients of a holomorphic modular form \(f\) of weight \(2k\) on \(\text{SL}_2(\mathbb Z)\). Finally, we show in the last section that if \(f\) is a Hecke eigenform, then the primitive coefficients (i.e., those at Gorenstein cubic rings) and the Hecke eigenvalues determine the rest of the coefficients and hence \(f\) (if \(f\) is a cusp form). This is the analog of the classical result that if \(f\) is a holomorphic cuspidal Hecke eigenform on \(\text{SL}_2(\mathbb Z\)), then \(f\) is determined by \(a_1(f)\) and its Hecke eigenvalues.

### MSC:

11F30 | Fourier coefficients of automorphic forms |

11F55 | Other groups and their modular and automorphic forms (several variables) |

PDFBibTeX
XMLCite

\textit{W. T. Gan} et al., Duke Math. J. 115, No. 1, 105--169 (2002; Zbl 1165.11315)

### References:

[1] | H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups , Comm. Algebra 18 (1990), 551–562. · Zbl 0717.20029 |

[2] | A. Borel, Linear Algebraic Groups , 2d ed., Grad. Texts in Math. 126 , Springer, New York, 1991. · Zbl 0726.20030 |

[3] | A. Borel and H. Jacquet, “Automorphic forms and automorphic representations” in Atomorphic Forms, Representations and \(L\)-Functions (Corvallis, Ore., 1977), Part 1 , Proc. Sympos. Pure Math. 33 , Amer. Math. Soc., Providence, 1979 189–207. · Zbl 0414.22020 |

[4] | A. Borel and J. Tits, Groupes réductifs , Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150. · Zbl 0145.17402 |

[5] | N. Bourbaki, Éléments de mathématiques, fasc. 37: Groupes et algèbres de Lie, chapitres II, III , Actualités Sci. Indust. 1349 , Hermann, Paris, 1972. |

[6] | W. Casselman, Canonical extensions of Harish-Chandra modules to representations of \(G\) , Canad. J. Math. 41 (1989), 385–438. · Zbl 0702.22016 |

[7] | H. Cohen, Sums involving the values at negative integers of \(L\)-functions of quadratic characters , Math. Ann. 217 (1975), 271–285. · Zbl 0311.10030 |

[8] | B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree , Trans. Math. Monogr. 10 , Amer. Math. Soc., Providence, 1964. · Zbl 0133.30202 |

[9] | M. Demazure, “Sous-groupes paraboliques des groupes réductifs” in Schémas en groupes, III , Séminaire de Géométrie Algébrique du Bois-Marie (SGA 3), Lecture Notes in Math. 153, Springer, New York, 1970, 426–517. |

[10] | W. T. Gan, An automorphic theta module for quaternionic exceptional groups , Canad. J. Math. 52 (2000), 737–756. · Zbl 0985.11024 |

[11] | –. –. –. –., A Siegel-Weil formula for exceptional groups , J. Reine Angew. Math. 528 (2000), 149–181. · Zbl 1005.11022 |

[12] | B. H. Gross, Groups over \(\mathbbZ\) , Invent. Math. 124 (1996), 263–279. · Zbl 0846.20049 |

[13] | B. H. Gross, “On the Satake isomorphism” in Galois Representations in Arithmetic Algebraic Geometry (Durham, England, 1996) , London Math. Soc. Lecture Note Ser. 254 , Cambridge Univ. Press, Cambridge, 1998, 223–237. · Zbl 0996.11038 |

[14] | B. H. Gross and W. T. Gan, Commutative subrings of certain non-associative rings , Math. Ann. 314 (1999), 265–283. · Zbl 0990.11018 |

[15] | B. H. Gross and N. R. Wallach, On quaternionic discrete series representations, and their continuations , J. Reine Angew. Math. 481 (1996), 73–123. · Zbl 0857.22012 |

[16] | J. W. Hoffman and J. Morales, Arithmetic of binary cubic forms , Enseign. Math. (2) 46 (2000), 61–94. · Zbl 0999.11021 |

[17] | J. S. Huang, P. Pandžić, and G. Savin, New dual pair correspondences , Duke Math. J. 82 (1996), 447–471. · Zbl 0865.22009 |

[18] | D. H. Jiang and S. Rallis, Fourier coefficients of Eisenstein series of the exceptional group of type \(G_2\) , Pacific J. Math. 181 (1997), 281–314. · Zbl 0893.11018 |

[19] | R. A. Rankin, Modular Forms and Functions , Cambridge Univ. Press, Cambridge, 1977. · Zbl 0376.10020 |

[20] | J.-P. Serre, A Course in Arithmetic , Grad. Texts in Math. 7 , Springer, New York, 1973. · Zbl 0256.12001 |

[21] | –. –. –. –., Exemples de plongements des groupes \(¶SL_2(\mathbbF_p)\) dans les groupes de Lie simples , Invent. Math. 124 (1996), 525–562. · Zbl 0877.20033 |

[22] | T. A. Springer, Linear Algebraic Groups , 2d ed., Progr. Math. 9 , Birkhäuser, Boston, 1998. · Zbl 0927.20024 |

[23] | D. Vogan, The unitary dual of \(G_2\) , Invent. Math. 116 (1994), 677–791. · Zbl 0808.22003 |

[24] | N. R. Wallach, Real Reductive Groups, II , Pure Appl. Math. 132 , Vol. II, Academic Press, Boston, 1992. · Zbl 0785.22001 |

[25] | –. –. –. –., “\(C^\infty\) vectors” in Representations of Lie Groups and Quantum Groups (Trento, Italy, 1993) , Pitman Res. Notes Math. Ser. 311 , Longman Sci. Tech, Harlow, 1994, 205–270. |

[26] | ——–, Generalized Whittaker vectors for holomorphic and quaternionic representations , preprint, 2000, |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.