We develop a theory of Fourier coefficients for modular forms on
the split exceptional group G2 over ℚ.
References
[ABS] H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), 551--562.
[Bo] A. Borel, Linear Algebraic Groups, 2d ed., Grad. Texts in Math. 126, Springer, New York, 1991.
[BoJ] 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.
[BoT] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55--150.
[B] N. Bourbaki, Éléments de mathématiques, fasc. 37: Groupes et algèbres de Lie, chapitres II, III, Actualités Sci. Indust. 1349, Hermann, Paris, 1972.
[C] W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $G$, Canad. J. Math. 41 (1989), 385--438.
[Co] H. Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), 271--285.
[DF] B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Trans. Math. Monogr. 10, Amer. Math. Soc., Providence, 1964.
[D] 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.
[Ga1] W. T. Gan, An automorphic theta module for quaternionic exceptional groups, Canad. J. Math. 52 (2000), 737--756.
[Ga2] —, A Siegel-Weil formula for exceptional groups, J. Reine Angew. Math. 528 (2000), 149--181.
[Gr1] B. H. Gross, Groups over $\mathbbZ$, Invent. Math. 124 (1996), 263--279.
[Gr2] 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.
[GG] B. H. Gross and W. T. Gan, Commutative subrings of certain non-associative rings, Math. Ann. 314 (1999), 265--283.
[GW] B. H. Gross and N. R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996), 73--123.
[HM] J. W. Hoffman and J. Morales, Arithmetic of binary cubic forms, Enseign. Math. (2) 46 (2000), 61--94.
[HPS] J. S. Huang, P. Pandžić, and G. Savin, New dual pair correspondences, Duke Math. J. 82 (1996), 447--471.
[JR] 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.
[R] R. A. Rankin, Modular Forms and Functions, Cambridge Univ. Press, Cambridge, 1977.
[Se1] J.-P. Serre, A Course in Arithmetic, Grad. Texts in Math. 7, Springer, New York, 1973.
[Se2] —, Exemples de plongements des groupes $\PSL_2(\mathbbF_p)$ dans les groupes de Lie simples, Invent. Math. 124 (1996), 525--562.
[Sp] T. A. Springer, Linear Algebraic Groups, 2d ed., Progr. Math. 9, Birkhäuser, Boston, 1998.
[V] D. Vogan, The unitary dual of $G_2$, Invent. Math. 116 (1994), 677--791.
[W1] N. R. Wallach, Real Reductive Groups, II, Pure Appl. Math. 132, Vol. II, Academic Press, Boston, 1992.
[W2] —, ``$C^\infty$ 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.