Abstract
This note is concerned with the representation constructed by Chinta and Gunnells in (Constructing Weyl group multiple Dirichlet series, J. Amer. Math. Soc. 23, 2010, 189–215), a representation of the Weyl group of an irreducible root system on an infinite-dimensional algebra over a base field. Chinta and Gunnells in (Constructing Weyl group multiple Dirichlet series, J. Amer. Math. Soc. 23, 2010, 189–215). The first group of remarks is that this result can, at least, in principle, be constructed and understood from the point of view of the representation theory of local metaplectic groups. The original proof is by means of generators, relations and computer algebra, and so a representation-theoretical proof makes the construction and verification more “natural.” The second group of remarks concerns the application of this local theorem to the global problem of determining the Fourier–Whittaker coefficients of metaplectic theta functions and the closely related problem of the distribution of the values of Gauss sums and their generalizations. These applications are still very preliminary, but the prospects are encouraging.
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Notes
- 1.
The condition that − 1 be an nth power is used in [8, Sect. 3] in the formula γ(j)γ( − j) = q − 1 for \(j\not\equiv 0 ({\rm mod} n)\) (their notations). This holds true if ( − 1, π) n, F = 1. If ( − 1, π) n, F = − 1, we can replace γ(j) by \({i}^{{j}^{2} }\gamma (j)\) with i 2 = − 1. From this, one can derive a representation of the same type as [8, Theorem 3.2] when ( − 1, π) n, F = − 1.
- 2.
The precise nature of T is unclear. In the case of the cubic theta function and in Wellhausen’s conjectures in the case n = 6, we find a factor (2π)1 − 1 ∕ n Γ(1 ∕ n). If this were also the case when n = 4, the constant of [10, pp. 240,251] which was numerically estimated as 0. 14742376—note that a digit was omitted on p. 240—could be \({(\frac{1} {4}{(2\pi )}^{\frac{3} {4} }\gamma (1/4))}^{2}\) This is numerically \(0.1475425748\ldots \). This is close but not close enough in view of the accuracy of the calculations of [10]. A much better estimate is \(\frac{{(2\pi )}^{3}} {128\Gamma {(1/4)}^{2}}\) which is numerically \(0.1474237606\ldots \). This is very puzzling.
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Patterson, S.J. (2012). Excerpt from an Unwritten Letter. In: Bump, D., Friedberg, S., Goldfeld, D. (eds) Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol 300. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8334-4_14
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