2020
DOI: 10.1112/plms.12390
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Exponential sums and total Weil representations of finite symplectic and unitary groups

Abstract: We construct explicit local systems on the affine line in characteristic p>2, whose geometric monodromy groups are the finite symplectic groups normalSp2nfalse(qfalse) for all n⩾2, and others whose geometric monodromy groups are the special unitary groups normalSUnfalse(qfalse) for all odd n⩾3, and q any power of p, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one‐parameter families of exponential sums of a very simple, that is, ea… Show more

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Cited by 6 publications
(4 citation statements)
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“…Following Barthel's work [1] in constructing the 2-cocycle from Sp(W ) to GSp(W ), we can easily extend the Weil representation from an affine symplectic group to its similitude group. Some recent papers on various aspects of Weil representations can also be found here: [5], [7], [8], [14], [15], [16], [17], [18], [24], [25], [30], [34], etc.…”
Section: Chun-hui Wangmentioning
confidence: 99%
“…Following Barthel's work [1] in constructing the 2-cocycle from Sp(W ) to GSp(W ), we can easily extend the Weil representation from an affine symplectic group to its similitude group. Some recent papers on various aspects of Weil representations can also be found here: [5], [7], [8], [14], [15], [16], [17], [18], [24], [25], [30], [34], etc.…”
Section: Chun-hui Wangmentioning
confidence: 99%
“…See e.g. [KT6,§6]. As Z(G) acts via scalars in Φ, we can extend θ to a character of Z(G)L, which is still G-invariant.…”
Section: The Characteristic Of Hypergeometric Sheavesmentioning
confidence: 99%
“…The cases (a)-(e) are indeed shown to occur. Namely, the respective hypergeometric sheaves H (in characteristic p in (b)-(e)) are explicitly constructed in Theorem 9.3 for case (a), in [KT5] and Theorem 8.6 for case (b), in [KT6] for type (α) in (c) and for (d) with 2 ∤ q, in [KT7] for type (β) in (c) and for (e), and in [KT8] for case (d) with 2|q. The extraspecial normalizers, and the sporadic and non-generic cases in (f), are handled in [KT8] and [KRL], [KRLT1]- [KRLT4].…”
Section: Converse Theoremsmentioning
confidence: 99%
“…In the case where p$p$ is an odd prime number, a detailed proof of this theorem is given in [3]. The results in [6] have interesting applications as in [7, 9] and [10].…”
Section: Introductionmentioning
confidence: 99%