A Hecke algebra on the double cover of a
Chevalley group over ℚ2

Karasiewicz, Edmund

doi:10.2140/ant.2021.15.1729

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…Thus, it remains to analyze the case , and it is here where we do detailed computations: in § 2.5, we produce an explicit (non-maximal) compact open subgroup of that splits into the double cover. Our result in this direction can be considered an extension of some work of [Kar21], who considers the simply laced case.…”

Section: Introductionsupporting

confidence: 53%

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Modular forms of half-integral weight on exceptional groups

Leslie,

Pollack

2024

Compositio Math.

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We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by ${\pm }1$ . We analyze the minimal modular form $\Theta _{F_4}$ on the double cover of $F_4$ , following Loke–Savin and Ginzburg. Using $\Theta _{F_4}$ , we define a modular form of weight $\tfrac {1}{2}$ on (the double cover of) $G_2$ . We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$ -torsion in the narrow class groups of totally real cubic fields.

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Section: Introductionsupporting

confidence: 53%

“…The proof of the first claim is immediate from the lemma and our normalization that for long roots, so that for our short root. The second claim follows precisely as in the proof of [Kar21, Lemma 3.1] with .…”

Section: Group Theorymentioning

confidence: 73%

Modular forms of half-integral weight on exceptional groups

Leslie,

Pollack

2024

Compositio Math.

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“…In fact, since η = ½ by our assumption, the K 2 -extension G over F arises from a K 2 -extension of G over O F (see [Wei16,Theorem 4.3]), which also entails the splitting of K. Note that for non-tame covers the groups K and I may not split. For examples of double covers of G over É 2 , see [Kar21].…”

mentioning

confidence: 99%

Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications

Gao¹,

Gurevich²,

Karasiewicz³

2022

Preprint

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We study the Iwahori-component of the Gelfand-Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro-p level. Thus to begin we study the structure of a genuine pro-p Iwahori-Hecke algebra, establishing Iwahori-Matsumoto and Bernstein presentations. With this structure theory we first describe the pro-p part of the Gelfand-Graev representation and then the more subtle Iwahori part.For the first application we relate the Gelfand-Graev representation to the metaplectic representation of Sahi-Stokman-Venkateswaran, which conceptually realizes the Chinta-Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series; for the third we do the same for unitary unramified principal series.

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A Hecke algebra on the double cover of a Chevalley group over ℚ2

Cited by 2 publications

References 16 publications

Modular forms of half-integral weight on exceptional groups

Modular forms of half-integral weight on exceptional groups

Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications

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