Stokes matrices for Airy equations

Hohl, Andreas; Jakob, Konstantin

doi:10.2748/tmj.20210506

Cited by 2 publications

(3 citation statements)

References 22 publications

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“…Recently, the first author and Hohl gave an explicit description of the Stokes data of GL 𝑛 Airy connections using the Fourier-Laplace transform for -modules [14]. Explicit determination of the Stokes data for Ĝ-Airy sheaves is an interesting open problem.…”

Section: Stokes Datamentioning

confidence: 99%

“…For a subgroup 𝐻 ⊂ 𝐺(𝐹) of finite codimension in some parahoric subgroup of 𝐺(𝐹) we write Bun 𝐻 ∶= [𝐿 − 𝐺∖𝐿𝐺∕𝐻]. Recall the Birkhoff decomposition (14).…”

Section: Geometry Of Relevant Orbitsmentioning

confidence: 99%

“…As mentioned above, the study of asymptotics of the classical Airy equation goes back to Stokes. Recently, the first author and Hohl gave an explicit description of the Stokes data of

\mathrm{GL}_n

Airy connections using the Fourier–Laplace transform for

\mathcal {D}

‐modules [14]. Explicit determination of the Stokes data for

\widehat{G}

‐Airy sheaves is an interesting open problem.…”

Section: Introductionmentioning

confidence: 99%

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Airy sheaves for reductive groups

Jakob

Kamgarpour

2022

Proceedings of London Math Soc

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We construct a class of ℓ$\ell$‐adic local systems on double-struckA1$\mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y′′(z)=zy(z)$y^{\prime \prime }(z)=zy(z)$. We employ the geometric Langlands correspondence to construct the sought‐after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GLn$\mathrm{GL}_n$, we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞$\infty$. These conjectures, in particular, imply cohomological rigidity of Airy sheaves.

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Section: Stokes Datamentioning

confidence: 99%

Section: Geometry Of Relevant Orbitsmentioning

confidence: 99%

“…As mentioned above, the study of asymptotics of the classical Airy equation goes back to Stokes. Recently, the first author and Hohl gave an explicit description of the Stokes data of

\mathrm{GL}_n

Airy connections using the Fourier–Laplace transform for

\mathcal {D}

‐modules [14]. Explicit determination of the Stokes data for

\widehat{G}

‐Airy sheaves is an interesting open problem.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Airy sheaves for reductive groups

Jakob

Kamgarpour

2022

Proceedings of London Math Soc

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Airy sheaves for reductive groups

Jakob¹,

Kamgarpour²,

Yi³

2021

Preprint

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We construct a class of ℓ-adic local systems on A 1 that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y ′′ (z) = zy(z). We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GLn, we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behaviour of the local systems at ∞. These conjectures in particular imply cohomological rigidity of Airy sheaves.

show abstract

Stokes matrices for Airy equations

Cited by 2 publications

References 22 publications

Airy sheaves for reductive groups

Airy sheaves for reductive groups

Airy sheaves for reductive groups

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