Geometric Langlands for hypergeometric sheaves

Abstract

I want to also thank Zhiwei Yun and Konstantin Jakob for discussing and sharing their work on rigid automorphic data with me.Last, I want to thank my parents for their unconditional love, support, and encouragement.

Euphotic representations and rigid automorphic data

“…Compared to the notion of epipelagic representations in [21], we have relaxed the condition on ψ: it is only required to have a closed orbit under L P and not required to have finite stabilizer under L P . Functionals on V P with closed orbit that are not stable are also encountered in work of Kamgarpour and Yi on the geometric Langlands correspondence for hypergeometric sheaves [12].…”

“…We expect case (2) to correspond to hypergeometric local systems with slope 1/m at ∞ and unipotent monodromy at 0. Rigid automorphic data corresponding to hypergeometric local systems are constructed in the work of Kamgarpour and Yi [12].…”

Airy sheaves for reductive groups

“…Since $$J^{+}=I(h+2)$$ is a Moy–Prasad subgroup of the standard Iwahori subgroup, $$\Omega$$ acts on $$\mathrm{Bun}_{J^{+}}$$ by changing the level structure. For $$\alpha \in \Omega$$ this allows us to identify the connected components of $$\mathrm{Bun}_{J^{+}}$$ via isomorphisms $$$$\begin{align} \mathbf {T}_\alpha : \mathrm{Bun}_{J^{+}}^{0} \simeq \mathrm{Bun}_{J^{+}}^{\alpha }, \end{align}$$$$see also [20, section 4.3.3. ].…”

Euphotic representations and rigid automorphic data

“…We expect case (2) to correspond to hypergeometric local systems with slope 1/m at ∞ and unipotent monodromy at 0. Rigid automorphic data corresponding to hypergeometric local systems are constructed in the work of Kamgarpour and Yi [KY20]. 8.2.…”

“…Compared to the notion of epipelagic representations in [RY14], we have relaxed the condition on ψ: it is only required to have a closed orbit under L P and not required to have finite stabilizer under L P . Functionals on V P with closed orbit that are not stable are also encountered in work of Kamgarpour and Yi on the geometric Langlands correspondence for hypergeometric sheaves [KY20].…”