Automorphic forms on covering groups

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1. Representations

Iwahori-Matsumoto presentation

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Problem 1.1.
[M. Weissman] What is the “Iwahori-Matsumoto Presentation” of Iwahori-Hecke algebra for $\widetilde{G}_F$ for split $G_F$ ?
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Problem 1.2.
[W. Gan] Construct supercuspidal representations of $\widetilde{G}_{F}$ .
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Problem 1.3.
[K. Choiy] What is the category of depth-zero unipotent representations for $\widetilde{G}_{F}$ for a $p$ -adic field $F$ ?
Steinberg representation

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Problem 1.4.
[G. Savin] Define Steinberg representation of $\widetilde{G}_{F}$ . What is the formal degree of Steinberg representation of $\widetilde{G}_{F}$ when $G$ is split over $F$ ?

What is analogue of the trivial representation?
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Problem 1.5.
[W. Gan] What is theta correspondence for $\widetilde{G}$ ?

e.g.) $SL_2 \times \widetilde{SL}^{(3)}_2 \hookrightarrow \widetilde{G_2}^{(3)}$ (G-R-S)

$SL_3 \times \widetilde{SL}^{(2)}_3 \hookrightarrow \widetilde{F_4}^{(2)}$

$\widetilde{Spin}^{(2)}_{2a+1} \times \widetilde{Spin}^{(2)}_{2b} \hookrightarrow \widetilde{Spin}^{(2)}_{2(a+b)+1}$ (Loke-Savin)

$\widetilde{Spin}^{(2)}_{5} \times \widetilde{Spin}^{(2)}_{4} \hookrightarrow \widetilde{Spin}^{(2)}_{9}$

Cite this as: AimPL: Automorphic forms on covering groups, available at http://aimpl.org/autoformcovergp.