Speaker: Clifton Cunningham

Room: MS 337

This is a continuation of the talk from last week on admissible representations of $p$-adic G(2) associated to cubic unipotent Arthur parameters.

We have seen how the subregular unipotent orbit in the L-group for split G(2) determines a unipotent Arthur parameter and thus an unramified infinitesimal parameter $\lambda : W_F \to \,^LG(2)$.

Using the Voganish conjectures (\texttt{https://arxiv.org/abs/1705.01885v4}) we find that there are exactly 8 admissible representations with infinitesimal parameter $\lambda$.

Last week Qing Zhang interpreted $\lambda$ as a Langlands parameter for the split torus in $p$-adic G(2) and worked out the corresponding quasi-character $\chi : T(F) \to \mathbb{C}^*$ using the local Langlands correspondence.

We expect that all admissible representations in the composition series of $\mathop{Ind}_{B(F)}^{G(2,F)} \chi$ have infinitesimal parameter $\lambda$; we wonder if not all 8 admissible representations arise in this way.

In this talk I will calculate the multiplicity matrix that describes how these 8 admissible representations are related to 8 standard modules with infinitesimal parameter $\lambda$, assuming the Kazhdan-Lusztig conjecture as in appears in Section 10.2.3 of the preprint above.

To make this calculation I will use the Decomposition Theorem to calculate the stalks of all simple $H_\lambda$-equivariant perverse sheaves on the mini-Vogan variety $V_\lambda$, following the strategy explained in Section 10.3.3 of the preprint.