Restriction to a hyperspecial subgroup: containing Steinberg

Let $G$ be an unramified $p$-adic reductive group, $K$ a hyperspecial maximal compact subgroup, and $K/K^+\cong G(\mathbb F_q)$. Restricting an irreducible representation $V$ of $G(F)$ to $K$ gives a $G(\mathbb F_q)$-representation $V^{K^+}$. Prasad asked which $V$ have $V^{K^+}$ containing the Steinberg representation (or more generally any $\pi_\lambda$ attached to a Weyl group representation $\lambda$). This talk answers the first question completely. We also show that the irreducible components of $K^+$-fixed subspace lies in the Harish-Chandra principal series. The method also extends to the general case, providing a computable framework for the further study.

The talk will be streamed through "Tencent": https://meeting.tencent.com/dm/X6fMLc1Q60b3.
meeting code: 148-207-215
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