On certain elements in the Bernstein center of $\mathbf {GL}_2$

Abstract:

Let $F$ be a nonarchimedean local field of residue characteristic $p$, and let $r$ be an odd natural number less than $p$. Using the work of Moy and Tadić, we find an element $z$ of the Bernstein center of $G = \mathbf {GL}_2(F)$ that acts on any representation $\pi$ of $G$ by the scalar $z(\pi ) = \operatorname {tr} \left (\operatorname {Frob} ; \left ( \operatorname {Sym}^r \circ \varphi _{\pi }\right )^{I_F} \right )$, the trace of any geometric Frobenius element $\operatorname {Frob}$ of the absolute Weil group $W_F$ of $F$, acting on the inertia-fixed points of the representation $\operatorname {Sym}^r \circ \varphi _{\pi }$ of $W_F$, where $\varphi _{\pi } : W_F \rightarrow \hat {G}$ is the restriction to $W_F$ of the Langlands parameter of $\pi$. This element $z$ is specified by giving the functions obtained by convolving it with the characteristic functions of a large class of compact open subgroups of $G$, that includes all the groups of both the congruence and the Iwahori filtrations of $G$ having depth at least one.