The test function conjecture for pro-p Iwahori local models of general linear groups and general symplectic groups

Abstract

Building on the enlarged local models of ${\rm GL}_n$ and ${\rm GSp}_{2g}$ at $\Gamma_1(p)$-level constructed in [HLS26+], and employing nearby cycles on these models, we prove that for suitable $m$'s, the function $\tau_{\mu,m}^{ss}$ in the center of the $\Gamma_1(p)$ Hecke algebra, defined geometrically via the semisimple trace, coincides with the function $z_{\mu}^{ss}$ obtained from semisimple local Langlands parameters [FS24] and the theory of the stable Bernstein center [Hai14]. This provides the first verification of the test function conjecture at $\Gamma_1(p)$-level, valid for all cocharacters $\mu$.