On the formal arc space of a reductive monoid

Let $X$ be a scheme of finite type over a finite field $k$, and let $\mathcal L X$ denote its arc space; in particular, $\mathcal L X(k) = X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of $\mathcal L X$ in the neighborhood of non-degenerate arcs, we show that a canonical "basic function" can be defined on the non-degenerate locus of $\mathcal L X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an "$L$-monoid". Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second-named author.