  According to a program of Braverman, Kazhdan and Ng\^o Bao Ch\^au, for a
large class of split unramified reductive groups $G$ and representations $\rho$
of the dual group $\hat{G}$, the unramified local $L$-factor $L(s,\pi,\rho)$
can be expressed as the trace of $\pi(f_{\rho,s})$ for a suitable function
$f_{\rho,s}$ with non-compact support whenever $\mathrm{Re}(s) \gg 0$. Such
functions can be plugged into the trace formula to study certain sums of
automorphic $L$-functions. It also fits into the conjectural framework of
Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term
basic functions; this is supposed to lead to a generalized
Tamagawa-Godement-Jacquet theory for $(G,\rho)$. In this article, we derive
some basic properties for the basic functions $f_{\rho,s}$ and interpret them
via invariant theory. In particular, their coefficients are interpreted as
certain generalized Kostka-Foulkes polynomials defined by Panyushev. These
coefficients can be encoded into a rational generating function.
