Let $G$ be a reductive group over $\mathbb{Q}$. Let $K$ be a maximal compact subgroup of $G(\mathbb{C})$. Let $S$ be a finite set of primes. For each prime $p$ not in $S$, let $Frob_p$ be a conjugacy class in $G(\mathbb{Q})$ whose eigenvalues are algebraic numbers with absolute value $p^{w/2}$. Let $\theta_p$ be the conjugacy class in $K$ of the semisimplification of $Frob_p/p^{w/2}$.
Is there a natural probability measure $\mu$ on $K$, depending only on $G$ and $S$, such that the sequence $\theta_p$ is equidistributed in $K$ according to $\mu$? If so, can we give an explicit description of $\mu$?
One natural candidate for the measure $\mu$ is the Haar measure on $K$. However, this is not always the correct measure. For example, if $G = GL_2$ and $S = \{2\}$, then the sequence $\theta_p$ is equidistributed according to the Sato-Tate measure, which is not the Haar measure on $U(2)$.
