Let $E/F$ be an unramified quadratic extension of $p$-adic fields. The Jacquet-Rallis fundamental lemma states that $1_{S_n(O_F)}$ and $(1_{U_n(O_F)}, 0)$ are transfer of each other, see ”On the Gross-Prasad conjecture for unitary groups“ by Jacquet and Rallis. In general, the fundamental lemma seems to roughly say that under some unramifiedness assumption, characteristic function of hyperspecial subgroup of $G(\mathbb Q_p)$ and $G'(\mathbb Q_p)$ are transfer of each other.

I wonder whether there are some descriptions for transfer of characteristic function of congruent subgroups or parahoric subgroups? For example, what is the transfer of $1_{Ker(S_n(O_F) \rightarrow S_n(O_F/\pi_F^n))}$ in the Jacquet-Rallis case (althought $S_n$ is not a group but only a symmetric space, we can still define $Ker(S_n(O_F) \rightarrow S_n(O_F/\pi_F^n))$ as the preimage of $Id$)?

Let $E/F$ be an unramified quadratic extension of $p$-adic fields. The Jacquet-Rallis fundamental lemma states that $1_{S_n(O_F)}$ and $(1_{U_n(O_F)}, 0)$ are transfer of each other, see ”On the Gross-Prasad conjecture for unitary groups“ by Jacquet and Rallis. In general, the fundamental lemma seems to roughly say that under some unramifiedness assumption, characteristic function of hyperspecial subgroup of $G(\mathbb Q_p)$ and $G'(\mathbb Q_p)$ are transfer of each other.

I wonder whether there are some descriptions for transfer of characteristic function of congruent subgroups or parahoric subgroups? For example, what is the transfer of $1_{Ker(S_n(O_F) \rightarrow S_n(O_F/\pi_F^n))}$ in the Jacquet-Rallis case (althought $S_n$ is not a group but only a symmetric space, we can still define $Ker(S_n(O_F) \rightarrow S_n(O_F/\pi_F^n))$ as the preimage of $\text{Id}$)?