Fundamental lemma and transfer of characteristic functions of congruent subgroups

Question

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 $\text{Id}$)?

Answer 1

There are a couple of reasons one typically works with just fundamental lemmas for spherical functions:

1) the unramified comparison suffices for a trace formula comparison (well, a fundamental lemma for the whole unramified Hecke algebra at almost all places), together with an abstract smooth matching statement, so there is no need to worry about the more complicated case of explicitly comparing characteristic functions of smaller compact subgroups

2) the unramified comparison is nice, e.g., a statement of the form $1_K$ matches $1_{K'}$, because for the transfers typically considered locally unramified representations correspond to one another. For a more general $K$, what should $1_K$ match with? Well, it will depend on (or if you prefer, will explain) the local correspondence between $G$ and $G'$ for representations with $K$-fixed vectors. However, the nature of the local correspondence in terms of conductors (in the sense of $K$-fixed vectors) is quite delicate in general and not well understood (it is not predicted by Langlands, or apparent from the theta correspondence).

One case where such a "fundamental lemma with conductor" has been worked out is for Jacquet's first relative trace formula toric periods of GL(2). I did this with David Whitehouse: https://math.ou.edu/~kmartin/papers/mw.pdf

However, in the article that we eventually published on that topic, we obtained more general results without doing such an explicit comparison of matching functions.

That said, even for GL(2) and a quaternion algebra, the way that conductor behaves along the local correspondence is not entirely clear. See Conductor of quaternionic representation.

In summary, I think if one can do what you propose in some generality, it will be interesting and possibly can be used to describe how conductors behave along functorial correspondences, and has possible applications to explicit trace formulas. However, the precise matching will be quite difficult and delicate, and you cannot expect such a simple matching form $1_K$ matches $1_{K'}$, but rather you might expect linear combinations of such functions to match.