Simple trace formula with different spectral footprint?

A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $$f=\prod_vf_v$$ to obtain a simple trace formula. This has been addressed in several previous questions, for example: related question.

As discussed in the linked question, one of the standard assumptions is that $$f_v$$ is a supercusp form at some finite place $$v$$, which is a function with no non-zero constant terms. This ensures that only cuspidal automorphic representations appear on the spectral side of the trace formula, which dramatically simplifies the analysis.

My question:

can we similarly reduce to one of the other components of the spectral decomposition by imposing finitely many local constraints?

To be more precise, I would like to choose a test function $$f=\prod_v f_v$$ such that when plugged into the trace formula, only the contribution of representations induced from cusp forms on the Levi subgroup of a single maximal parabolic subgroup $$P=MN$$. Naturally, this would require Arthur's analysis of these terms to work with, but that's fine for my purposes.

Are there simple local conditions to place on the test function to produce such a simplification? Are there examples in the literature of such simple trace formulas being studied?

The applications I have in mind are actually for a relative trace formula, but any examples would be great!

Some simple remarks:

1. The main construction of super cusp forms is to take $$f_v$$ to be a matrix coefficient of a supercuspidal representation. We obviously can't naively use this idea for other parts of the spectral decomposition as the matrix coefficients will not be compactly supported. Perhaps we can approximate such matrix coefficients by compactly supported test functions, but it isn't clear how this can lead to an answer.

2. If we slacken the definition of a supercusp form to allow for only some constant terms, this helps but I don't see how to reduce things further to get only one parabolic subgroup.

• Just as with cuspidal-data Eisenstein series, inducing from (super-) cuspidal data on the Levi component of a parabolic produces a repn with no non-zero homs to induced reps from any other parabolics except those containing the "associates" of the given one. So using (super-) cuspidal data on a maximal proper parabolic, perhaps self-associate and associate to nothing else, would be interesting. E.g., all the maximal proper parabolics in Sp4 (=4x4 matrices) have this feature. – paul garrett Aug 13 '19 at 21:25
• I agree! the case I was thinking of was the self-associate $GL(2)\times GL(2)$ Levi in $GL(4)$, but the rank two case would also be clarifying. – WSL Aug 14 '19 at 12:51