I was reading the two Repka papers where he computes the leading and subleading Shalika germs for $GL_n$ and I was wondering, where are we since then? Have these germs (and the integrals) been computed somewhere? Especially if there is a reference using similar methods or something relatively simple.

The following may be a too general question, but what do we know to date about Shalika germs? What is your viewpoint of them?


This 2015 paper (also on arXiv) by Frechette, Gordon, and Robson can serve as a summary of the status with pointers to the literature:

Shalika germs first appeared in the papers of Shalika and Harish-Chandra. For a survey of their role in harmonic analysis on $p$-adic groups we refer to the beautiful article by Kottwitz (in particular, sections 6, 17, and 27).
Shalika germs, by definition, are functions on the set of regular semisimple elements in a Lie algebra, yet except for those defined on a few Lie algebras of small rank, their exact values elude computation. Here we use a general theorem about uniform bounds for motivic functions to estimate the absolute values of the Shalika germs in a uniform way over all local fields of a given (sufficiently large) residue characteristic.


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