Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 \ldots 0\; 1)$.

A natural question would then be whether other reductive groups have subgroups which behave in a similar way, ie: allows for useful "Fourier-like" expansions leading towards "multiplicity one"-type results, and enable period integrals to be rewritten as Euler products. This latter property if crucial for the development of the theory of standard L-functions for $\mathrm{GL}_n\times \mathrm{GL}_m$.

These are but a few of the first applications of the mirabolic subgroup to the representation theory of $\mathrm{GL}_n$, but already one can see that it isn't an overstatment that this subgroup is "miraculous".

Anyway, I was told read through a paper from the 70's by Piatetski-Shapiro [1], in which the author seeks to understand the type of subgroup which would generalize the mirabolic, calling it an Euler subgroup. In this context, PS proves a fairly general set up for when an Euler subgroup exists and allows for natually defined zeta integrals to give rise to Euler products.

However, aside from this paper, I can't find many references for Euler subgroups. The MathSciNet review by Stephen Gelbart seems to indicate that these ideas have not been as effective in more generality; in particular the connection between Euler subgroups and "multiplicity one" results does not generalize.

My question is: what fails when trying to use Euler subgroups to extend the $\mathrm{GL}_n$ theory to more general groups? Are there explicit examples where the hoped for properties of the $\mathrm{GL}_n$ setup fail to generalize?

I would greatly appreciate any references if they exist. Any ideas which have replaced these in the intervening years will also be of interest!

[1]I.I. Piatetski-Shapiro, Euler subgroups, Proc. Summer School Bolya-Janos Math. Soc. (Budapest, 1970), Halsted, New York, 1975, 597-620.Math. Soc. (Budapest, 1970), Halsted, New York, 1975, 597-620.


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