Regularized orbital integrals for representations of \({\mathbf{S} \mathbf{L}}(2)\) (Q2781417)

The motivation of the paper under review is to explain the local invariance of certain distributions that appear in the zeta function of the space of binary quartic forms without resorting to the methods used by Shintani (for the real field) and by Datskovsky and Wrigth (for other local fields). NEWLINENEWLINENEWLINELet \(F\) be a non-Archimedean local field and let \(G\) be the group \(\text{SL}(2,F)\) acting on a finite-dimensional \(F\)-vector space \(V\) via linear representations. The main results of the paper are the following. NEWLINENEWLINENEWLINEFor each orbit \(o\), there is a canonical (up to a scalar) \(G\)-invariant measure \(\mu_ o\) on \(o\), the integral \(\int_ o f d\mu_o\) converges for the functions \(f\) that vanish on the boundary of \(o\) in the usual topology on \(V\), and there exists an invariant distribution that agrees with \(f\mapsto \int_o f d\mu_o\) at all functions \(f\) that vanish on the (topological) boundary of an orbit \(o\), if the normalizer of some maximal \(F\)-split torus fixes the boundary of that orbit. NEWLINENEWLINENEWLINEIn the real case, the situation is more complicated. The author proves that, if \(G=\text{SL}(2,\mathbb R)\) acts on a finite dimensional real vector space \(V\) and if \(o\) is an orbit whose boundary is not fixed by the normalizer of any maximal split torus, then the integral \(\int_o f d\mu_o\) converges for functions \(f\) that vanish on the boundary of \(o\), but no invariant distribution on \(V\) agrees with \(f\mapsto \int_o f d\mu_o\) at all such \(f\). NEWLINENEWLINENEWLINEThe paper is very well written and very pleasant to read.