Regular primes of harmonic analysis mod \(\ell\) of a \(p\)-adic reductive group (Q2720959)

Suppose that \(F\) is a finite extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers, and that \(G\) is the group of rational points of a reductive \(F\)-group. One denotes by \(X\) either the group \(G\) or its Lie algebra \(G\). The group \(G\) acts on \(X\) by conjugation or by the adjoint action. Let \(A\) be a commutative ring. The most interesting cases are \(A=\mathbb{Z}[1/p]\), \(A= \overline{F}_\ell\) the algebraic closure of the finite field with \(\ell\) elements, for a prime \(\ell\neq p\). A \(G\)-invariant \(A\)-distribution on \(X\) is a \(G\)-invariant linear form on the \(A\)-module of locally constant compactly supported functions \(f:X\to A\). An \(A\)-orbital integral on \(X\) is a \(G\)-invariant \(A\)-distribution on \(X\) with support a \(G\)-orbit. Many properties of classical harmonic analysis remain true for \(G\)-invariant \(\mathbb{Z}[1/p]\)-distributions on \(X\). These properties have been used to prove congruences between automorphic forms by the first author. NEWLINENEWLINENEWLINEThe main results of this paper are NEWLINENEWLINENEWLINE-- Any \(G\)-orbit of \(X\) is the support of a \(\mathbb{Z}[1/p]\)-orbital integral (a Deligne-Rao theorem in the complex case). NEWLINENEWLINENEWLINE-- There exists a certain finite set of primes \(\ell\neq p\) called irregular primes, where the Harish-Chandra theory for orbital itegrals, and characters of irreducible representations of \(G\), remains valid \(\operatorname {mod}\ell\), i.e. over \(\overline{F}_\ell\). The determination of irregular primes is difficult in general. For instance the results of M. Assem show that 13 is banal and irregular for \(\text{Sp}(4,\mathbb{Q}_3)\). A prime which is not irregular is called regular. NEWLINENEWLINENEWLINE-- For \(\text{SL} (n, \mathbb{Q}_p)\), the prime numbers \(\ell\neq p\) and \(\ell> n\) are regular.