On the computability of some positive-depth supercuspidal characters near the identity (Q2889334)

This work is part of a program whose aim is to obtain harmonic analysis results for reductive \(p\)-adic groups in a ``field-independent fashion'' (\textit{motivic} point of view).NEWLINENEWLINEMore precisely the authors tackle the problem of the computability, for positive depth irreducible representations of such groups, of values of their Harish-Chandra characters. This has already been considered by various authors for depth zero representations.NEWLINENEWLINEFor technical reasons, they restrict to the following particular case: NEWLINE{\parindent=6mmNEWLINE\begin{itemize}\item[--]they assume the representations supercuspidal of positive depth, \item[--]they impose tameness conditions so that the supercuspidal representations are described by \textit{J.-K. Yu}'s construction [``Construction of tame supercuspidal representations'', J. Am. Math. Soc. 14, No. 3, 579--622 (2001; Zbl 0971.22012)], NEWLINE\item[--]they impose the supercuspidal representations to belong to a certain restricted class where the Harish-Chandra character may be described (near the identity) by a Murnaghan-Kirillov type formula, NEWLINE\item[--]they assume the reductive group to be either symplectic or split special orthogonal; in that case the conjugacy classes of regular semisimple elements are easy to parametrize. NEWLINENEWLINE\end{itemize}}NEWLINELet \(\mathbb G\) be a reductive group as above, defined over a non-Archimedean local field \(K\). In order that the motivic point of view make sense, the authors fix a depth \(r>0\) and consider an equivalence relation on this set of restricted supercuspidal representations of \({\mathbb G}(K)\) in the following manner. Two representations are said to be \(r\)-equivalent if their Harish-Chandra characters agree on regular semisimple elements ``sufficiently close'' to the identity. This set of equivalence classes may be seen as the points over \(k_K\) (the residue field of \(K\)) of a certain ``geometric'' object, a definable subassignment as defined by \textit{R. Cluckers} and \textit{F. Loeser} in their motivic integration theory [``Constructible motivic functions and motivic integration'', Invent. Math. 173, No. 1, 23--121 (2008; Zbl 1179.14011)]. The regular topologically nilpotent elements of \({\mathbb G}(K)\) may be themselves seen as the \(k_K\)-rational points of a certain subassigment.NEWLINENEWLINEThe main result of the article may be resumed as follows. For each \(r\)-equivalence class \(y\) of restricted supercuspidal representations, there exists an explicit exponential motivic function in the sense of [\textit{R. Cluckers} and \textit{F. Loeser}, ``Constructible exponential functions, motivic Fourier transform and transfer principle'', Ann. Math. (2) 171, No. 2, 1011--1065 (2010; Zbl 1246.14025)] on the subassigment of regular topologically nilpotent elements, with the following property. The restriction of its specialization to a neighbourhood of the identity (depending on \(r\)) gives the values of the Harish-Chandra characters of the representations of \({\mathbb G}(K)\) in the equivalence class corresponding to \(y\).NEWLINENEWLINEThe authors expect several future consequences of this theorem. First they believe that their construction should give a proof of the local integrability of the Harish-Chandra character when the local field \(K\) has positive characteristic. They also think that it should give rise to a computer program giving a large part of the character table of the symplectic and split orthogonal groups. Finally they hope to get rid of the tameness conditions and to be able to consider values of characters at elements that are not necessary to be close to the identity.