On the set of maximal nilpotent supports of supercuspidal representations
This article makes a valuable contribution to our understanding of the wave-front set of supercuspidal representations of split classical \(p\)-adic groups. According to an important result of \textit{C. Moeglin} and \textit{J. L. Waldspurger} [Math. Z. 196, 427--452 (1987; Zbl 0612.22008)], the wave-front set of an irreducible representation of a \(p\)-adic group, which is a set of nilpotent orbits in the Lie algebra, may be understood either in terms of the Fourier transform of the character distribution, or in terms of degenerate Whittaker functionals. The latter is the view taken in this paper. Thus, the general context is the problem of computing \(\mathcal{N}_{\text{Wh}, \max}(\pi)\), defined as in Moeglin-Waldspurger, for \(\pi\) an irreducible representation of a \(p\)-adic group. The class of representations considered is that of supercuspidals \(\pi(T, \chi)\) attached to pairs \((T, \chi)\) where \(T\) is a minisotropic torus and \(\chi\) is a regular character of \(T\), according to a construction given in [\textit{J. D. Adler}, Pac. J. Math. 185, No. 1, 1--32 (1998; Zbl 0924.22015)]. (The terminology ``minisotropic'' and ``regular'' may be found in [\textit{M. Reeder}, J. Reine Angew. Math. 620, 1--33 (2008; Zbl 1153.22021)]; for semisimple groups, minisotropy reduces to anisotropy.) Such representations were organized into packets in [Reeder, loc. cit.], by varying \((T, \chi)\) in a family corresponding to stable conjugacy of tori. So, the more specific context of the present work is the problem of computing \(\mathcal{N}_{\text{Wh}, \max}(\pi(T, \chi))\), given \(T\) and \(\chi\), and addressing how it varies in an \(L\)-packet. This problem is partly solved in [\textit{S. DeBacker} and \textit{M. Reeder}, Compos. Math. 146, No. 4, 1029--1055 (2010; Zbl 1195.22011)], where it is shown that \(\pi(T, \chi)\) is generic if and only if the torus \(T\) corresponds to a hyperspecial point in the Bruhat-Tits building. The present paper addresses the case when the group in question is \(\mathrm{Sp}(2n)\), split \(\mathrm{SO}(2n)\) or split \(\mathrm{SO}(2n+1)\). As the author explains, in this case, stable conjugacy classes of anisotropic maximal tori are parametrized by partitions of \(n\). Moreover, if one fixes a partition \(\mu\), then conjugacy classes of anisotropic maximal tori in the stable conjugacy class of \(\mu\) may be parametrized by sub-partitions \(\mu'\) of \(\mu\), obtained by deleting some of the parts of \(\mu\). In the special orthogonal case, the number of parts deleted must be even. Clearly, the sum of the elements of \(\mu'\) will be less than or equal to \(n\) with equality only if \(\mu' = \mu\). Let \(i = i(\mu)\) denote the difference. The parametrization is done in such a way that the torus attached to \(\mu'\) determines a hyperspecial point in the Bruhat-Tits building only when \[ i (\mu')\notin I_{\text{nsp}}:= \begin{cases} \{ 2, \dots, n \} , & \mathrm{SO}(2n+1),\\ \{ 2, \dots, n-2\}, & \mathrm{SO}(2n),\\ \{ 1, \dots, n-1\}, & \mathrm{Sp}(2n). \end{cases} \] As it happens, over an algebraically closed field, nilpotent orbits of classical groups are also parametrized by partitions: this time partitions of the dimension of the standard representation (i.e., \(2n\) or \(2n+1\) as appropriate) with some additional conditions depending on the case (\(\mathrm{SO}(2n), \mathrm{SO}(2n+1)\) or \(\mathrm{Sp}(2n)\)). This then attaches a partition to each nilpotent orbit over an arbitrary field. For \(i \in I_{\text{nsp}}\) define \[ \lambda^i = \begin{cases} [2i-1, 2n-2i+1, 1], & \mathrm{SO}(2n+1),\\ [2i-1m2n-2i-1,1,1], & \mathrm{SO}(2n),\\ [2i, 2n-2i], & \mathrm{Sp}(2n). \end{cases} \] Then the main result of the paper is as follows: Theorem: If \(T\) is in the stable conjugacy class attached to \(\mu\), and is attached to the sub-partition \(\mu'\), and if \(i(\mu') \in I_{\text{nsp}}\) then \(\mathcal{N}_{\text{Wh}, \max}(\pi(T, \chi))\) contains at least one orbit attached to the partition \(\lambda^{i(\mu')}\).
scientific article; zbMATH DE number 6382635