Paquets stables de représentations tempérées et de réduction unipotente pour \(\text{SO}(2n+1)\). (Stable packets of tempered representations and unipotent reduction for \(\text{SO}(2n+1)\).)
Let \(p\) be a prime number, and let \(F\) be a finite extension of \(\mathbb Q_p\). Let \(\mathbf G_{iso}\) (resp. \(\mathbf G_{an}\)) denote the special orthogonal group of a form with anisotropic kernel of dimension \(1\) (resp. \(3\)). Hence the group \(\mathbf G_{iso}\) is \(F\)-split, and the group \(\mathbf G_{an}\) is a non-split inner form of \(\mathbf G_{iso}\). For \(\#\in\{iso,an\}\), let \(G_{\#}\) be the group \(\mathbf G_{\#}(F)\) of the \(F\)-points of \(\mathbf G_{\#}\). Let \(\text{Irr}_u^{G_{\#}}\) denote the set isomorphism classes of irreducible admissible representations of \(G_{\#}\) with \textit{unipotent reduction}, that is, the subset of the set of isomorphism classes of level zero irreducible admissible representations of \(G_{\#}\) such that the corresponding cuspidal representation of the reductive quotient (a finite group of Lie type) of the parahoric subgroup is \textit{unipotent}. Lusztig has proved that this set is in bijective correspondence with the set \(\underline\Psi_u^{\#}\) of all \(\text{Sp}(2n,\mathbb C)\)-conjugacy classes of pairs \((\psi,\varepsilon)\) with \(\psi\colon W_F\times\text{SL}(2,\mathbb C)\to\text{Sp}(2n,\mathbb C)\) an homomorphism such that its restriction to \(W_F\) (the Weil group of \(F\)) is semisimple and trivial on the inertia subgroup of \(W_F\), and its restriction to \(\text{SL}(2,\mathbb C)\) is algebraic, and \(\epsilon\colon\,A(\psi)=\mathbb Z_{\text{Sp}(2n,\mathbb C)}(\psi)/\mathbb Z_{\text{Sp}(2n,\mathbb C)}^0(\psi)\to\{\pm1\}\) is a character such that its composition with the natural homomorphism from the centre of \(\text{Sp}(2n,\mathbb C)\) to \(A(\psi)\) is equal to \(1\) (resp. \(-1\)) if \(\#=iso\) (resp. \(an\)). (Here \(\mathbb Z_{\text{Sp}(2n,\mathbb C)}(\psi)\) denotes the centralizer of the image of \(\psi\), and \(\mathbb Z_{\text{Sp}(2n,\mathbb C)}^0(\psi)\) the connected component of the latter.) The authors consider the subset \(\text{Irr}_{utemp}^{G_{\#}}\) of the isomorphism classes of tempered irreducible admissible representations with unipotent reduction, and deduce from the bijection defined by Lusztig a bijection \((\psi,\epsilon)\mapsto\pi_{\psi,\epsilon}\) of \(\underline\Psi_{utemp}^{G_{\#}}\) with \(\text{Irr}_{utemp}^{G_{\#}}\), where \(\underline\Psi_{utemp}^{G_{\#}}\) is the subset of \(\underline\Psi_u^{G_{\#}}\) of the \(\text{Sp}(2n,\mathbb C)\)-conjugacy classes of pairs \((\psi,\epsilon)\) such that, in addition, \(\psi(W_F)\) is a relatively compact subgroup of \(\text{Sp}(2n,\mathbb C)\). Note that this bijection is not the restriction of Lusztig's bijection, but the composition of the latter with the involution defined by Schneider, Stuhler and the reviewer. For \(\psi\) as above (in particular, \(\psi(W_F)\) is assumed to be relatively compact), one can consider the following sum (which is the sum in \(\mathbb C(\text{Irr}_{utemp}^{G_{\#}})\) of the elements in the \(L\)-packet parameterized by \(\psi\)): \[ \pi_\psi^{st,G_{\#}}=\sum_{\epsilon\atop(\psi,\epsilon)\in\underline\Psi_{utemp}^{G_{\#}}}\pi_{\psi,\epsilon}. \] The main results of the paper under review are the following (assuming \(p\geq 6n+4\)): (i) \(\pi_\psi^{st,G_{\#}}\) is a stable distribution. (ii) The family \((\pi_\psi^{st,G_{\#}})_\psi\) (where the \(\pi_\psi^{st,G_{\#}}\) are supposed to be non-zero) gives a basis of the subspace of the stable distributions which belong to \(\mathbb C(\text{Irr}_{utemp}^{G_{\#}})\). (iii) \(-\pi_\psi^{st,G_{an}}\) is the endoscopic transfer of \(\pi_\psi^{st,G_{iso}}\). The authors were assuming in the paper (see 2.11) an analog for the group \(O(2m)\) (which is non-connected and not always split) over a finite field of a conjecture of Lusztig. This conjecture has now been proved by \textit{J.-L.Waldspurger} in [Mémoires S.M.F., vol. 96 (2004)].
scientific article; zbMATH DE number 1947265