Supercuspidal representations and preservation principle of theta correspondence
The paper studies the``first occurrence of supercuspidal representations'' in relation with the local theta correspondence (also known as the Howe correspondence) for representations of reductive groups over \(p\)-adic fields. The main themes are the connections between the general case and the depth-zero case, and the interplay with the situation for finite fields. Before we can state the results more precisely, we must establish notation for the Howe correspondence. Let \(F\) be a \(p\)-adic field, and let \(D\) be either \(F\) or a quadratic extension of \(F\). Let \((G(V), G(V'))\) be a reductive dual pair attached to a pair or Hermitian or anti-Hermitian spaces over \(D\). More precisely, fix \(\epsilon = \pm 1\), let \(V\) (resp. \(V'\)) be a finite-dimensional nondegenerate \(\epsilon\)-Hermitian (resp. \((-\epsilon)\)-Hermitian) space over \(D\). Let \(G(V)\) and \(G(V')\) denote the corresponding groups of isometries, and let us assume that \((G(V), G'(V))\) is a reductive dual pair in \(\mathrm{Sp}(V \otimes_D V')\). Let \(\mathrm{Irr}(G(V))\) and \(\mathrm{Irr}(G(V'))\) denote the admissible duals of \(G(V)\) and \(G(V')\). The Howe correspondence determines a map \(\mathrm{Irr}(G(V)) \to \{0\} \cup \mathrm{Irr}(G(V'))\), which we will denote by \(\pi \to \theta(\pi, V')\). It is injective on the set of \(\pi \in \mathrm{Irr}(G(V))\) such that \(\pi \to \theta(\pi, V') \neq 0\). The author assumes \(p \neq 2\) throughout, in which case the correspondence is due to \textit{J. L. Waldspurger} [Isr. Math. Conf. Proc. 2, 267--324 (1990; Zbl 0722.22009)]. A central notion of the paper is that of ``first occurrence of supercuspidal representations'' in the correspondence. Let us describe it briefly. Fix an anisotropic \((-\epsilon)\)-Hermitian space \(V'_\mathrm{an}\), and for each nonnegative integer \(k\), fix a vector space \(V'_{k}\) given by the direct sum of \(k\) hyperbolic planes. Now let \(\pi\) be a supercuspidal irreducible representation of \(G(V)\), and let \(k\) be the smallest nonnegative integer such that \(\pi'=\theta(\pi, V'_\mathrm{an}\oplus V'_k)\) is nonzero. Then by work of Howe and Kudla, the representation \(\pi'\) must be supercuspidal; furthermore, for \(\ell >k\), the representation \(\theta(\pi, V'_\mathrm{an}\oplus V'_\ell)\) is nonzero, but non-supercuspidal. See [\textit{S. S. Kudla}, Invent. Math. 83, 229--255 (1986; Zbl 0583.22010)] and Proposition 2.2 of the paper under review. In that situation, one says that ``the correspondence \(\pi\leftrightarrow\pi'\) is a first occurrence of supercuspidal representations in the Howe correspondence''. We can now state two of the paper's main results. We will state them only in the case where \(G(V)\) and \(G(V')\) are unitary groups, as in \S 5 of the paper. The corresponding statements for orthogonal and symplectic groups are in \S 6. The first theorem describes a way to produce a first occurrence of supercuspidal representations, starting from other first occurrences involving smaller reductive dual pairs, assuming one of the occurrences concerns depth-zero supercuspidal representations. It provides an important connection to the finite group case. The result applies to the ``tame'' supercuspidal representations constructed by \textit{J.-K. Yu} [J. Am. Math. Soc. 14, No. 3, 579--622 (2001; Zbl 0971.22012)]. It uses the notion of depth, which is preserved by the correspondence by classical work of the author [Duke Math. J. 113, No. 3, 531--592 (2002; Zbl 1011.22009)]. To state the theorem, suppose \begin{itemize} \item \(V\) and \(V'\) admit orthogonal decompositions \(V=V^{(1)} + V^{(2)}\) and \(V=V'^{(1)} + V'^{(2)}\), where \(V^{(1)}\) and \(V'^{(1)}\) are \emph{one-dimensional} spaces (over \(D\), which is now a quadratic extension of \(F\)); \item \(\sigma \leftrightarrow \sigma'\) is a first occurrence of supercuspidal representations \textit{of positive depth} for the pair \((G(V^{(1)}), G(V'^{(1)}))\) attached to these \textit{one-dimensional} subspaces; \item \(\sigma, \sigma'\) are constructed by Yu from ``generic supercuspidal data'' \(\Psi^{(1)}, \Psi'^{(1)}\) (see \S 5.1 for the notion); \item and \(\tau \leftrightarrow \tau'\) is a first occurrence of \textit{depth-zero} supercuspidal representations for the pair \((G(V^{(2)}), G(V'^{(2)}))\). \end{itemize} It is known that \(\tau, \tau'\) are tame, and therefore constructed by Yu from generic supercuspidal data \(\Psi^{(2)}, \Psi'^{(2)}\). Now let \(\pi\) and \(\pi'\) be the tame supercuspidal representations of \(G(V)\) and \(G(V')\) constructed by Yu from \({\Psi^{(1)} \boxtimes \Psi^{(2)}}\) and \({\Psi'^{(1)} \boxtimes \Psi'^{(2)}}\) (where the product data are defined in \S 5.1). Then the author proves that the theta correspondence sends \(\pi\) to \(\pi'\), and \(\pi\leftrightarrow \pi'\) is a first occurrence of supercuspidal representations. This is Theorem 5.2 of the paper. Its proof is highly nontrivial and occupies all of sections 7 and 8. Its importance becomes clear if one realizes that \(G(V^{(1)})\) and \(G(V'^{(1)})\) must be isomorphic because of the dimension-one condition \(-\) therefore \(\sigma\) and \(\sigma'\) must, in a sense, be identical. The behavior of the theta correspondence on \(\pi\) and \(\pi'\) can therefore be studied through the case of depth-zero representations, whose relationship with the representations of finite groups is deep and well-known. The reader might notice that the relationship between the representations \(\pi\), \(\pi'\) of the conclusion, and the representations \(\tau, \tau'\) of the hypothesis, is not immediate from the theorem's statement. In fact, there is a natural operation \(\partial\) on tame supercuspidal representations, due \textit{J. Hakim} and \textit{F. Murnaghan} [IMRP, Int. Math. Res. Pap. 2008, Article ID rpn005, 166 p. (2008; Zbl 1160.22008)], which sends \(\pi\) to \(\sigma \otimes \tau\) and \(\pi'\) to \(\sigma' \otimes \tau'\). A corollary of the above theorem, highlighted by the author as being perhaps the main result of the paper (see \S 1.4), is that \(\sigma \otimes \tau\) and \(\sigma' \otimes \tau'\) must correspond under the theta correspondence, and furthermore that the natural compositions of \(\partial\) and \(\theta\) form a commutative square. See Corollary 5.4 and Remark 5.5. The analogue of the latter result for finite groups, due to \textit{A.-M. Aubert} et al. [Duke Math. J. 83, No. 2, 353--397 (1996; Zbl 0856.22027)], is an important ingredient of the present paper \(-\) both as motivation and as a technical tool. The results described above appear in \S 5.1 and \S 5.2 for unitary groups, with detailed proofs in \S 7-8, and in \S 6.1-6.2 for orthogonal and symplectic groups, with the modifications for the argument described in \S 9. We come now to the second theme of the paper, the ``preservation principle'' mentioned in the title. We still assume that \(G(V)\) is a unitary group; see \S 2.3, 4 and 6.3 for the orthogonal and symplectic cases. Let \(V^{'+}_{\mathrm{an}}\) be \(\{0\}\) and let \(V^{'-}_{\mathrm{an}}\) be a two-dimensional anisotropic space. Let \(n^+(V)\) be the smallest possible dimension for \(V^{'+}=V{'+}_\mathrm{an}\oplus V'_k\) such that \(\theta(\pi, V') \neq 0\). Define \(n^-(V)\) analogously starting from \(V^{'-}_{\mathrm{an}}\). The ``preservation principle'' is the fact that \(n^+(V)+n^-(V)\) does not depend on \(\pi\) (in the present case it is equal to \(2\dim(V)+2\)). The principle enters the author's work because it implies that every irreducible supercuspidal representation \(\pi\) appears in a chain \((\pi_i)_{i \in \mathbb{Z}}\) of supercuspidal irreducible representations of groups \(G(V_i)\), with the property that for all \(i\), the correspondence \(\pi_i\leftrightarrow \pi_{i+1} \otimes \mathrm{sgn}\) is a first occurrence of supercuspidal representations for the dual pair \((G(V_i), G(V_{i+1}))\). See \S 2.3. One can arrange for \(V_0\) to be the vector space with minimum dimension among all the \(V_i\), then all spaces \(V_i\) and groups \(G(V_i)\) are completely determined. The paper's main contribution to this topic is a complete determination of the sequence \((\pi_i)_{i \in \mathbb{Z}}\), under the assumption (for the case of unitary groups) that \(V\) is one-dimensional. This is essentially done in three steps: (1) use the theorems described in the first part of this review to reduce matters to the depth-zero case; (2) prove that the case of depth zero can actually be deduced from that of finite groups; and (3) solve the analogous problems for finite groups, using the Aubert-Michel-Rouquier work already mentioned and a reduction to unipotent cuspidal representations due to Lusztig. These results appear in \S 5.3 for unitary groups and \S 6.3 for orthogonal and symplectic groups. The reduction from depth-zero representations to finite group representations is in \S 4, and the detailed results on the problem for finite groups are in \S 3.4--3.5.
representations of \(p\)-adic reductive groups
scientific article; zbMATH DE number 7051099
scientific article