Degenerate principal series representations for quaternionic unitary groups
Let \(F\) be a non-Archimedean local field with uniformizer \(\pi\). Let \(\omega= |\cdot|\) be the absolute value such that \(|\pi|^{-1}\) is the order of the residue field. Let \(D\) be a quaternion \(F\)-algebra and let \(W= D^{2n}\) be a (right) \(D\)-vector space with basis \(\{e_1,\dots, e_n,\, f_1,\dots, f_n\}\). For \(\varepsilon=\pm 1\), we define an \(\varepsilon\)-skew Hermitian (sesquilinear) form \(\langle,\rangle\) on \(W\) such that \(\langle e_i,e_\rangle= \langle f_i,f_j\rangle= 0\) and \(\langle e_i, f_j\rangle= \delta_{ij}\). Let \(G= G_n\) be the isometric subgroup of \(\text{GL}_{2n}(D)\) stabilizing the Hermitian form. Then \(G_n\) is of type \(D_{2n}\) (resp. \(C_{2n}\)) if \(\varepsilon= 1\) (resp. \(\varepsilon= -1\)). Let \(P= P_n= M_n N_n\) denote the maximal parabolic subgroup of \(G\) stabilizing the span of the \(e_i\)'s. Here \(M_n= \text{GL}_n(D)\) and let det be the reduced norm on it. Let \(\chi: F^\times\to\mathbb{C}\) and we will continue to use \(\chi\) to denote the character \(\chi\circ\text{det}\) on \(P\). The main result of this paper is to study the reducibility and irreducible subquotients of the normalized induced degenerate principal series representation \(I_n(\chi)= \text{Ind}^{G_n}_{P_n}\chi\). Using a normalized intertwining operator \(M^*(\chi): I_n(\chi)\to I_n(\chi^{-1})\), the author is able to show that for \(\chi\) outside a finite set, \(I_n(\chi)\) is irreducible. In order to study the irreducible quotients of a reducible \(I_n(\chi)\), the author uses the dual pair correspondences of the Weil representation. In this review, we will only state the main results for \(\varepsilon=1\). First \(I_n(\chi)\) is reducible if and only if \(\chi^2= \overline\omega^{2j-1}\) for \(-n+ 1\leq j\leq n\). In the reducible case, \(I_n(\chi)\) is of the form \(I_n(\omega^{j-{1\over 2}})\otimes \widetilde\mu\) where \(\widetilde\mu\) is a quadratic character of \(G_n\). Let \(H\) be the isometry group of a right \(D\)-vector space of dimension \(n\) with a non-degenerate Hermitian form. Let \(R_{n,m}\) be the \(H\)-coinvariants of the action of the dual pair \(G_n\times H\) on the Weil representation. Then \(R_{n,m}\) is irreducible and it embeds into \(I_{n,m}:= I(\omega^{m-n+{1\over 2}})\) as its unique irreducible submodule. In the reducible case, i.e. \(0\leq m< 2n\), we have \(I_{n,m}/R_{n,m}\simeq R_{n,m_0}\) where \(m_0= 2n- 1- m\). In particular \(I_n(\chi)\) has at most two irreducible subquotients. We refer to the paper for the case \(\varepsilon= -1\). In Section 8, the author discusses the module structures of the corresponding degenerate principal series of the isometry group \(O(k, k)\) of a split quadratic \(F\) vector space of dimension \(2k\). Section 9 is a summary of previously known results of Johnson, Sahi, Ørsted and Zhang, Zhang when \(F=\mathbb{R}\). This is a carefully written and well-organized paper.
scientific article; zbMATH DE number 6108373
scientific article