Distinguished tame supercuspidal representations and odd orthogonal periods (Q2889352)

Let \(G\) be a reductive group over a local nonarchimedean field, and \(\theta\) an involution of \(G\). Let \(G^\theta\) be the subgroup of \(G\) fixed under \(\theta\). An irreducible admissible representation \(\pi\) of \(G\) is distinguished by \(G^\theta\) if there is a nontrivial \(G^\theta\) invariant linear form on the space of \(\pi\). A fundamental question is to classify the set of distinguished representations. In an important paper by \textit{J. Hakim} and \textit{F. Murnaghan} [IMRP, Int. Math. Res. Pap. 2008, Article ID rpn005, 166p. (2008; Zbl 1160.22008)], a general theory was developed to classify the distinguished supercuspidal representations based on the cuspidal datum associated to the representations, through the works of Howe, Moy and Prasad, Yu and others.NEWLINENEWLINEThe current paper further develops the theory of Hakim and Murnaghan, and applies it to the special case when \(G\) is a general linear group and \(G^\theta\) is an orthogonal group. Let \(\Psi\) be the cuspidal datum associated to the irreducible supercuspidal representation \(\pi\), one can construct from \(\Psi\) compact groups \(K^\circ\subset K\subset G\) and a representation \(\xi\) of \(K\). The group \(G\) acts on the set of involutions on \(G\) through NEWLINE\[NEWLINEg\cdot\theta (x)=g^{-1}\theta(gxg^{-1})g,\,\,x\in G.NEWLINE\]NEWLINE Let \(\langle \theta,\pi\rangle\) be the dimension of the space of \(G^\theta\) invariant linear forms on the space of \(\pi\). The first main result is an identity NEWLINE\[NEWLINE\langle \theta,\pi\rangle=\sum m_K(\Theta)\langle \Theta,\xi\rangle_KNEWLINE\]NEWLINE where the sum is over the \(K-\)orbits \(\Theta\) of involutions in the \(G-\)orbit of \(\theta\); \(m_K(\Theta)\) is an integer that is a power of \(2\), and \(\langle \Theta,\xi\rangle_K\) is the dimension of the space of \(K^{\theta'}\)-invariant linear forms on the space of \(\xi\) (with \(\theta'\in \Theta\)). In fact, this formula corrects a mistake in a prior formula claimed in the work of Hakim and Murnaghan.NEWLINENEWLINEThe authors then establish a finer formula: NEWLINE\[NEWLINE\langle \theta,\pi\rangle=\sum m_{K^\circ}(\Theta)\langle \Theta,\pi\rangle_{K^\circ}NEWLINE\]NEWLINE where the sum is over the \(K^\circ-\)orbits \(\Theta\) of involutions in the \(G-\)orbit of \(\theta\); \(m_{K^\circ}(\Theta)\) is a certain multiplicity, and \(\langle \Theta,\pi\rangle_{K^\circ}\) is the dimension of the space of \(K^{\circ,\theta'}\)-invariant linear forms on the space of a certain representation of \(K^\circ\) (with \(\theta'\in \Theta\)). The point is that the calculation of this dimension is essentially a calculation over a finite field, where one can apply a prior work of Lusztig.NEWLINENEWLINEFor the case \(G=\roman{GL}(n)\) and \(G^\theta\) being an orthogonal group, and when \(n\) is odd, the authors were able to show that \(m_{K^\circ}(\Theta)\) is either \(0\) or \(1\), and they are able to calculate \(\langle \Theta,\pi\rangle_{K^\circ}\) by a generalization of Lusztig's work. The main result is that the supercuspidal representation \(\pi\) constructed from \(\Psi\) (a tamely supercuspidal representation) is \(G^\theta\) distinguished if and only if \(\omega(-1)=1\) where \(\omega\) is the central character of \(\pi\) and \(G^\theta\) is a quasi-split orthogonal group. In a later work the authors also considered the case \(n\) being even and got a classification of distinguished supercuspidal representations.