Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial Representation

In this paper, we give a complete picture of Howe correspondence for the setting (O(E, b), Pin(E, b),π), where O(E, b) is a real orthogonal group, Pin(E, b) is the two-fold Pin-covering of O(E, b), and π is the spinorial representation of Pin(E, b). More precisely, for a dual pair (G,G′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(G, G^{\prime })$\end{document} in O(E, b), we determine explicitly the nature of its preimages (G~,G′~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\tilde {G}, \tilde {G^{\prime }})$\end{document} in Pin(E, b), and prove that apart from some exceptions, (G~,G′~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\tilde {G}, \tilde {G^{\prime }})$\end{document} is always a dual pair in Pin(E, b); then we establish the Howe correspondence for π with respect to (G~,G′~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\tilde {G}, \tilde {G^{\prime }})$\end{document}.