An explicit matching theorem for level zero discrete series of unit groups of \(\mathfrak p\)-adic simple algebras. (English) Zbl 1087.22012
The theory of types due to C. J. Bushnell and P. C. Kutzko [Proc. Lond. Math. Soc., III. Ser. 77, No. 3, 582–634 (1998; Zbl 0911.22014)] assigns to each twist class of a discrete series representation a conjectural simple type. In case of level zero representations of the unit group of a \({\mathfrak p}\)-adic simple algebra, this has been realized in M. Grabitz, A. Silberger and E.-W. Zink [J. Number Theory 91, No. 1, 92–125 (2001; Zbl 1009.22016)]. In a foregoing paper [Can. J. Math. 55, No. 2, 353–378 (2003; Zbl 1028.22017)] of the authors, a list of types has been chosen for each \({\mathfrak p}\)-adic simple algebra of fixed reduced degree and an assignment between types that is compatible with the abstract matching theorem (AMT) of discrete series representations due to P. Deligne, D. Kazhdan and M.-F. Vigneras [Représentations des algèbres centrales simples \(p\)-adiques. Représentations des groupes réductifs sur un corps local, 33–117 (1984; Zbl 0583.22009)].
These types were pairs \(({\mathfrak A}^\times,\tau)\), where \({\mathfrak A}^\times\) is a suitable parahoric subgroup of the unit group \(A^\times\) of a \({\mathfrak p}\)-adic simple algebra \(A\) and \(\tau\) is a representation of this subgroup which appears in each level zero discrete series representation of the unramified twist class which they classify. The authors now consider extensions of such representations to suitable compact-mod-center subgroups such that each level zero discrete series representation \(\Pi\) of \(A^\times\) is determined by the containment of a specified extended type representation and contains a unique simple type component \(\widetilde\Sigma(\Pi)\). They call the resulting pairs maximal level zero extended simple types and refine the assignment of level zero types to an assignment of maximal level zero extended simple types realizing the AMT for level zero discrete series representations.
The proof is by finding certain regular elliptic elements, where the characters of \(\widetilde\Sigma(\Pi)\) and \(\Pi\) are the same. They compute the character values at these elements by using a version of T. Shintani’s [J. Math. Soc. Japan 28, 396–414 (1976; Zbl 0323.20041)] descent. The AMT is characterized by a character identity for discrete series representations of various algebras.
These types were pairs \(({\mathfrak A}^\times,\tau)\), where \({\mathfrak A}^\times\) is a suitable parahoric subgroup of the unit group \(A^\times\) of a \({\mathfrak p}\)-adic simple algebra \(A\) and \(\tau\) is a representation of this subgroup which appears in each level zero discrete series representation of the unramified twist class which they classify. The authors now consider extensions of such representations to suitable compact-mod-center subgroups such that each level zero discrete series representation \(\Pi\) of \(A^\times\) is determined by the containment of a specified extended type representation and contains a unique simple type component \(\widetilde\Sigma(\Pi)\). They call the resulting pairs maximal level zero extended simple types and refine the assignment of level zero types to an assignment of maximal level zero extended simple types realizing the AMT for level zero discrete series representations.
The proof is by finding certain regular elliptic elements, where the characters of \(\widetilde\Sigma(\Pi)\) and \(\Pi\) are the same. They compute the character values at these elements by using a version of T. Shintani’s [J. Math. Soc. Japan 28, 396–414 (1976; Zbl 0323.20041)] descent. The AMT is characterized by a character identity for discrete series representations of various algebras.
Reviewer: Martin Grabitz (Berlin)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11R39 | Langlands-Weil conjectures, nonabelian class field theory |
