Weak explicit matching for level zero discrete series of unit groups of \(\mathfrak p\)-adic simple algebras. (English) Zbl 1028.22017
Let \(A_1\) and \(A_2\) be central simple algebras over a \(p\)-adic field having the same reduced degrees \(>1\). It is known that there is a bijection (the “Jacquet-Langlands correspondence”) between the sets of irreducible discrete series representations of the unit groups \(A_1^\times\) and \(A_2^\times\). This correspondence preserves character values for regular elliptic elements.
The authors describe the above correspondence explicitly for the class of level zero representations (see M. Grabitz, A. J. Silberger and E.-W. Zink [J. Number Theory 91, No.1, 92-125 (2001; Zbl 1009.22016)]). A parametrization is given for the set of unramified twist classes of level zero discrete series which does not depend on the algebras \(A_1\) or \(A_2\) and is invariant under the Jacquet-Langlands correspondence.
The authors describe the above correspondence explicitly for the class of level zero representations (see M. Grabitz, A. J. Silberger and E.-W. Zink [J. Number Theory 91, No.1, 92-125 (2001; Zbl 1009.22016)]). A parametrization is given for the set of unramified twist classes of level zero discrete series which does not depend on the algebras \(A_1\) or \(A_2\) and is invariant under the Jacquet-Langlands correspondence.
Reviewer: Anatoly N.Kochubei (Kyïv)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11R39 | Langlands-Weil conjectures, nonabelian class field theory |
