Allan J.
Silberger and Ernst-Wilhelm Zink
LEVEL
ZERO TYPES AND HECKE ALGEBRAS FOR LOKAL CENTRAL SIMPLE ALGEBRAS
Preprint series: Institut für Mathematik, Humboldt-Universität
zu Berlin (ISSN 0863-0976)
-
MSC:
-
22E50 Representations of Lie and linear algebraic groups over local
fields
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11S45 Algebras and orders, and their zeta functions, See Also {11R52,
11R54, 16H05,
16Kxx}
Abstract: Let D be a central
division algebra and Ax = GLm(D) the unit
group of a central simple algebra over a p-adic field F. The purpose
of
this paper is to give types (in the sense of Bushnell and Kutzko)
for all level zero Bernstein components of Ax and to establish
that the Hecke algebras associated to these types are isomorphic to
tensor
products of Iwahori Hecke algebras. The types which we consider are
lifted from cuspidal representations \tau of M(kD), where
M is a
standard Levi subgroup of GLm and kD is the residual
field of D. Two
types are equivalent if and only if the corresponding pairs (M(kD),\tau)
are conjugate with respect to Ax. The results are basically
the
same as in the split case Ax = GLn(F) due to
Bushnell and
Kutzko. In the non split case there are more equivalent types and the
proofs
are technically more complicated.
Keywords: p-adig groups, types, Hecke algebras