Abstract

We consider certain asymptotic properties of smooth $p$-adically valued functions and representations of a $p$-adic reductive group $G$. First, we continue the study of the so-called $p$-tempered and $p$-discrete representations, as defined in a former paper, and apply this to get a classification of "locally integral" representations, i.e., those representations such that for any open compact subgroup $H$, the $H$-invariant subspace admits Hecke-invariant lattices. Then we show that the space of square-integrable smooth functions, as defined in the text, is an algebra under convolution to which the action of the Hecke algebra on any $p$-tempered representation extends naturally. We formulate a Plancherel-like formula but prove it only for $SL(2)$.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 227-255
Launched on MUSE
2009-01-30
Open Access
No
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