We discuss two versions of the Hecke algebra of a locally profinite group G, one that is complex valued and one that is p-adic valued. We reproduce several results which are well known for the complex valued Hecke algebra for the p-adic valued Hecke algebra. Specifically we show the equivalence of smooth representations of G and smooth modules of the Hecke algebra of G. We specialize to the group G=GLn(F) for F an extension of Qp, and show that the spherical Hecke algebra of G is finitely generated, and exhibit its generators. This is a standard fact for the complex valued Hecke algebra that we reproduce for the p-adic valued case. We then show that the spherical Hecke algebra of SLnF is isomorphic to a subalgebra of the spherical Hecke algebra of GLnF. Then a character of the spherical Hecke algebra ofGLn(F) can also be viewed as a character of the spherical Hecke algebra of SLn(F). Therefore such a character has two induced modules, one for the Hecke algebra of GLn(F) and another for the Hecke algebra of SLn(F). Theorem 3.4.3 and corollary 3.4.4give a condition under which the coinduced and induced modules of such a character areisomorphic as vector spaces.
| Identifer | oai:union.ndltd.org:siu.edu/oai:opensiuc.lib.siu.edu:theses-3768 |
| Date | 01 September 2020 |
| Creators | Roberts, Jeremiah |
| Publisher | OpenSIUC |
| Source Sets | Southern Illinois University Carbondale |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses |
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