[For the entire collection see Zbl 0742.00067.]In the first part some general results on Hecke algebras are recalled; the structure constants corresponding to the standard basis are defined; in the following the example of the commuting algebra of the Gelfand- Graev representation of the general linear group $GL(2,F)$ is examined; here $F$ is a finite field of $q$ elements; the structure constants are explicitly determined first for the standard basis and then for a new basis obtained via a Mellin-transformation. Using the character table of the group $GL(2,F)$ two identities envolving Gaussian sums over finite fields are obtained. One of them is a formal analogue of the classical Barnes’ First Lemma; this lemma involves the classical gamma-function which is in analogy with the Gaussian sum function. Three more finite identities are given and several open questions are brought into discussion.Let us mention that meanwhile a parallel proof of the finite a!

EUDML-ID : urn:eudml:doc:220123
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@article{701492,
     title = {On the structure constants of certain Hecke algebras},
     booktitle = {Proceedings of the Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1991},
     pages = {[179]-188},
     mrnumber = {MR1151904},
     zbl = {0756.20003},
     url = {http://dml.mathdoc.fr/item/701492}
}

Helversen-Pasotto, Anna. On the structure constants of certain Hecke algebras, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books,  (1991), pp. [179]-188. http://gdmltest.u-ga.fr/item/701492/