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**On the structure constants of certain Hecke algebras.**
*(English)*
Zbl 0756.20003

Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 179-188 (1991).

[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 and of the classical identity has been elaborated by the author and P. SolĂ©; this represents a new proof of the classical identity and will appear in Can. Math. Bull. under the title “Barnes’ First Lemma and its Finite Analogue”.

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 and of the classical identity has been elaborated by the author and P. SolĂ©; this represents a new proof of the classical identity and will appear in Can. Math. Bull. under the title “Barnes’ First Lemma and its Finite Analogue”.

Reviewer: A.Helversen-Pasotto (Nice)

### MSC:

20C33 | Representations of finite groups of Lie type |

20G40 | Linear algebraic groups over finite fields |

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |

20G05 | Representation theory for linear algebraic groups |

20C15 | Ordinary representations and characters |