# Exponential sums and representations of finite groups of Lie type

SPEAKER: Charles Curtis

TITLE: Exponential sums and representations of finite groups of Lie type.

ABSTRACT The irreducible representations of a finite dimensional semisimple commutative algebra over the complex numbers are obtained as eigenvalues of matrices whose entries are structure constants of the algebra with respect to a basis. This idea was used by Frobenius to define characters of finite groups in 1896 before representations of finite groups had appeared in print. In this talk, the idea will be applied to obtain spherical functions of a finite group of Lie type associated with a multiplicity free representation, beginning with the case of SL_2(q) where the main result is proved using Gauss sums. Some remarks about extensions to groups of higher rank are included at the end if there’s time.