Analytic properties of the Chevalley map, and Harish-Chandra's integrability theorem for GLn(Fq((t)))

The Chevalley map for GLn  is the map sending an n by n matrix to its characteristic polynomial. We consider it for the field Fq((t)) - a local field of positive characteristic. Very recently, we showed that the push forward via this map of every smooth compactly supported measure on the vector space of matrices is a measure on the affine space of monic polynomials of degree n whose density belongs to $L^q$ for every finite $q$.

 

From this we deduced that Harish-Chandra's characters of irreducible cuspidal representations of GLn(Fq((t))) are locally integrable  functions.

 

I will explain all the terms I just used, clarify the formulation of the two theorems, and their role in representation theory, and say a couple of words about the proofs.