Local Summability of Characters on $p$-adic Reductive Groups

In this paper we study the complex representations of reductive groups over local non-Archimedean fields. We use the building of the reductive group to give upper-bounds for the absolute value of the character of an admissible representation and for the Weyl integration formula for certain regular elements. The upper-bound for the character of a representation is based on the alternative description, depending on the building, of the character as given by R. Meyer and M. Solleveld [MS12]. Once the character and the Weyl integration formula are related to the building, the upper-bounds will follow from a similar argument. Both upper-bounds generalize the upper-bounds given by Harish-Chandra [HC70] to groups defined over fields of positive characteristic. At last following Harish-Chandra's method we combine both upper-bounds to show that for a maximal torus $T$ containing a maximal split torus the character is locally summable on $\{ gtg^{-1} : g\in G,t\in T\}$.