This is joint work with M.-F. Vignéras in Paris. Let $p$ be a prime number and $F$ a finite extension of $Q_p$ or $F_p((T))$, with residue field of cardinality $q$. Let $G$ be the group of $F$-points of a reductive group over $F$, and $C$ a field of coefficients. We consider a smooth admissible irreducible representation $\pi$ of $G$ on a $C$-vector space $V$, and investigate its behaviour near identity. In the case where $G=GL_n(D)$ for a central division $F$-algebra $D$ with finite degree, we obtain precise results when the characteristic of $C$ is not $p$. Namely there are integers $c_\pi(P)$ attached to association classes of parabolic subgroups $P$ of $G$ such that $\pi$ and $\sum c_\pi(P)Ind_P^G(1)$ agree on a small enough open compact subgroup $K$ of $G$. As a consequence the asymptotic behaviour of the dimension of fixed points under congruence subgroups of $G$ is given by an explicit polynomial formula in $q$, with integer coefficients. When $C$ has characteristic $p$, similar information is available only for $GL_2(Q_p)$, but for split groups $N$. Abe has recently made progress towards the dimension of fixed points in the important case where $\pi$ is the Steinberg representation of $G$.

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