Here,$F$ denotes a non-Archimedean local field (with finite residue field) and $G$ the group of $F$-points of aconnected reductive algebraic group defined over $F$. Let $\frak R(G)$ denote the category of smooth, complexrepresentations of $G$. Let ${\cal B}(G)$ be the set of pairs $(L,\sigma)$, where $L$ is an $F$-Levi subgroupof $G$ and $\sigma$ is an irreducible supercuspidal representation of $L$, taken modulo the equivalencerelation generated by twisting with unramified quasi characters and $G$-conjugacy. To $\frak s\in {\cal B}(G)$,one can attach a full (abelian) sub-category $\frak R^\frak s(G)$ of $\frak R(G)$; the theory of the Bernsteincentre shows that $\frak R(G)$ is the direct product of these $\frak R^\frak s(G)$. The object of the paper isto give a general method for describing these factor categories via representations of compact open subgroupswithin a uniform framework. Fix $\frak s\in {\cal B}(G)$. Let $K$ be a compact open subgroup of $G$ and $\rho$an irreducible smooth representation of $K$. The pair $(K,\rho)$ is an $\frak s$-type if it has the followingproperty: an irreducible representation $\pi$ of $G$ contains $\rho$ if and only if $\pi\in \frak R^\fraks(G)$. Let ${\cal H}(G,\rho)$ be the Hecke algebra of compactly supported $\rho$-spherical functions on $G$;if $(K,\rho)$ is an $\frak s$-type, then the category $\frak R^\frak s(G)$ is canonically equivalent to thecategory ${\cal H}(G,\rho) \text{-Mod}$ of ${\cal H}(G,\rho)$-modules. Let $M$ be a Levi subgroup of $G$; thereis a canonical map ${\cal B}(M)\to {\cal B}(G)$. Take $\frak t\in {\cal B}(M)$ with image $\frak s\in {\calB}(G)$. The choice of a parabolic subgroup of $G$ with Levi component $M$ gives functors of {\it parabolicinduction\/} and {\it Jacquet restriction\/} connecting $\frak R^\frak t(M)$ with $\frak R^\frak s(G)$. Weassume given a $\frak t$-type $(K_M,\rho_M)$ in $M$; the paper concerns a general method of constructing fromthis data an $\frak s$-type $(K,\rho)$ in $G$. One thus obtains a description of these induction andrestriction functors in terms of an injective ring homomorphism ${\cal H}(M,\rho_M) \to {\cal H}(G,\rho)$. Themethod applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observedin certain particularly interesting cases, one can go some distance to describing ${\calH}(G,\rho)$ explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism,and this in turn implies powerful intertwining theorems for the types.

1991 Mathematics SubjectClassification: 22E50.