This work is concerned with type theory for reductive groups over a non Archimedean field. Given such a field F, and a division algebra D of finite dimension over its center F, we obtain results concerning the construction of simple types for the group GL(m, D), $m\geqslant1$. More precisely, for each simple stratum of the matrix algebra M(m, D), we produce a set of β-extensions in the sense of Bushnell and Kutzko.

We use some fundamental work of Bernstein to study parabolic induction in reductive p-adic groups. In particular, we determine when parabolic induction from a component of the Bernstein decomposition of a Levi subgroup to the corresponding component of the full group is an equivalence of categories.

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.

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.