This is an account of joint work with P. C. Kutzko, mainly concerned with ${\rm GL}(N)$ ([1,4]), but there are some similar results [2,3] for ${\rm SL}(N)$ and the beginnings [5] of a general philosophy.
We start with the ${\rm GL}(N)$-theory [1]. Throughout, $F$ is a non-Archimedean local field with discrete valuation ring ${\germ o}_F$ and finite residue field $k_{F$. Also $V$ is an $F$-vector space of dimension $N<\infty$, with $G={\rm Aut}_F(V)$ and $A={\rm End}_F(V)$. The crucial notion is that of a simpel type in $G$; we review the definition briefly.
Let ${\germ A}$ be a hereditary ${\germ o}_F$-order in $A$, with Jacobson radical ${\germ B}$. We write $U({\germ A})$ for the parahoric subgroup ${\germ A}^x$ of $G$ and $U^m({\germ A})=1+{\germ B}^m$, $m\geq 1$, for its congruence subgroups. A stratum (over ${\germ A})$ is a 4-tuple $[{\germ A},n,mb]$ where $n>m$ are integers and $b\in {\germ B}^{-n}$. Two such strata $[{\germ A},n,m,b_i]$ are equivalent if $b_1\equiv b_2$ $\mod {\germ B}^{-m}$). A stratum $[{\germ A},n,m,\beta]$ is pure if the algebra $F[\beta]$ is a field, $\beta\notin{\germ B}^{1-n}$, and $x^{-1}{\germ A}x={\germ A}$ for all $x\in F[\beta]^\times$. It is simple if, in addition, it satisfies a certain condition "$m>-k_{\germ o}(\beta,{\germ A})$" (see [1] (1.4)).
A simple stratum of the form $[{\germ A},n,0,\beta]$ (in fact its equivalence class) gives rise to a pair of ${\germ o}_F$-orders ${\germ H}(\beta,{\germ A})\subset {\germ I}(\beta,{\germ A})\subset{\germ A}$, and hence to two families of compact open subgroups of $G$ by $H^m(\beta,{\germ A})={\germ H}(\beta,{\germ A})\cap U^m({\germ A})$, $J^m(\beta,{\germ A|)={\germ I}(\beta,{\germ A})\cap U^m({\germ A})$, for $m\geq 1$. We also write $J={\germ I}^\times={\germ I}\cap U({\germ A})$. The group $G$ acts on the set of (equivalence classes of) simple strata, and the constructions of ${\germ H}$, ${\germ I}$ are compatible with this.
By direct construction, the simple stratum $[{\germ A},n,0,\beta}$ gives rise to a certain finite set ${\scr C}({\germ A},\beta)$ of simple characters $\theat$ of $H^1(\beta,{\germ A})$. These character sets have a remarkable "intertwing implies conjugacy" property:
Theorem 1: For $i=1,2$, let $[{\germ A},n_i,0,\beta_i]$ be a simple stratum in $A$ and let $\theta_i\in {\scr C}({\germ A},\beta_i)$. Suppose that the $\theta_i$ intertwine in $G$. There then exists $x\in U({\germ A})$ such that ${\scr C}({\germ A},\beta_2)={\scr C}({\germ A},x^{-1}\beta_1x)$ $(={\scr C}({\germ A},\beta_1)\circ {\rm Ad}(x))$ and $\theta_2=\theta_1\circ ${\rm Ad}(x)$.
(It is worth noting in passing that the set ${\scr C}({\germ A},\beta)$ does not determine the equivalence class of $[{\germ A},n,0,\beta]$: at the level of conjugacy classes, a simple character $\theta$ determines a finite set $\hat{\scr C}(\theta)$ of conjugacy classes of equivalence classes of simple strata, and each $[\cdots,\beta]\in\hat{\scr C}(\theta)$ gives a finite set $\overline{\scr C}(\beta)$ of conjugacy classes of simple characters. We have $\#\hat{\scr C}(\theta)=\#\overline{\scr C}(\beta)$ for $\theta\in {\scr C}({\germ A},\beta)$, and this number is almost always greater than 1. (See [4].))
Given $[{\germ A},n,0,\beta]$ and $\theta\in{\scr C}({\germ
A},\beta)$, ther is a unique irreducible representation $\kappa$ of
$J(\beta,{\germ A})$ which extends $\eta$ and is intertwined by every
element of $B^\times$. (It is not unique, but no matter.) We now
assume that ${\germ A}$ is a principal order. This gives us
$$J/J^1\cong {\rm GL}(f,k_E)^\epsilon,\tag1$$
with the obvious abbrevations, and where $ef[E:F]=N$. Let $\sigma_0$
be an irreducible cuspidal representation of ${\rm GL}(f,k_E)$. We
form the $e$-fold tensor power of $\sigma_0$ and inflat it to a
representation of $J$. Set $\lambda=\kappa\otimes\sigma$. The pair
$(J,\lambda)$ is what we call a simple type in $G$. There is also a
degenerate case to be dealt with: if ${\germ A}$ is a principal order,
then $U({\germ A})/U^1({\germ A}\cong {\rm GL}(f,k_F)^e$, $ef=N$. We
construct a representation $\sigma$ of $U({\germ A})$ from an
irreducible cuspidal representation of ${\rm GL}(f,k_F)$ as before, to
get a simple type $(U({\germ A}),\sigma)$. (For example, the trivial
character of an Iwahori subgroup of $G$ is a simple type of this
form.) In either case, a simple type $(J,\lambda)$ is called maximal
if $e=1$.
Theorem: Let $(J,\lambda)$ be a simple type in
$G$. Then the Hecke algebra ${\scr H}(G,\lambda)$ is isomorphic to the
affine Hecke algebra ${\scr H}(e,(\#k_E)^f)$.
The simple modules over affine Hecke algebras are classified in [7],
so this theorem implies a classification of the irreducible smooth
representations $\pi$ of $G$ which contain $\lambda$ (in the sense
that ${\rm Hom}_J(\pi,\lambda)\not\{0\}$). There is, mreover, a strong
uniqueness property:
Theorem: For $i=1,2$, let $(J_i,\lambda_i)$ be a
simple type in $G$. Suppose that the $\lambda_i$ intertwine in $G$.
There then exists $g\in G$ such that $J_2=g^{-1}J_1g$ and
$\lambda_2\cong \lambda_1\circ {\rm Ad}(g)$. In particular, an
irreducible smooth representation $\pi$ of $G$ can contain at most one
simpel type, up to conjugacy.
The case $e=1$ is particularly interesting.
Theorem: (i) Let $\pi$ be an irreducible
supercuspidal representation of $G$. Then $\pi$ contains a maximal
simple type $(J,\lambda)$. There is a unique representation $A$ of
$\tilde J$ such that $\Lambda|J=\lambda$ and $\pi\cong c$-${\rm
Ind}(\Lambda)$. (Thus every irreducible supercuspidal representation
of $G$ is induced from an open compact mod centre subgroup.)
(ii) Let $(J,\lambda)$ be a maximal simple type in $G$, and let $\pi$ be an irreducible smooth representation of $G$ which contains $\lambda$. Then $\pi$ is supercuspidal.
One can characterise the irreducible representations containing a
given simple type. If $\pi_i$ is an irreducible smooth representation
of ${\rm GL}(N_i,F)$, $i=1,2$ we write $\pi_1\tims \pi_2$ for the
representation of ${\rm GL}(N_1+N_2,F)$ obtained, in the usual way, by
parabolic induction.
Theorem: Let $(J,\lambda)$ be a simple type in
$G$, and define $\epsilon$ by (1). There exists an irreducible
supercuspidal representation $\pi_0$ of ${\rm GL}(N/e,F)$ with the
following property: an irreducible smooth representation of $G$
contains $\lambda$ if and only if it is equivalent to a subquotient of
$\pi_0\chi_1\times\cdots\pi_0\chi_e$, for unramified quasicharacters
$\chi_j$ of ${\rm GL}(N/e,F)$.
This relates the classification via the Hecke algebra isomorphism above to the Zelevinsky classification [11]. One can then deal with the irreducible representations containing no simple type via a process of irreducible parabolic induction, as in [1] (8.5). This gives a complete classification of the irreducible smooth representations of $G$, extending the Zelevinsky classification of the non-supercuspidal ones.
There is, however, an alternative approach. One can use the last theorem to define the notion of simple type. More generally, suppose given an irreducible supercuspidal representation $\pi_i$ of ${\rm GL}(N_i,F)$, $1\leq i\leq r$, with $\sum_iN_i=N$. A semisimple type (for $(\pi_1,\cdots,\pi_r)$) is then an irreducible smooth representation $\pi$ of $G$ contains $\rho$ if and only if it is a subquotient of $\pi_1\chi_1\times\cdots\times \pi_r\chi_r$, for unramified quasicharacters $\chi_i$. The explicit constructions of [1] thus give the existence of a (particularly nice) semisimple type $(K,\rho)$ when the $\pi_i$ are all equvialent. This approach (as in [5, 6]) has the advantage of yielding a description of the category of all smooth represenations of $G$, rather than just the irreducible representations. it can equally be formulated for any reductive $p$-adic group $G$, rather than just ${\rm GL}(N)$. The results of [8] (for classical groups), [9] (for `level zero' representations of arbitrary groups) and [10] (for quite general groups) appear, on the basis of conversations with the authors, to be consistent with it.
Bibliography: