Abstract: One of the cornerstones of the representation theory of
reductive groups over non-archimedean local fields is the seminal work
of Bernstein and Zelevinsky in the 1970s. Much of their work is
concentrated on the general linear group. It culminated in Zelevinsky's
classification of the (complex, smooth) irreducible
representations of $GL_n(F)$ (where $F$ is a non-archimedean local
field) in terms of multisegments --
an almost purely combinatorial object. Later on this classification was
extended to inner forms of the general linear groups, namely to the
groups $GL_n(D)$ where $D$ is a local non-archimedean division algebra.
There are two classification schemes, "à la Zelevinsky" and "à la
Langlands". The Langlands classification is valid for any reductive
group, but in the case of linear groups it can be refined and made more
In this talk we will explain these classifications and a new approach,
due to Lapid-Minguez-Tadic, giving a uniform proof of the
classifications in all cases.