**Santosh Nadimpalli**

TIFR

November 12, 2015

**Typical representations for certain Bernstein components of $GL_n(F)$**:
The indecomposable blocks of the category of smooth representations
of a $p$-adic reductive group $G$ are determined by Bernstein and
these blocks are called Bernstein components. Let $F$ be a non-Archimedean local field with ring of integers $integers{F}$ and finite residue field. If $G$ is $g{n}{F}$ and $s$ is any given Bernstein component it follows from the Bushnell and Kutzko's work that there exists irreducible smooth representation $ au_s$ of $g{n}{integers{F}}$ such that for any irreducible smooth representation $pi$ of $G$
$$ho_{g{n}{integers{F}}}( au_s, pi)
eq 0$$
if and only if $pi$ belongs to the Bernstein component $s$. For
applications in arithmetic it was required to classify such
representations $ au_s$ usually called typical representation. We will try to present such a classification result for ``non-cuspidal'' Bernstein components.