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 $\mathcal O_F$ and finite residue field. If $G$ is $GL_{n}{F}$ and $s$ is any given Bernstein component it follows from the Bushnell and Kutzko's work that there exists irreducible smooth representation $\tau_s$ of $GL_n (\mathcal O_F)$ such that for any irreducible smooth representation $\pi$ of $G$ $$\mathrm{Hom}_{GL_n \mathcal O_F}(\tau_s, \pi) \neq 0$$ if and only if $\pi$ belongs to the Bernstein component $s$. For applications in arithmetic it was required to classify such representations $\tau_s$ usually called typical representation. We will try to present such a classification result for ``non-cuspidal'' Bernstein components.
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 $\mathcal O_F$ and finite residue field. If $G$ is $GL_{n}{F}$ and $s$ is any given Bernstein component it follows from the Bushnell and Kutzko's work that there exists irreducible smooth representation $\tau_s$ of $GL_n (\mathcal O_F)$ such that for any irreducible smooth representation $\pi$ of $G$ $$\mathrm{Hom}_{GL_n \mathcal O_F}(\tau_s, \pi) \neq 0$$ if and only if $\pi$ belongs to the Bernstein component $s$. For applications in arithmetic it was required to classify such representations $\tau_s$ usually called typical representation. We will try to present such a classification result for ``non-cuspidal'' Bernstein components.