Colloquium abstracts

Manish Mishra
Hebrew University, Jerusalem
August 22, 2013

The Struture of an Unramified $L$-Packet and Related Results:  Let $\mathrm G$ be a connected reductive group over a nonarchimedean local field $F$, such as $\mathbb{Q}_p$. The local Langlands conjectures predict that the (isomorphism classes of) irreducible `admissible' representations of $\mathrm G(F)$ (in vector spaces over $\mathbb{C}$) can be partitioned in a certain natural way into equivalence classes, called $L$-packets. Here we will consider an `unramified' $\mathrm G$, and the unramified representations of $\mathrm G(k)$. Such representations can be parameterized by pairs $(K, \lambda)$, where $K$ is a so called `hyperspecial' compact subgroup of $\mathrm G(k)$ and $\lambda$ is a character of a maximally split maximal torus $\mathrm T$ of $\mathrm G$. Let $\tau_{K, \lambda}$ be the representation associated to a pair $(K, \lambda)$. We will first describe when two such representations $\tau_{K, \lambda}$ and $\tau_{K', \lambda'}$ are isomorphic. Now given an unramified $L$-packet we can : (i) think of its elements as $\tau_{K, \lamb