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\textbf{Double cosets and $K$-types for $p$-adic GL(3)} \\

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\textbf{Peter Campbell, University of Bristol}
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Let $\mathfrak{o}$ be the ring of integers of a $p$-adic field $F$  with 
prime ideal $\mathfrak{p}$, then $K=\mathrm{GL}(3,\mathfrak{o})$ is a 
maximal compact subgroup of the $p$-adic group $G=\mathrm{GL}(3,F)$.  
For each positive integer $i$ define $B_{i}$ to be the subgroup of 
matrices in $K$ whose lower triangular entries lie in 
$\mathfrak{p}^{i}$.  Onn, Prasad and Vaserstein have recently shown that 
whenever $i>2$ the double coset space $B_{i}\backslash K/B_{i}$ depends 
on the residue field of $F$.  We extend this to the general case 
$H_{1}\backslash K/H_{2}$ where $H_{1}$ and $H_{2}$ are finite index 
subgroups of $K$ containing the upper triangular matrices.   
Consequently, we are able to describe the decomposition of an unramified 
principal series representation of $G$ on restriction to $K$.  (Joint 
work with Monica Nevins).

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