langlands conjectures - examples of admissible representations of $GL_{n}(K)$ over p-adic field

I second L Spice's recommendation of the book by Bushnell and Henniart, called "The local Langlands conjectures for GL(2)."

After you master the principal series representations, it's not too hard to tinker with some supercuspidals. Easiest among these are the tamely ramified supercuspidals. To construct these, let's start with the unramified quadratic extension $L/K$, with corresponding residue fields $ell/k$. Choose a character $theta$ of $L^times$ which has these properties:

(a) The character $theta$ is trivial on $1+mathfrak{p}_L$, so that $thetavert_{mathcal{O}_L^times}$ factors through a character $chi$ of $ell^times$.
(b) $chi$ is distinct from its $k$-conjugate. (In other words, $chi$ does not factor through the norm map to $k^times$.)

It's a standard fact that there's a corresponding representation $tau_chi$ of $text{GL}_2(k)$, characterized by the identity $text{tr}tau_chi(g)=-(chi(alpha)+chi(beta))$ whenever $gintext{GL}_2(k)$ has eigenvalues $alpha,betainellbackslash k$. (This is somewhere in Fulton and Harris, for instance.)

Inflate $tau_chi$ to a representation of $text{GL}_2(mathcal{O}_K)$, and extend this to a representation $tau_theta$ of $K^timestext{GL}_2(mathcal{O}_K)$ which agrees with $theta$ on the center. Finally, let $pi_theta$ be the induced representation of $tau_theta$ up to $text{GL}_2(K)$; then $pi_theta$ is an irreducible supercuspidal representation.

By local class field theory, our original character $theta$ can be viewed as a character of the Weil group of $L$. In the local Langlands correspondence, $pi_theta$ lines up with the representation of the Weil group of $K$ induced from $theta$. All the supercuspidals of $text{GL}_2(K)$ arise by induction from an open compact-mod-center subgroup, but the precise construction of these is a little more subtle than the above example.

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