Question

I've been reading about the Langlands program (the paper by Torsten Wedhorn "Local langlands correspondence for GL(n) over p-adic fields, to be precise), and I want to get my hands dirty with examples. What are some interesting (yet not too nasty) examples of admissible representations of $GL_{n}(K)$, where $K$ is a $p$-adic field? Taking Schur functors of an admissible representation, should still generally give you something admissible, right?

Answer 1

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 $\theta\vert_{\mathcal{O}_L^\times}$ 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 $g\in\text{GL}_2(k)$ has eigenvalues $\alpha,\beta\in\ell\backslash k$. (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 $K^\times\text{GL}_2(\mathcal{O}_K)$ 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.

Answer 2

Ila, I think that the result on admissibility of irreducible, smooth representations is due to Jacquet.

Probably the best way to get some hands-on experience is to start with the principal series. These are the representations constructed by parabolic induction from the characters of the diagonal torus; they are irreducible when the character is regular, i.e., distinct from its non-trivial Weyl conjugates. Of course these representations sweep out only a small part of the admissible spectrum, but they can offer a start.

To understand the 'rest' of the (irreducible) admissible spectrum, even for $GL_2$ (where all that's left is the discrete series), is already something; it's the subject of the lovely book of Bushnell and Henniart.

Answer 3

This seems more like a comment than an answer, but it might be a nice way to think of things.

For any open compact subgroup $K_0$ of $GL_n(K)$, the trivial representation is admissible. However, it is NOT true that the induced representation from $K_0$ to $GL_n(K)$ is admissible. This is due to the fact that $GL_n(K)/K_0$ may not be compact. In fact when you consider any admissible representation $(\rho,V)$ of a closed subgroup $H$ inside $G$, then if $G/H$ is compact, inducing $\sigma$ from $H$ to $G$ results in an admissible representation.

Also, any irreducible smooth representation of $GL_n(K)$ is admissible. This follows from Harish-Chandra.