Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is supercuspidal if $\pi/\langle\pi(n)v-v\rangle = 0$. This condition is equivalent to the fact that their matrix coefficients have compact support modulo $Z(\mathbb{Q}_p)$, the centre of $G(\mathbb{Q}_p)$.
Let $K$ be a maximal compact subgroup of $G(\mathbb{Q}_p)$, we say that $\pi$ is spherical or unramified if $\pi^K$, the space of $K$-fixed vectors of $\pi$ has dimension bigger than $0$.
We say that $\pi$ is a discrete series representation if the matrix coefficients of $\pi$ are $2$-integrable modulo $Z(\mathbb{Q}_p)$.
Usually I have found that people divide the admissible representations into three disjoint sets: Supercuspidals, non supercuspidals and discrete series and spherical. Is this decomposition true? Can a supercuspidal representation be spherical? (If not, why not?) Can a discrete series be spherical? (If not, why not?) Why the classical principal series representation (induction of characters of the Torus) are not discrete series?
Best Answer
As it happens, (admissible) supercuspidals cannot be spherical, because (by Borel–Casselman–Matsumoto) admissible repns with Iwahori-fixed vectors have non-trivial maps to and from unramified principal series. The Jacquet-module vanishing condition for supercuspidals is exactly that (via Frobenius Reciprocity, etc.) they admit no (non-zero) homomorphisms to principal series.
Many people would count supercuspidals as discrete series.
For $p$-adic groups, I myself do not know much about discrete series that are not supercuspidals.
And, again, if a discrete series repn were spherical, it would be a sub and quotient of principal series, which is not possible (EDIT … (thanks @Amitay for comments) for hyperspecial maximal compacts. For $\operatorname{GL}_n$, all maximal compacts are conjugate, and are hyperspecial. For $\operatorname{SL}_n$, they are all hyperspecial, but there are $n$ conjugacy classes. At least every (EDIT: thanks @LSpice) reductive group that splits over an unramified extension has at least one (conjugacy class of) hyperspecial maximal compact, but/and some (EDIT: split groups!) do have non-hyperspecials: for example, the affine apartments of $\operatorname{Sp}_4$ have two different types of vertices, one with fewer edges touching it. The latter is not hyperspecial.
Principal series are generically irreducible. For classical groups, we can explicitly compute the $L^2$ norm (mod center) of the (essentially unique) spherical vector, (EDIT for hyperspecial maximal compact) and it's not finite….
Yes, there are some square-integrable subrepns of principal series, at some special values of the parameters. The archimedean case has the well-known examples of holomorphic discrete series subreps of principal series far away from the "unitary range".