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 $\pi/\left<\pi(n)v-v\right> = 0$. 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?

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?