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I'm reading the following notes on Brian Conrad's website. There are a couple of statements there which seem too good to be true. There is Proposition 3.6, which states: (Bernstein): Every irred …

Originally posted on math.stackexchange. Let $G$ be a semisimple algebraic group over a field $k$. Let $A_0$ be a maximal split torus of $G$, and $\mathfrak a_0 = \operatorname{Hom}(X(A_0), \mathbb …

Finite dimensional, irreducible representations of simply connected, complex semisimple algebraic groups can be classified by their highest weight. I was wondering if there is an analogous classifica …

Let $P_0$ be a minimal parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field $k$. Let $P$ be a parabolic subgroup containing $P_0$ with Levi decomposition $P = MN$. Let $N^-$ …

In the first chapter of W. Casselman's unpublished notes on representation theory, there is at least one stated result which is not true: A counterexample to this last result is given in the questi …

There is a claim in a proof in Casselman's notes on representation theory which I have not been able to verify. I have asked several people and nobody knows why it is true. The thing I can't figur …

Let $(V,\pi)$ be an irreducible, admissible, supercuspidal representation of $G = \operatorname{GL}_n(F)$ for $F$ a $p$-adic field. Let $B = TU$ be the usual Borel subgroup, maximal torus, and unipot …

Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in repres …

Let $F$ be a $p$-adic field, $G = \operatorname{Gal}(\overline{F}/F)$ and $W$ the Weil group of $F$. The inclusion map $W \subset G$ is continuous with dense image, so $\rho \mapsto \rho|_W$ defines …

Let $G = \operatorname{GL}_n(F)$ with the usual Borel subgroup $P = TU$. Let $\chi = \chi_1 \otimes \cdots \otimes \chi_n$ be an unramified character of $T$. Suppose that $\chi$ is regular, which is …

$\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Ind}{Ind}$I have a question on the details of the Bernstein-Zelevinsky classification. This classification allows us to obtain irreducible repre …

Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee …

Let $P$ be a parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field. Let $M$ be a Levi subgroup of $P$, and let $N$ be the unipotent radical of $P$. If $(\pi,V)$ is a smooth, i …

In Bill Casselman's notes on root systems (http://www.math.ubc.ca/~cass/courses/tata-07a/Roots.pdf), I am confused about the proof of the result $s_{\alpha}(\beta)^{\vee} = s_{\alpha^{\vee}}(\beta^{\v …

Let $G$ be a connected, reductive group over a $p$-adic field $k$, $A_0$ a maximal split torus of $G$, and $P = MU$ a parabolic subgroup with $M$ containing $A_0$. Let $\overline{P} = M \overline{U}$ …