14 problems

Let $F$ be a non-Archimedean local field of characteristic zero, let $l$ be a positive integer, and let $\mathrm{GL}_l(F)$. Let $\pi,\pi'$ be irreducible generic representations of…

Let $k$ be a nonarchimedean local field, let $n$ be a positive integer, and let $\rho$ and $\rho'$ be irreducible generic admissible representations of ${\rm SO}_{2n+1}(k)$. For ea…

Let $F$ be a non-archimedean local field of characteristic $0$ with residual characteristic $p$, and let $N$ be an odd positive integer. Let $\pi$ and $\pi'$ be self-dual supercusp…

Let $F$ be a non-archimedean local field of characteristic $0$ and residual characteristic $p$, and let $\mathrm{G}(F)$ be the group of $F$-points of a split symplectic or special…

Local converse conjecture. If

Let $F$ be a $p$-adic field. Let $\pi,\pi'$ be two irreducible $\psi$-generic representations of $\mathrm{SO}_{2n}(F)$ with the same central character. For an irreducible generic r…

Let $F$ be a local non-Archimedean field, let $n$ be a positive integer, and let $\pi_1$ and $\pi_2$ be irreducible unitarizable supercuspidal representations of…

Let $F$ be a $p$-adic field, and suppose the $G_2$ multiplicity-one conjecture for $\widetilde I(\chi,\psi)$ holds. Let $\Pi_1,\Pi_2$ be irreducible generic representations of…

Let $n$ be prime and let $\omega$ be a character of $\mathbb F_{p^n}^{\times}$; write $S(\chi)$ for the corresponding Gauss sum. Let $\hat k=k(p^n-1)/(p-1)$. Prime-characteristic G…

Let $E/{\mathcal F}$ be the quadratic extension implicit in the distinction notation, with $n\ge3$. Let $\pi$ be a supercuspidal representation of $\operatorname{GL}_n(E)$, and for…

Let $n$ be prime, and let $\chi_1,\chi_2$ be regular characters of $\mathbb F_{q^n}^{\times}$. For each character $\eta$ of $\mathbb F_q^{\times}$, regard $\eta$ as inflated to…

Let $r$ be a prime divisor of $n$. For a regular character $\chi$ of $\mathbb F_{q^n}^{\times}$, let $\pi_\chi$ be the associated primitive cuspidal representation of…

Let $n>m$. Let $\pi$ and $\tau$ be irreducible cuspidal representations of $\operatorname{GL}_n(\mathbb F_q)$ and $\operatorname{GL}_m(\mathbb F_q)$, corresponding to regular chara…

Local converse conjecture. If