34 problems

Corrected local model flatness conjecture. The scheme $M$ is flat over

Let $\operatorname{SO}_{2m-1}$ and $\operatorname{SO}_{2m}$ denote the special orthogonal groups of the indicated dimensions, and let $\operatorname{cd}(G)$ denote the canonical di…

Let ${\mathfrak g}=\operatorname{so}(2n+1)$ and let ${\mathfrak h}=\operatorname{so}(2k+1)\times\operatorname{so}(2n-2k+1)$. Let $Y$ and $Z$ be regular semisimple elements, and sup…

Gross's conjecture. The following assertions hold: $\pi^{K_{n,m}}=0$ for $0\leq m<c_\pi$ and $\dim_\mathbb{C}\pi^{K_{n,c_\pi}}=1$; the action of $J_{n,c_\pi}/K_{n,c_\pi}$ on…

Let $H$ be a $p$-adic symplectic group or split special orthogonal group, and in the special even orthogonal case let $H^+$ be the corresponding full even orthogonal group. Let…

Let $F/F_0$ be a quadratic extension, let $(V,\phi)$ be a symmetric space of dimension $2n+2$, and let $I\subset [0,n]$ be non-empty. Let $\RM^\pm_I$ be the spin local model over…

Let $F/F_0$ be a quadratic extension, let $(V,\phi)$ be a symmetric space of dimension $2n+2$, and let $I\subset [0,n]$ be non-empty. Let $\RM^\naive_I$ be the naive local model ov…

Let $n>2$, let $f:[-1,1]\to\mathbb{R}$ satisfy $f(1)=1$, and define $\widehat f:SO(n)\to\mathbb{R}$ by $\widehat f(\gamma)=f(\gamma_{11})$. Assume that $\widehat f$ defines a posit…

Let $SO(n)$ be the special orthogonal group, and let $SO(n-1)$ be the subgroup fixing the first coordinate. A class-one representation means an irreducible representation admitting…

Let $F$ be a discretely valued field with ring of integers $\CO$, let $k$ be its residue field, and let $n\geq 1$. For a non-empty subset $I\subset[0,n]$, let $\Lambda_I$ be the as…

Let $F$ be a discretely valued field with ring of integers $\CO$, and let $G$ be the split even orthogonal group of rank $2n$. Let $\CL$ be a self-dual lattice chain, and let…

Let $F$ be a complete discretely valued field with ring of integers $\mathcal{O}$, uniformizer $\pi$, and residue field $k$ of characteristic $p>2$. Let $V=F^{2n}$ with a split non…

Let $G$ be either the finite symplectic group $\operatorname{Sp}_{N}(q)$ or the finite minus-type orthogonal group $\operatorname{O}^{-}_{N}(q)$, and let $c$ be a Singer cycle in…

Let $G$ be a finite subgroup of $\operatorname{O}(n)$. Say that $G$ has the amphichiral linear centralizer property (LCP) if its centralizer in $\operatorname{O}(n)$ contains an el…

Let $\varphi$ be a level-one cuspidal quaternionic modular form, let $\Pi$ be the irreducible automorphic representation generated by $\varphi$, and let…

Fix open subgroups $U^p\subseteq G(\mathbb{A}^{p,\infty})$ and $U_p\subseteq G(\mathbb{Q}_p)$ containing $G(\mathbb{Q}_p)^+$, and put $U=U^pU_pG(\mathbb{R})^+$. For a coset…

Let $G$ be the orthogonal group and let $\Gamma$ be the arithmetic subgroup under consideration. Write $s$ for the relevant cohomological degree, let $\mathcal{RMC}(\Gamma)$ denote…

Let $R$ be a commutative ring and let $(M,q)$ be a quadratic module over $R$. Write $M^{\perp}$ for the radical of the polar form of $q$, and let an orthogonal map mean an $R$-line…

Let $G_n={\rm SO}_{2n+1}$ be the split odd special orthogonal group of rank $n\geq 1$ over $F$, let $U_n\subset G_n$ be its upper triangular maximal unipotent subgroup, and let…

Let $G$ be a unitary group or an even orthogonal group, and let … be an elliptic $A$-parameter for $G$. Let $\pi$ be an irreducible representation of $G(\mathbb A)$ such that the…

Critical orthogonal-group conjecture. For every $k>1$ with $s=1/\sqrt{k}$,

For $d\geq 2$, let $\mathcal U_{d;\leq n}$ be the domain of the moving frame and let … be the moving frame defined in the preceding section. Almost-polynomial moving-frame conjectu…

Let $\mathcal{G}$ be a closed subgroup of the orthogonal group $\mathcal{O}(d)$, and let $\operatorname{conv}(\mathcal{G})$ denote its convex hull. For…

Let $V_{m+1}$ be an orthogonal space of dimension $m+1$, and let $V_m$ be a non-degenerate codimension-one subspace. Assume that…

Let $m\geq 2$ be even, let $R\in SO(p)$ be an involution, and suppose that $R$ has level $m$. A sign-change matrix is a matrix in $\mathcal I_p^+$, and $R$ is sign-change reducible…