73 problems

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…

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…

Kudla–Rallis conservation relation. The first occurrence indices satisfy

Let $\lambda_O$ be the map in the displayed comparison diagram from … to … The diagram compares this map with the corresponding general linear maps and the maps $\alpha$ and…

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…

Let $\mathrm{O}(n)$ be the group of real orthogonal $n\times n$ matrices, and let the least statistically-dependent basis (LSDB) be the basis obtained by optimizing statistical ind…

Let $lat$ be the lattice, let $m=\dim_{\mathbb{R}}[lat]$, let $ind(lat)$ denote its index, and let $rot$ and $rot^{\times}$ be the rotation groups defined in the surrounding discus…

Let $\mathcal O(r)$ be the real square orthogonal group, and define … A signed permutation matrix is an orthogonal matrix with exactly one nonzero entry, equal to $1$ or $-1$, in e…

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 $\RM^\naive_I$ 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 $I(r)$ be the rate function governing the large-deviation estimate for the mean eigenvalue parameter $r$, with the preceding analysis establishing $I(r)=2r^2$ for…

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 toric cycles be the cycles attached to algebraic tori in orthogonal groups, and let the rigid meromorphic cocycles of Darmon–Gehrmann–Linowski be evaluated on such cycles. Darm…

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

Large-image conjecture. For almost all primes $p$, the image of $\rho_p^{\mathrm{proj}}$ should contain

Let $O_N$ denote the classical orthogonal group, and let $O_N^*$ and $O_N^+$ denote its half-liberated and free orthogonal quantum-group liberations, respectively. A liberation of…

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…