24 problems

Complete type-edge-pair conjecture. The union over all type-edge pairs equals the solution set:

Let $\mathbf M^{\mathrm{naive}}_{GSp_{2g}}$ denote the naive local model for the symplectic similitude group $GSp_{2g}$ in the ramified setting considered in the paper. Flatness co…

For the compact symplectic group $Sp(n)$, let $\operatorname{cat}(Sp(n))$ denote its Lusternik–Schnirelmann category. Symplectic-group category conjecture. … This is described in t…

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…

For a positive integer $n$, let $q$ be the residue-field cardinality and let $s$ be a complex variable. Let $\mathcal{Z}_{n,n,\mathfrak{o}}^{\mathsf{C}}(s)$ denote the symplectic l…

Let $F$ be a non-Archimedean local field, let $\boldsymbol{o}$ be its ring of integers, and use the notation $B(\gamma)^{\mathrm{irred}}$, $B(\gamma)^{\mathrm{split}}$, $d_i$,…

Let $N\in\mathbb{N}$, and let $S^*$ denote the unit circle. A function … is $\frac{1}{N}$-admissible if it is a finite sum $\sum_{\vec m}c_{\vec m}(x)z^{\vec m}$, where the exponen…

Uniform bounded generation conjecture. There is a minimal constant $K:=K(\Phi)$ proportional to ${\rm rank}(\Phi)$ and independent of $R$ and $I$ such that

Critical symplectic-group conjecture. For these parameters,

Let $X$ be a finite dimensional reduced Stein space, and let $f\colon X\to \operatorname{Sp}_{2m}(\mathbb{C})$ be a holomorphic mapping that is null-homotopic. Let $M_1,\ldots,M_K$…

Let $\mathcal{C}_m(\mathbb{R}^3)$ and the symplectic determinant $D_S:\mathcal{C}_m(\mathbb{R}^3)\to\mathbb{C}$ be defined from the Hopf-lift polynomials $p_\alpha$ using the order…

Symplectic LIC. The polynomials $p_\alpha$, for $\alpha\in I$, are linearly independent over $\mathbb{C}$. This is the symplectic analogue of the Atiyah-Sutcliffe LIC for the root…

Cohomological-building-block conjecture. There exist a $\theta$-stable parabolic $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ such that

Assume that $\dim(U)=k<n$. Let $Sp$ be the finite symplectic group, let $O_{\beta+}$ and $O_{\beta-}$ correspond to the two isomorphism classes of inner products of rank $k$, and l…

Let $r$ be odd with $r<2n$, and let $\sigma_{2n-r+1}^{(2r)}$ and $\Theta_{2n-r+1}^{(2r)}$ denote the representations constructed in the paper on the metaplectic symplectic group of…

Let $r$ be odd and let $r<2n$. Write $2n=a(n,r)r+b(n,r)$ with $a(n,r)$ and $b(n,r)$ nonnegative and $0\leq b(n,r)\leq r-1$. For a partition of $2n$, let its $Sp$ collapse be the gr…

Let $\phi=\boxplus_{i=1}^r(\tau_i,1)\in\widetilde{\Phi}_2({\mathrm{Sp}}_{2n})$ be a generic global Arthur parameter. Here $\widetilde{\Pi}_{\phi}({\mathrm{Sp}}_{2n})$ denotes its g…

Malkoun's conjecture 2 for $Sp(n)$. For every such configuration,

Malkoun's conjecture 1 for $Sp(n)$. For every such configuration, the polynomials $p_1,q_1,\ldots,p_n,q_n$ are linearly independent over $\mathbb{C}$.

Conjecture 4.2. For every $\psi\in\widetilde{\Psi}_2(\operatorname{Sp}_{2n})$:

Let $k$ be a field, let $n>2$, and let $C_i(2n,k)$ denote the number of isomorphy classes of involutions of Type $i$ in $\operatorname{SP}(2n,k)$, for $1\leqslant i\leqslant 4$. Le…

Let $p$ be the characteristic of the ground field, let $R^{S\!p(n)}$ be the invariant ring for $n\times n$ generic matrices under the symplectic group, and let $K_{\infty}$ denote…

Let $n\ge 1$, let $\mathfrak{g}=\mathfrak{sp}(2n)$, and let $V$ be the vector representation. For a partition $\mu$, write $\overline{2\mu}$ for the partition obtained by doubling…

Let $\operatorname{Sp}(n)$ be the compact symplectic group, let $\mathcal O_k$ be the distinguished conjugacy class associated with the $k$-th vertex of the fundamental alcove, and…