15 problems

Polynomial dimension-and-frequency conjecture. There exist $k\geq 1$, polynomials

Let $F$ be an algebraically closed field. Here $\mathfrak{p}_n(F)$ denotes the primitive part of the homology group associated with $\operatorname{GL}_n(F)$, and $l$ is a prime. Th…

Let $\pi$ be an irreducible representation of $\mathrm{GL}_n$, and let $M_n$ be the mirabolic subgroup. Use the paper's definition of a homogeneous representation and write…

Let $\pi$ be an irreducible representation of $\mathrm{GL}_n$ of depth $d$. Regard its highest derivative $\pi^-$ as a representation of $M_{n-d+1}$, and let … be its increasing so…

Let $G_n$ be the relevant general linear group, let $G_n'$ be its distinguished subgroup, and let $pi$ be an irreducible $G_n'$-distinguished representation of $G_n$ with depth $d$…

Explicit general-linear basis conjecture. The set $\mathcal{B}_m(n)$ consisting of these elements forms an $\mathbb{F}_q$-basis of ${\mathcal{Q}}_m(n)^{G_n}$. The conjecture is the…

Onn's weak conjecture. There is an isomorphism of group algebras

Partition-function bound conjecture. For $n\geq 7$,

Root bound conjecture. For $n\geq 6$, the maximum real root of $v_q(h(\operatorname{GL}_n(k)))$ is less than $2$.

Gurevich–Howe exhaustion conjecture. Suppose $k<\left\lfloor\frac{n}{2}\right\rfloor$. Then, up to twist by a character, every irreducible representation of $U$-rank $k$ of $GL_n$…

Let $F$ be a local field. For an Arthur type representation, let its associated Arthur parameter be a finite representation of $WD_F\times \mathrm{SL}_2(\mathbb C)$; two such param…

Final unstable homology conjecture. For every $m\geqslant 1$, the group $H_m(\operatorname{GL}_{2m}(\mathbb{F}_2);\mathbb{F}_2)$ is a single copy of $\mathbb{F}_2$ generated by the…

Let $G=\mathbf{GL}_{n}$ and $H=\mathbf{GL}_{m}$ with $n\leq m$. Let $K(\mathcal{X})$ and $K(DP_{I_{H}\times I_{G}}(\Pi(F)))$ be the associated bimodules under the extended affine H…

Let $G_n$ denote the relevant general linear group, and let $\mathcal{M}_{\infty}^{d_i}(G_{n_i})$ be the class of smooth representations of degree $d_i$ for $i=1,2$. Set … and let…

Adduced representation conjecture. The adduced representation of $\pi$ is the parabolic product of the adduced representations of its basic factors: