28 problems

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…

Let $q=p^r$, let $GL_n(\mathbb F_q)$ be the general linear group over the finite field with $q$ elements, and let $k$ be the coefficient field. Consider the cohomology groups…

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 $\pi$ be an irreducible representation of $\mathrm{GL}_n$ in the archimedean setting. A Casselman–Wallach representation is called homogeneous if every $M_n$-subrepresentation…

Let $\pi$ be an irreducible representation of $\mathrm{GL}_n$ of depth $d$, and let $\pi^-$ denote its highest derivative, regarded as a representation of the mirabolic group…

Let $\Phi_{\lambda}(G)$ be the set of $L$-parameters with fixed infinitesimal character, identified with the corresponding orbits in a Vogan variety. The geometric closure order on…

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 $d$…

Bernstein's Zelevinsky-invariance conjecture.

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

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 $GL_n$…

Gurevich–Howe agreement conjecture. For $GL_n(\mathbb{F}_q)$, U-rank and tensor rank coincide for values in that range.

Conjectural automorphic category. For every $G=\operatorname{GL}_n(\mathbb{A})$, there is such a category with realization functors and compatibility properties; in particular, the…

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…

Let $k$ be a local field, let $M_A$ and $N_A$ be A-parameters of ${\rm GL}_n(k)$ and ${\rm GL}_{n-1}(k)$, respectively, and let $\pi_M$ and $\pi_N$ be the associated irreducible re…

Let $R$ be a commutative ring with identity, let ${\rm GL}_n(R)$ be the general linear group, and let ${\rm E}_n(R)$ be the subgroup generated by elementary matrices. Consider the…

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 $A\Delta(4m,k,s)$ denote the adduced representation associated with the representation $\Delta(4m,k,s)$ of the relevant general linear group. It is known that … when $k\neq m$.…

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: