19 problems

Non-linearity conjecture. If $W$ is Gromov-hyperbolic and $q>\!\!>1$, then $\Lambda$ is not linear.

Let $A=(a_{ij})_{n\times n}$ be a Cartan matrix of indefinite type with $n\geq3$, and let $G(A)$ be the associated Kac-Moody group. Infinite-generation conjecture. The rational coh…

Let $K(A)$ be a Kac-Moody group of infinite type, and let $G(A)$ denote the associated group. For a prime number $p$, consider the integral cohomology ring $H^*(BG(A);\mathbb Z)$.…

Let $A$ be an $n\times n$ Cartan matrix of infinite type. Let $P$ be the ring of invariants associated with the Weyl group of $A$, and let $P_i$ be the corresponding invariant subr…

Let $A$ be a generalized Cartan matrix and let $\widehat{G}$ be the associated local pro-$p$ completion. An irreducible principal minor of $A$ is a principal submatrix that is irre…

Let $A$ be the rank-$2$ generalized Cartan matrix considered in the source, with $\max(a_{21},a_{12})\leq -2$ and without assuming $a_{21}=a_{12}$. Let $\widehat{G}$ denote the loc…

Let $A$ be the rank-$2$ generalized Cartan matrix considered in the source, with $\max(a_{21},a_{12})\leq -2$ and without assuming $a_{21}=a_{12}$. Let $\widehat{G}$ denote the loc…

Let $G$ be the hyperbolic Kac--Moody group, let $K$ be its relevant compact subgroup, let $T^{ad}_{\mathbb R}$ be the real adjoint torus, and define $U_{i,\infty}^\pm=U_W^\pm U_i$…

Let $B_{i,\infty}^\pm$ be the stabiliser of the corresponding end, with subgroups $B_W^\pm$, $U_i$, and $W^{even}$ as defined in the paper. Structure decomposition conjecture. For…

Let $A^7=g\cdot A^7$ for some $g\in B_{i,\infty}^7$, where $A^7$ is the standard apartment and $U_i$ is the corresponding unipotent subgroup. Transitivity conjecture. For each such…

Small-representations conjecture. Kac–Moody groups possess a minimal unitary representation realizable automorphically. For $E_n(\mathbb{R})$, it is obtained by inducing from $P_1$…

Gabber–Kac simplicity conjecture. The group $\mathfrak{G}_A^{pma}(k)$ is GK-simple.

Luna's conjecture. Mapping ${\mathcal H}\subseteq{\mathcal G}$ to the homogeneous spherical datum of ${\mathcal G}/{\mathcal H}$ induces a bijection between the set of conjugacy cl…

Let $W$ be a Coxeter group, let $J$ be a subset of its simple generators, and let $w\in W^J$. Let $P_w^J$ denote the corresponding parabolic Kazhdan–Lusztig polynomial, and call it…

Let $E_s(g)$ be the Eisenstein series induced by the parameter $s$, and let $A'$ be the cone used in the paper. Let $K$ be the unitary form and $N$ the unipotent subgroup in the ra…

Everywhere-convergence conjecture. $E_\nu(g)$ converges absolutely for $g\in KA'N$ and $\nu$ satisfying $\mathrm{Re}~\nu(h_{\alpha_i})<-1$, $i\in I$. The proved result gives almost…

Let $\mathcal{D}$ be a Kac–Moody root datum with indecomposable generalized Cartan matrix $A$, let $\mathbf{F}_q$ be the finite field of characteristic $p$ with $q$ elements, and l…

Let $G$ be a Kac–Moody group, let $P$ be a parabolic subgroup, and let $X=G/P$. Let $W$ be the Weyl group, and let $\Lambda$ be the dominant weight associated with $P$. Let…

Let $(W,S)$ be an irreducible non-spherical and non-affine Coxeter system with $S$ finite, let $X$ be a building of type $(W,S)$, and let $G$ be a group acting on $X$ by automorphi…