68 problems

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…

Finite-part factorization conjecture. There are positive integers $d_1,d_2,\ldots,d_l$, with $l\leq n$, such that

Let $G$ be a Kac–Moody group over an algebraically closed field of characteristic $0$. An automorphism of $G$ may be inner, diagonal, graph, field, or sign, where these are the sta…

Homotopy-generator conjecture.

Polynomiality conjecture. $\bar Q_A(q)$ and $\bar R_A(q)$ are polynomials in $q$, and they correspond to the finite part and the infinite polynomial part of $H^*(F(A))$.

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

Ol'shanskii's conjecture. The other unitary groups considered in the paper should also be type I.

Foley's conjecture. One has

Twin-masure retraction conjecture.

Let $\mathcal{G}$ be a Kac–Moody group with standard Borel subgroup $\mathcal{B}$, maximal torus $T\subset\mathcal{B}$, and Weyl group $W$. For $w\in W$, let…

Amalgam conjecture. The group $G(\mathfrak m)$ arises naturally as an amalgam of the $\operatorname{GL}_2$ subgroups and pro-unipotent subgroups discussed in the preceding construc…

Amalgam conjecture. The group $G(\mathfrak m)$ is an amalgam of automorphism groups of Lie algebras related to $\mathfrak m$.

Dense embedding conjecture. The subgroup $U^+$ embeds as a dense subgroup of $\widehat{U}^+$.

Let $G_{twin}$ act on the positive masure ${{\mathscr I}}_{\oplus}$, let $(G_{twin})^+_{\oplus}:=\{g\in G_{twin}\mid g.0_{\oplus}\geq 0_{\oplus}\}$, and let $J^+$ and…

Let ${\boldsymbol{\bvar}}$ be the valuation ring setting described in the paper, let $(G_{\mathcal O})_{\oplus}^+=\{g\in G_{\mathcal O}\mid g.0_{\oplus}\geq 0_{\oplus}\}$, and let…

Let $\Omega\subseteq\Delta_+^{\operatorname{re}}$ be a reasonable choice of positive real roots. Define … where $U_\alpha=\{\chi_\alpha(t)\mid t\in\mathbb{Q}\}$ and…

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…

The groups under consideration are finitely generated simple Kac–Moody groups whose Weyl groups have Coxeter diagrams containing entries equal to $\infty$. Abramenko's conjecture.…

Galashin–Karp–Lam conjecture. The totally nonnegative part of ${\mathcal P}_K$ is a regular CW complex, and $Q_K$ is its face poset.

$E_{10}(\mathbb{Z})$ invariance conjecture. In analogy with $SL_2(\mathbb{Z})$, the group $E_{10}(\mathbb{Z})$ is conjectured to be the discrete invariance group for certain functi…

The paper considers a Kac–Moody group whose Kac–Moody matrix has size $2$, a parameter $\tau\file$ in the relevant set, the principal series representation $I_\tau$, and the associ…

Finite presentability conjecture. If $(W,S)$ is not 2-spherical, meaning that there exist $s,t\in S$ with $|st|=\infty$, then $G$ is never finitely presented.

Let $A$ be a generalized Cartan matrix, let $K(A)$ be the associated Kac-Moody group, and let $F(A)$ be its flag manifold. Let $d_t$ be the positive integers occurring in the facto…