19 problems

Multiplicity and packet conjecture. For every $v\in S$, there is a representation $\Pi_v^-$ of $\mathrm{G}_2(\mathbb{Q}_v)$, different from $\mathcal{L}_\alpha(\pi_v,1/10)$, such t…

Non-positive-curvature $G_2$ freeway basis conjecture. The set of $G_2$ freeways of non-positive curvature with $n$ endpoints is a basis for

Let $V_2$ be the two-dimensional complex Cartan subalgebra for the Lie algebra of type $G_2$, and let $V_2^{\rm reg}$ be the complement of the six root-kernel lines. Let $W_{G_2}$…

Let $G_2$ have basis $C_1,\ldots,C_{14}$, with subgroups $S$, $\Sigma$, and maximal torus $T$ generated respectively by $\{C_1,C_2,C_3\}$, $\{C_8,C_9,C_{10}\}$, and…

Let $G_2$ be the reductive group under consideration, and let $\pi$ be an irreducible unitarizable representation of $G_2$. The Aubert–Zelevinsky duality of $\pi$ is denoted by…

Let $\Pi(f)$ be the Gan–Gurevich lift associated with the Hecke eigenform $f$. Discrete automorphy conjecture. The representation $\Pi(f)$ should always be a discrete automorphic r…

Mundy's conjecture. The representation $\Pi$ occurs in $L^2_{\rm disc}(G_2(\mathbb Q)\backslash G_2(\mathbb A))$ with multiplicity zero or one, and it occurs with multiplicity one…

Let $F$ be an archimedean local field, let $G$ be the group of type $\mathsf{G}_2$, and let $K/F$ be the quadratic algebra used to define $G'_J$. Let $\chi:K^\times\to S^1$ be a un…

Let $\operatorname{Sh}_\mathbf{G}$ be the Shimura variety associated with $\mathbf{G}$. In the setting where the Tate conjecture predicts a cycle giving the trivial representation…

Let $G$ be a group of type $G_2$, let $\sigma$ be a cuspidal automorphic representation of $G(\mathbf{A})$, and let $\pi$ be its lift to $\operatorname{PGSp}_6(\mathbf{A})$. Let…

Functorial lifting conjecture. There is a self-dual automorphic representation $\widetilde\Pi$ of $GL_7(\mathbb A)$ satisfying all five listed properties: matching unramified Satak…

Cuspidal occurrence conjecture. The $\Pi_{F,f}^v$-isotypic component of the cuspidal spectrum is

The subregular Arthur-sheaf conjecture. If $\operatorname{char}(k)\neq 2$ or $3$, then there exists a unique sheaf $\mathcal F_{\mathrm{sr}}$ on $\operatorname{Bun}_G$ such that, o…

Let $F$ be a $p$-adic field, and suppose the $G_2$ multiplicity-one conjecture for $\widetilde I(\chi,\psi)$ holds. Let $\Pi_1,\Pi_2$ be irreducible generic representations of…

Let $F$ be a $p$-adic field, let $\Pi$ be an irreducible selfdual representation of $G_2(F)$, let $\psi$ be a nontrivial additive character of $F$, and let $\widetilde\pi$ be an ir…

Let $F$ be a $p$-adic field, let $\Pi$ be an irreducible generic representation of $G_2(F)$, let $\psi$ be a nontrivial additive character of $F$, and let $\widetilde I(\chi,\psi)$…

For the group $G_2$, the nilpotent orbit $A_{1_s}$ has complex dimension $4$ and is not in the image of the Spaltenstein map. Let $\mathcal{W}_{\mathcal{S}}$ label one D5 brane, wi…

Let $\lambda=m_1\varpi_1+m_2\varpi_2\in\mathcal{P}_+$, and let $\boldsymbol{G_2^{\varpi_1}}(1)$ and $\boldsymbol{G_2^{\varpi_2}}(1)$ be the polytopes defined by the displayed inequ…

G2 DT-invariant conjecture. There exist Calabi-Yau 3-folds $X$ with $h_{1,1}(X)=1$ whose invariants $\Omega(\gamma)$ are captured by Fourier coefficients of automorphic forms attac…