29 problems

Equivalence conjecture. For every simple simply connected Lie group $G$ and every level $k\geq 0$, the categories $\mathrm{Rep}^k(LG)$ defined via textit{a.} or textit{b.}, and via…

Gaitsgory–Kazhdan irreducibility conjecture. The object $\Pi\in \operatorname{Rep}(\BG)$ is irreducible.

The Verlinde algebra isomorphism conjecture. For every homogeneous degree $z^n$ with $n\neq 0$, the map

Let $Sp(n)$ be the compact symplectic group, let $U(i)$ be the unitary group, and let $K.\Omega Sp(n)$ denote the K-homology ring of its based loop space. For $1\leq i\leq n$, the…

With $X$, $G$, $\ul{x}$, and $\ul{\operatorname{Shv}}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})$ as above, regard this object as an object of…

Let $X$ be a smooth complete curve over $k$, let $\ul{x}$ be a finite set of $k$-points, and let $\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}}$ be the moduli stack of $G$-b…

Let $G$ be a reductive group and let $\fL(G)\operatorname{Cat}_{\operatorname{restr}}$ be the restricted 2-category of categories acted on by its loop group. Let…

Let $G$ be a reductive group with Langlands dual $cG$, let $\fL(G)$ be its loop group, and let $\fL(G)\operatorname{Cat}_{\operatorname{restr}}$ be the restricted 2-category of cat…

Let $G$ be a reductive group, let $\xi=(M,\lambda,L,\mathcal{O},\mathcal{L})$ be a polar-cuspidal datum for $G$, and let $\mathbf{Q}\subset LM$ be a parahoric subgroup containing…

Representability conjecture. The fpqc sheaf $\prescript{}{1}{\mathcal{C}_1^{\leq\mu}}$ is represented by a normal $k$-scheme. This predicts that the indicated truncated double quot…

Centralizer transitivity conjecture. The group $(LG)_{\mathfrak g}$ acts transitively on $\operatorname{Gr}_{\mathfrak g,\mathcal O}$. The paper notes that this fiber is nonempty a…

Let $g\in VMO(S^1,SU(2))$. Let $A(g)$ and $A_1(g)$ be the operators associated with $g$, and let a triangular factorization mean the factorization specified in (II.3). A root subgr…

Let $k_2$ be the loop associated with a sequence $\zeta=(\zeta_n)$ satisfying $\zeta\in l^2$, and suppose the phases of the $\zeta_k$ are uniform and independent random variables.…

Tempered positive-energy conjecture. The $K$-group $K_\bullet(\Gamma_0(\tau_G^G)\rtimes G)$ reflects the tempered, positive energy representations of $LG$ in a suitable sense, and…

Let $\widetilde{\Omega G}$ be the centrally extended based loop group, and let $H$ be a representation of it at level $k$. Call $H$ locally normal when it belongs to the category o…

Let $G$ be the compact, simply connected Lie group and $k$ the level appearing in the representation categories. Let $\mathrm{Rep}(\cA_{G,k})$ be the category of representations of…

Let $\mathfrak G$ be a loop reductive group scheme over $R_n$, and suppose that all semisimple quotients of $\mathfrak G$ are isotropic. Isotropic loop reductive torsor conjecture.…

Let $G$ be a connected Lie group and let $k\in H^4(BG,\mathbb{Z})$ be a positive level. Consider the categories $\operatorname{Rep}^k(LG)$ defined via vertex operator algebras and…

Converse root subgroup factorization conjecture. If $g_1$ is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds: $g_1$ has a root subgroup…

Let $G$ be the group considered in the paper, let $BG$ be its classifying space, and let $\mathcal{L}BG$ denote its free loop space. Write $G(\mathbb{F}_q)$ for the corresponding f…

Let $\G$ be a loop group, let $\bN$ be its loop maximal unipotent subgroup, and let $\C$ be a category equipped with a $\bG$-action. Let $\chi$ be the character of $\bN$ defined fr…

Let $m$ be an integer, and let $P_m(d)$ be the free abelian group generated by the isomorphism classes of irreducible positive-energy representations of the universal central exten…

Let $\Omega$ be the space of totally positive loop-group elements. For $X\in\Omega$, let $J(X)$ be the set of equivalence classes $[\mathbf i]$ of infinite reduced words…

Let $\mathbf i$ be an infinite reduced word, and let $e_{\mathbf i}$ be its associated factorization map. Say that $\mathbf i$ is minimal in its block when it is minimal in the rel…

Let $\Omega$ be the space of totally positive loop-group elements. For $X\in\Omega$, let $I(X)$ be the set of equivalence classes $[\mathbf i]$ of infinite reduced words…