99 problems

Let $\mathfrak{g}$ be the Lie algebra of an exceptional simple algebraic group, let $\0$ be a special nilpotent orbit, and let ${\mathcal P}(\0)$ be its special piece. Let $H$ be t…

Let $\mathrm G$ be a connected reductive group over $F$ and let $G=\mathrm G(F)$. Assume a local Arthur-packet theory for $G$. Let $\Psi^+(G)$ be the set of local Arthur parameters…

Let $R=R(\eta,\gamma)$ be a dominant sequence of rectangles, and let $R'$ be obtained from $R$ by replacing a subsequence … by … where $|\alpha|=|\beta|$ and $\alpha^+$ dominates…

Let $G$ be a complex reductive Lie group, and let $X=G\cdot\lambda=G/G(\lambda)$ be a nilpotent coadjoint orbit. Suppose an irreducible $G$-equivariant local system on $X$ is given…

Arthur–Barbasch–Vogan conjecture. $X$ is unitarizable.

Achar–Sommers' conjecture. The maximal element in this expression is $h^{\vee}$, the semisimple element of a Jacobson–Morozov triple of $\mathcal{O}^{\vee}$. The paper states that…

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field of characteristic $0$, and let $x\in\mathfrak{g}$. By $C_{\mathfrak{g}}(x)$ denote the centralizer…

Let $G$ be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a$ be its extended affine Weyl group, and let…

Let ${\mathfrak g}={\mathfrak{sl}}_n$ and let ${\mathcal V}$ be an orbital variety, that is, an irreducible component of ${\mathcal O}\cap{\mathfrak n}$ for a nilpotent orbit…

Sommers's cell-region decomposition conjecture.

Let $\bar{p}: X \rightarrow \overline{\0}$ be a covering of a nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ such that $R[1]=0$. Suppose that $\pi: Z\rightarrow X$ is a sym…

Let $\mathcal U$ be the relevant set of rigid orbits, and suppose that $G(1)$ is not $O(l)$ for any $l\geq2$. Unipotent-exhaustivity conjecture. The set…

Let $G$ be either an orthogonal group or a metaplectic group. Let $\mathcal O_{\bold D}$ be the orbit indexing $\Pi_{\mathcal O_{\bold D}}(G)$, let $\mathcal O_{\bold d}$ be a rigi…

Let ${^{\scriptscriptstyle L\!}}{\mathcal O}$ be a nilpotent orbit in the Lie algebra of the Langlands dual group, let $\mathcal O=d({^{\scriptscriptstyle L\!}}{\mathcal O})$ be th…

Let $G$ be a complex reductive group with nilpotent cone $ilcone$, let $K_G( ilcone)$ be the Grothendieck group of $G$-equivariant coherent sheaves on $ilcone$, and let…

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$, let $\mathfrak g^d=\mathfrak g\oplus\mathfrak g$, and let $\mathbf e=(e_1,e_2)\in\mathfrak g^d$ be a…

Let $N\geq 1$, and let $V(\Delta)$ be the boundary hypertoric vertex operator superalgebra for the minimal nilpotent orbit closure of $\mathfrak{sl}_N$. Weight conjecture. After re…

Let $\fg$ be a simple simply-laced Lie algebra, let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbb{h}}}=m\geq1$, and let $L_k(\fg)$ be the simple affine vertex algebra. Let…

For every $n\geqslant3$, let ${\mathcal O}(\boldsymbol{\lambda})\subset{\mathfrak{sl}}_n$ be the unique nilpotent orbit with … where ${\mathfrak b}$ is a Borel subalgebra and…

Let ${\mathcal N}\subset{\mathfrak{sl}}_n$ be the nilpotent cone, let $e\in{\mathcal N}$, and define ${\mathcal O}^{\langle k\rangle}:=SL_n\cdot e^k$ for $k\geqslant1$. Power conje…

Let $G$ be an algebraic group with Lie algebra ${\mathfrak g}$, let ${\mathcal N}$ be the nilpotent cone, and let $e\in{\mathcal N}$. Write ${\mathfrak g}^e$ for the centraliser of…

Let $H$ be a split classical group over $F$, let $u$ denote the nilpotent orbit of the image of … under the derivative of the relevant $\mathrm{SL}_2(\mathbb{C})$-factor, and let…

Let $\mathfrak{g}$ be a Lie algebra, let $e$ be a distinguished nilpotent element, and let $Q=Q(e)$ be the finite centralizer of an associated $sl_2$-triple. Let $\mathcal{S}_e$ be…

Let $\mathfrak{g}$ be a Lie algebra, let $e\in\mathfrak{g}$ be distinguished nilpotent, and let $Q=Q(e)$ be the finite centralizer of an $sl_2$-triple associated with $e$ in the si…

One-step geometric wavefront set speculation. Either