30 problems

Eswara Rao's conjecture. Every non-trivial irreducible Harish-Chandra module over $\mathcal{V}ir$ comes from either a quotient of a module of tensor fields on an $(n+1)$-dimensiona…

Vogan's conjecture. The induced topologies on the cohomology groups $H^{\text{p}}(\mathfrak{n},M_{\text{glob}})$ are Hausdorff, and there are natural isomorphisms of $L_{0}$-repres…

Let $G$ be a connected semisimple Lie group with finite center, let ${\bf g}$ be its Lie algebra, and let $K$ be a maximal compact subgroup. Let $(\pi,V_{\pi,K})$ be a Harish-Chand…

Let $G'$ be the set of regular elements of $G$, let $A^{-}$ be the subset of $A$ defined by … and let $(MA)^{-}$ be the interior in $MA$ of the set of elements $g$ such that … for…

Let $G=SL_4(\mathbb{C})$, $K=S(GL_2 \times GL_2)$, $B \subset G$ be the standard Borel subgroup of upper triangular matrices, and let $X=G/B$ be the variety of complete flags. For…

Let $G$ be a reductive Lie group and let $\pi$ be a Harish-Chandra module representation of $G$. Denote by $\operatorname{AV}(\pi)$ its associated variety and by…

Uniform finiteness conjecture. One has

Weight-dependence conjecture. There exists a constant $C=C_G>0$ depending only on $G$ such that

Uniform finiteness conjecture. For any given weight $w$ on $G$ one has

Let $G$) be a real reductive group with complexified Lie algebra $\mathfrak g$, let $K$ be a maximal compact subgroup, and let $\mathfrak q=\mathfrak l\oplus\mathfrak u$ be a…

Suppose that $G_{\mathbb{R}}$ is split modulo center, let $H_s$ be a maximal torus with $\Delta(G,H_s)=\Delta_{\mathbb{R}}(G,H_s)$, and set … Here $\mathcal{N}_{\theta}$ is the K-n…

Let $G_{\mathbb{R}}$ be a real reductive group, let $K_{\mathbb{R}}$ be a maximal compact subgroup, and let $M$ be the Harish-Chandra module of a unipotent representation of…

Cohen–Macaulay conjecture. The $\mathbb{C}[\OO_K]$-module

Generic-character simplicity conjecture. The module $Q(0,\chi)$ is simple.

Let $\widehat{\mathfrak{g}}$ be an affine Kac–Moody Lie superalgebra obtained from a basic classical Lie superalgebra by adjoining the degree derivation. Consider the family of bou…

Let $G_{\mathbb{R}}$ be a real reductive group with maximal compact subgroup $K$, complexified Lie algebra $\mathfrak{g}$, and associated decomposition…

Let $M$ be an irreducible $(\mathfrak{g},K_\mathbb{R})$-module in a coherent family. Write its associated cycle as … where the multiplicities $m_i(M)$ extend to polynomials on…

Let $G$ be a reductive group with symmetric subgroup $K=G^\theta$, let $P$ be a $\theta$-minimal parabolic with unipotent radicals $N$ and $N^-$, and let $M_K=K\cap P$. Write…

Let $i$ be the inclusion of the relevant $K$-orbit, let $phi$ be the corresponding local system, and let $i_!\phi$ and $i_+\phi$ denote its minimal and maximal extensions as…

Let $M_\lambda$ be an irreducible Harish-Chandra module equipped with its $\mathfrak u_\mathbb{R}$-invariant hermitian form $(\,\ )_{\mathfrak u_\mathbb{R}}$ and Hodge filtration ……

Let $\widetilde K$ be the spin cover of a maximal compact subgroup $K$, and let $\mathsf{I}_{\mathrm{Dir}}(X)$ denote the Dirac index of a finite-length Harish-Chandra module $X$.…

Let $G$ be a real reductive group with the notation above, let $G_{\mathrm{ell}}$ be the regular semisimple elliptic elements, let $C^{\mathrm{ell}}$ be the set of regular semisimp…

Let $\mathcal{H}(\chi)_{\mathbb O}$ be the Harish-Chandra category associated with a nilpotent orbit $\mathbb O$, and let ${}^\chi \mathcal{H}^K_{\mathbb O}$ be the full subcategor…

Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be an abelian subcategory of $\mathcal{A}$ with exact inclusion. For objects $M,N\in\mathcal{B}$ and $d\geq 0$, the i…

Let $X$ be an irreducible Harish-Chandra $(\mathfrak{g},K)$-module, and let … be the irreducible decomposition of its associated variety. The graph…