12 problems

Let $\mathcal O^{k,n-k}$ be the relevant parabolic category, and let $\mathcal U_i$ for $1\leq i\leq n-1$ be the projective functors defined by $\mathcal U_i=T_i\circ T^i$. Product…

Let $\theta_x^{(n)}$ be the projective functor indexed by $x\in\rm S_n$, and let $L_z^{(n)}$ be the simple highest weight module indexed by $z\in\rm S_n$ in the principal block of…

Let $\theta_x$ be an indecomposable projective functor and let $L(y)$ be a simple highest weight module in the principal block of the Bernstein–Gelfand–Gelfand category…

Let $L$ be a simple $\mathfrak{g}$-module. Let $\mathscr{A}(L)$ be the abelian category of finitely generated subquotients of modules $V\otimes_{\mathbb{C}}L$ with $V$ finite-dimen…

Let $L$ be a simple $\mathfrak{g}$-module, and let $\mathcal{X}_L$ be the class of simple modules linked to $L$ by projective functors. Define $M\triangleright N$ when $M$ occurs i…

Let $L$ be a simple $\mathfrak{g}$-module, and let $\mathbf{Y}^L$ be the birepresentation obtained from the projective-functor action on the subcategories generated by the translat…

Let $S_n$ be the symmetric group, let $x,y\in S_n$, let $L_y$ be the corresponding simple highest weight module, and let $\theta_x$ be the projective functor associated with $x$. I…

Let $\theta_x$ and $\theta_y$ be projective functors acting on the principal block $4\mathcal{O}_04$ of the BGG category for $4\mathfrak{g}4$, and let $L_y$ be the simple module in…

Let $W$ be the Weyl group with longest element $w_0$, let $\theta_x$ be the projective functor indexed by $x\in W$, and let $\Delta_z$ and $\nabla_z$ denote the standard and costan…

Let $\mathfrak{g}$ be a semisimple complex Lie algebra of type $A$, let $W$ be its Weyl group, let $L_y$ be the simple object of the principal block $\mathcal{O}_0$ indexed by…

Indecomposability conjecture. The module $theta L$ is either indecomposable or zero.

Steinberg's conjecture. The preceding assertion should hold for arbitrary parabolic subalgebras and, more generally, in the general case beyond the maximal-parabolic type-$A$ setti…