5 problems

Let $L_\theta(\lambda)$ be the crystal lattice, let $c(L_\theta(\lambda))$ denote its bar-conjugate lattice, and let $V_\theta(\lambda)^{\mathrm{up}}_{\mathbf A}$ be the dual latti…

Assume that no $i\in I$ satisfies $\theta(i)=i$, and use the lattice $L_\theta$, its bar-conjugate $L_\theta^-$, and the integral module $(V_\theta)_\mathbf A$ defined from the bar…

Lusztig's almost orthogonality conjecture. The global base $\{G(b)\mid b\in\mathcal B(\lambda)\}$ of $V(\lambda)$ satisfies the almost orthogonality property with respect to the ge…

Let $V_\theta(\lambda)$ be the symmetric-crystal module above, let $L_\theta(\lambda)$ be its crystal lattice, let $\overline{L_\theta(\lambda)}$ be its bar-invariant counterpart,…

Assume the crystal-basis data $L_\theta(\lambda)$ and $B_\theta(\lambda)$ for the symmetric-crystal module $V_\theta(\lambda)$, and let $L_\theta(\lambda)^-$ be the corresponding l…