8 problems

Let $G$ be a reductive group with involution $\theta$, let $K^*$ act on the symmetric variety $P=\{g^{-1}\theta(g)\mid g\in G\}$ by conjugation, and let … be the quotient morphism,…

Let $\sigma:G\to G$ be an algebraic involution, let $H=G^\sigma$, and let $X=G/H$ be a symmetric space over a field of characteristic different from $2$. Assume that $M=T^*X$ is hy…

Let $b_{m,n}$ denote the number of Borel orbits of the symmetric variety $SO_{2m+2n+1}/S(O_{2m} \times O_{2n+1})$. For fixed $n$, this number is a polynomial in $m$. Can–Uu{g}urlu'…

Let $K\subset G$ be the symmetric subgroup of a real form $G_{\mathbb R}\subset G$. Write $X({\mathcal K})$ for the corresponding symmetric variety, $G({\mathcal O})$ for the posit…

Let $(\operatorname{G},\mathrm{H})$ be a symmetric pair over $k$ such that $\operatorname{G}$ is quasi-split over $k$ and $\operatorname{G}_{\mathrm{der}}$ is simply connected. Set…

Let $\X$ be a symmetric variety and let the construction of an endoscopic variety associated to $\X$ be as in the paper. Endoscopic-variety conjecture. This construction should pla…

Pattern-avoidance equivalence conjecture. In types $AI$ and $AII$, an involution $\pi$ avoids all bad patterns in the sense of this section if and only if it avoids all bad pattern…

Let $p$ and $q$ be nonnegative integers, and let $b_{p,q}$ denote the number of Borel orbits in the corresponding symmetric variety of type $BI$. For each fixed $q$, regard…