16 problems

Let $(\mathfrak g,\mathfrak k)$ be a symmetric pair listed in Table 1 in Section. Let the relevant Nakajima varieties carry an involution $\sigma$, and let the associated Steinberg…

Let $(\mathfrak{g},\mathfrak{g}')$ be a reductive symmetric pair, let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra of $\mathfrak{g}_{\mathbb{C}}$, and write…

Assume an orbit embedding … and Steinberg maps … where $\iota^{\pm\theta}:\mathcal{N}_{\mathfrak{g}}^{\pm\theta}/K\to\mathcal{N}_{\mathfrak{G}}^{\pm\theta}/\mathbb{K}$ is the natur…

Let $\mathfrak{X}$ be a double flag variety of finite type, with projection … Let $\mathcal{N}_{\mathfrak{s}}$ denote the nilpotent variety in $\mathfrak{s}$. Nil-projection conjec…

Let $\Gamma$ be a Dynkin diagram, let $\theta$ be the involution determined by $w_0(\alpha_i)=-\alpha_{\theta(i)}$, and let $\mathfrak g_{\Gamma}$ be the associated simple Lie alge…

Let $\mathfrak{u}$ be a compact semisimple Lie algebra with $*$-compatible Chevalley generators $\{e_r,f_r,h_r\mid r\in I\}$ of $\mathfrak{g}=\mathfrak{u}^{\mathbb{C}}$. Let…

Fourier-transform Springer correspondence. If $\lambda$ has at least one odd part, then $\mathfrak F$ induces a bijection

Converse stability conjecture. If $(G,H,\theta)$ is p-stable, then $(G,H,\theta)$ is stable.

Januszewski's conjecture. In the setting of Kobayashi's conjecture, $m_{\tau}(\pi|_{G'})$ grows at most polynomially in the norm of the infinitesimal character of $\tau$, in the se…

Kobayashi's conjecture. The multiplicity $m_{\tau}(\pi|_{G'})$ is finite for every $\tau\in\hat{G}'$. This is a finite-multiplicity assertion for discretely decomposable restrictio…

Virtually symmetric-type bounded multiplicity conjecture. If $\mathfrak{q}$ is of virtually symmetric type, then the restriction

GK-pair conjecture for good pairs. Every good symmetric pair is a GK-pair.

GK-pair conjecture. Any connected symmetric pair over $\mathbb{C}$ is a GK-pair.

Good-pair GK conjecture. Every good symmetric pair is a GK pair.

van Dijk's GK conjecture. If $F=\mathbb{C}$, any connected symmetric pair is a GK pair.

van Dijk's conjecture. Any connected symmetric pair $(G,H)$ over $\mathbb{C}$ such that $G/H$ is connected is a Gelfand pair.