27 problems

van Dijk's conjecture. Every complex symmetric pair $(G,H)$ is a unitary Gelfand pair.

Panyushev's irreducibility conjecture. If the rank of the symmetric pair $(\mathfrak g,\mathfrak g_0)$ is greater than $1$, then $\mathfrak C(\mathfrak g_1)$ is irreducible.

Let $(G,K)$ be a symmetric pair, and let the corresponding invariant eigendistributions be solutions of a holonomic $\mathcal D$-module. Sekiguchi's conjecture. In the more general…

Aizenbud–Gourevitch's conjecture. All symmetric pairs are regular. In particular, this would imply van Dijk's conjecture, since all complex symmetric pairs are stable.

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…

Irreducibility and component-count conjecture. The varieties $P_q$ in the preceding proof are irreducible, and the number of irreducible components of $\mathfrak C(\mathfrak g_1)$…

The open-embedding conjecture. The morphism $\iota_\mathfrak{c}$ is an open embedding. This conjecture concerns the relation between the regular centralizer group scheme $J$ and th…

Let $\mathfrak{so}_{2n}$ and $\mathrm{O}_m$ act through the $(\mathfrak{so}_{2n},\mathrm{O}_m)$-duality on $\mathfrak{P}_{nm}$. On the $\mathrm{O}_m$ side, consider the cyclotomic…

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…

Consider the symmetric pair $(\mathfrak{g},\mathfrak{k})$ of type AIII induced by the longest word of the Weyl group of type $A_{\text{odd}}$. Let $\mathfrak{Z}(\mathbf{w})$ be the…

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.

Let $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$ be a symmetric decomposition, and let $\tilde{\mathfrak g}=\mathfrak g_0\ltimes\mathfrak g_1$ be the associated $\mathbb Z_2$-con…

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 multiplicity-growth conjecture. This dimension is finite and grows at most polynomially in the norm of the infinitesimal character of $\tau$. The statement is presented…

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.

Regularity conjecture. Any symmetric pair is regular.

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

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

Regularity conjecture. Any symmetric pair is regular.