29 problems

Let $B^{\vee,+}$ and $B^{\vee,-}$ be opposite Borel subgroups of $G^{\vee}$ with $B^{\vee,+}\cap B^{\vee,-}=T^{\vee}$. Set $\mathcal{W}=B^{\vee,-}B^{\vee,+}\subset G^{\vee}$, let…

Let $G$ be the reductive group underlying the affine Grassmannian, let $L^{\heartsuit}{\mathfrak g}$ be the space of elements under consideration, and for…

A generalized affine Springer fiber is the variety associated with a reductive group, a finite-dimensional representation, and an element of the corresponding loop-space representa…

Let $0\leq d_1\leq\cdots\leq d_1$, let $\beta=\operatorname{JM}_n^{d_1}\ldots\operatorname{JM}_2^{d_{n-1}}$, and set $J=J(d_1,\ldots,d_n)$, where … The notation…

Let $\mathbb{G}$ be a quasi-split generic group, and let the local systems, perverse filtrations, rational DAHA actions, and braid-group actions be defined as in the split case. Ob…

Let … where $z_i\in\mathbb{C}^*$ are pairwise distinct and $d_i\geq0$. Equivariant formality conjecture. The affine Springer fiber $\mathrm{Sp}_{\gamma}$ is equivariantly formal fo…

Let $G$ be a reductive group, let $\mathcal A_G=\bigoplus_{d=0}^{\infty}{}_0\mathcal A_d$ be the graded Coulomb branch algebra, and for $\gamma\in\mathfrak g$ define … Let…

Let $\gamma$ be an unramified element with associated link $L_{\gamma}$, let $\mathrm{Sp}_{\gamma}$ be its affine Springer fiber, and let $\mathrm{gr}_P$ be the specified perverse…

Let $P\subset G$ be a parabolic subgroup, let $\widetilde{N}=T^*(G/P)$ be the corresponding Springer resolution, and let $T\widetilde{N}$ be its tangent bundle. The dilation action…

Let $G=GL_n$, let $\gamma$ be the regular element used to define the affine Springer fiber $\Fl^{\gamma}_{\bI}$, let $\Lambda$ be the lattice acting on it, and let $P$ be the perve…

BBASV's conjecture. The embedding above is an isomorphism.

Center–cohomology isomorphism conjecture. The algebra embedding

Let $cmathfrak g=\operatorname{Lie}(GL_n)$ or $cmathfrak g=\operatorname{Lie}(SL_n)$, and let $gammabin g(cO)$ be elliptic, regular semisimple, and topologically nilpotent. Let…

For a regular semisimple $\gamma$, define the graded module … It is a module over the graded algebra $\oplus_{d=0}^{cinfty}{}_0\mathcal A_d$ and therefore defines a quasi-coherent sh…

For $G=GL_n$, let $gammabin g derbrace{llbracket trrbracket}{\mathcal O}$ be regular semisimple, let $cbeta=\beta(\gamma)$ be the associated braid, and let…

The construction associates to each $gammabin g(cK)$ a {cC}^-equivariant quasi-coherent sheaf $cFgamma$ on $cmathfrak C{ccheck G}$. Coherence conjecture. For any regular semi…

Let $C$ be an irreducible plane curve singularity, let $L$ be the corresponding algebraic knot, and let $\beta$ be the corresponding braid on $n$ strands. Let $X(\beta\Delta;w_0)$…

Let $\mathcal{P}$ be the Procesi bundle on $\operatorname{Hilb}^n(\mathbb{C}^2)$, let $\mathcal{P}^*$ denote its dual, and let $\mathcal{O}(d)$ be the indicated determinant-line-bu…

Let $C$ be the unique compactification with rational components and no other singularities of the curve $\{x^n=y^{dn}\}$. Let … The algebra $A_n$ is the algebra acting on this homo…

Let $\textup{Sp}_{q/p}$ be the affine-flag Springer fiber, and let $\epsilon,\xi_1,\ldots,\xi_p,e_2,\ldots,e_p$ be the equivariant classes and parameters defined in the source. Let…

Moment-graph formula conjecture. One has

Weyl-group component bound conjecture. For any regular semisimple $\gamma\in\mathfrak{g}$, the number of $\mathcal{T}$-orbits of components of…

Orbital-integral boundedness conjecture. For any $\gamma\in\mathfrak{g}$,

Orbital integrals for classical groups are expected to be related only to hyperelliptic curves. The precise formulation below concerns the cohomology of Hessenberg varieties. Hyper…

Let $F$ be the Laurent series field with valuation $v$, let $(G,G_0)$ be the symmetric pair considered in the paper, and let … be regular semisimple. Regard $XY$ as an element of…