arXiv subjects

Nikolay Grantcharov

Publications and source records attributed to Nikolay Grantcharov.

6 recordsLinked to original sources

arXiv2026

Cohomology on Cotangent Bundles of Partial Flag Varieties in Type A

Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $λ$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(λ)^*$. We express its cohomology as a direct limit of the cohomology of certain line bundles on a Bott--Samelson variety associated to an affine Kac-Moody group. This comparison yields vanishing of higher cohomology and shows that the global sections are generated over $\mathbf C[\mathfrak g^*]$ by their degree-zero part $V_G(λ)^*$. Using these results together with the Braverman--Kazhdan intertwiners constructed in earlier joint work with A. Slipper, we give an explicit generating set for $\mathbf C[T^*(\mathrm{SL}_n/[P,P])]$. Finally, in an appendix joint with Tom Gannon, we combine these results to show the affinization $\mathrm{Spec}(\mathbf{C}[T^*(SL_n/[P,P])])$ has terminal singularities.

math.AG↗

arXiv2026

Deformation theory of the Double Affine Hecke algebra of type $(C_n^\vee,C_n)$

We study the double affine Hecke algebra (DAHA) of type $(C_n^\vee,C_n)$ from the perspective of deformation theory. First, we provide a zeros-and-residues realization of this algebra, extending the construction of Ginzburg, Kapranov, and Vasserot to the non-reduced affine root system setting. Specializing the parameters of the DAHA to the base point gives the crossed product of a quantum torus algebra with the finite Weyl group of type $C_n$. We then show that for all $n$, the completed DAHA is the formal universal deformation of this crossed product algebra, extending Oblomkov's result for $n=1$. Our proof explicitly identifies the completed DAHA with the undeformed crossed product algebra equipped with a formal star product.

math.QA↗

arXiv2026

Quasi-Classical Braverman--Kazhdan Intertwiners via Quiver Varieties

We show that Braverman--Kazhdan normalized intertwiners for $SL_n(\mathbf{C})$ have a quasi-classical incarnation governed by type $A$ quiver varieties. More precisely, for standard parabolic subgroups $P$ and $P'$ with conjugate Levi subgroups, we construct $SL_n\times L^{\mathrm{ab}}$-equivariant isomorphisms $Φ(P,P'):\overline{T^*(SL_n/[P,P])}^{\mathrm{aff}}\rightarrow\overline{T^*(SL_n/[P',P'])}^{\mathrm{aff}}$ between the affinizations of the cotangent bundles of the corresponding Braverman--Kazhdan spaces, and we prove that these isomorphisms satisfy Coxeter relations. The construction uses $SL$-gauge analogues of Lusztig--Maffei--Nakajima reflection functors, thereby extending Wang's quiver-variety realization of the quasi-classical Gelfand--Graev action from the Borel case to arbitrary parabolic subgroups. In this way, we complete the quasi-classical Braverman--Kazhdan intertwiner story for $SL_n(\mathbf{C})$ and obtain a systematic source of non-isomorphic varieties whose affinized cotangent bundles are isomorphic.

math.RT↗

arXiv2024

Infinitesimal jet spaces of $\text{Bun}_G$ in positive characteristic

Given a semisimple reductive group $G$ and a smooth projective curve $X$ over an algebraically closed field $k$ of arbitrary characteristic, let $\text{Bun}_G$ denote the moduli space of principal $G$-bundles over $X$. For a bundle $P\in\text{Bun}_G$ without infinitesimal symmetries, we provide a description of all divided-power infinitesimal jet spaces, $J_P^{n,PD}(\text{Bun}_G)$, of $\text{Bun}_G$ at $P$. The description is in terms of differential forms on $X^n$ with logarithmic singularities along the diagonals and with coefficients in $(\mathfrak{g}_P^*)^{\boxtimes n}$. Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification of the configuration space, $\hat{X}^n$, is an isomorphism. Thus, we relate the two constructions of Beilinson-Drinfeld and Beilinson-Ginzburg, and as a consequence, give a connection between divided-power infinitesimal jet spaces of $\text{Bun}_G$ and the $\mathcal{L}ie$ operad.

math.AG↗

arXiv2020

Extension Quiver for Lie Superalgebra $\mathfrak{q}(3)$

We describe all blocks of the category of finite-dimensional $\mathfrak{q}(3)$-supermodules by providing their extension quivers. We also obtain two general results about the representation of $\mathfrak{q}(n)$: we show that the Ext quiver of the standard block of $\mathfrak{q}(n)$ is obtained from the principal block of $\mathfrak{q}(n-1)$ by identifying certain vertices of the quiver and prove a ''virtual'' BGG-reciprocity for $\mathfrak{q}(n)$. The latter result is used to compute the radical filtrations of $\mathfrak{q}(3)$ projective covers.

math.RT↗

arXiv2020

On BBW parabolics for simple classical Lie superalgebras

In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.

math.RT↗