arXiv subjects

Vladyslav Zveryk

Publications and source records attributed to Vladyslav Zveryk.

5 recordsLinked to original sources

arXiv2026

Virasoro extensions of generalized Kac-Moody algebras

Borcherds-Kac-Moody (BKM) Lie algebras are constructed by combining a number of $\mathfrak{sl}_2$-s and $3$-dimensional Heisenberg algebras in a certain way, generalizing the celebrated Kac-Moody algebras. Such algebras have a wide range of applications, including the examples of the monster and fake monster Lie algebras being BKM algebras. Among their main features are existence of an invariant bilinear form, denominator identity and Weyl-Kac character formulas for irreducible representations. We introduce a construction of closely related algebras that we call Virasoro-Borcherds-Kac-Moody (VBKM) algebras: if it happens that certain $3$-dimensional Heisenberg algebras inside a BKM algebra form the infinite-dimensional Heisenberg algebra, we replace it by the Virasoro algebra in the construction. We construct a family of Lie algebras over $\mathbb{P}^1$, where the generic fiber is a VBKM algebra and a special fiber is a BKM algebra. This allows us to visualize VBKM algebras as deformations of BKM algebras and prove structural properties about VBKM algebras similar to those of BKM algebras, despite the absence of an invariant bilinear form. Those include denominator identity and character formulas of irreducible representations. We give examples of realizations of such $\mathbb{P}^1$-families of VBKM algebras in lattice vertex operator algebras. In particular, we include the fake monster Lie algebra into such a family.

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arXiv2026

Eigenforms and graphs of Hecke operators with wild ramification

Hecke operators on moduli of bundles over a global function field become substantially more complicated in the presence of ramification. We show that far enough in the Harder-Narasimhan cone of $\mathrm{Bun}_G$, this extra complexity has a simple structure, which allows to reduce most of the study to the unramified case. Using the theory of graphs of Hecke operators, we transform this statement into a combinatorial condition. Utilizing the combinatorial language, we obtain tight bounds, and for generic eigenvalues exact formulas for the dimensions of Hecke eigenspaces with arbitrary ramification for $\mathrm{Bun}_{\mathrm{PGL}_2}$. We compare these formulas to the known results in the theory of Eisenstein series. Moreover, our methods allow to construct eigenforms explicitly.

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arXiv2026

Graphs of Hecke operators in mixed ramification

We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a choice of a divisor and subgroups of $G$ at every point of the divisor. Building on our previous work, we prove that, under mild regularity conditions, the action of a Hecke operator in the deep cusp of $\mathrm{Bun}_G$ in a highly complex ramification mimics an action in a much simpler ramification. This reduces the study to a smaller number of cases which, in particular, involve divisors supported at no more than two points. We demonstrate our methods by computing various examples for $G=\mathrm{PGL}_2$ and computing the dimensions of spaces of Hecke eigenforms for generic eigenvalues. We connect the obtained dimension formulas with the known results from the theory of Eisenstein series.

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arXiv2025

Stabilization of Kac polynomials along root strings

We study the stabilization behavior of cohomology groups associated with moduli spaces of quiver representations for a fixed quiver $Q$. Under mild conditions on a dimension vector $δ$, we show that the dimensions of these cohomology groups stabilize when a sufficiently large multiple of $δ$ is added. We derive explicit formulas for the stabilized dimensions and, in particular, obtain stabilization of the coefficients of Kac polynomials.

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arXiv2025

Dynkin automorphism actions on Gaudin algebras

We study the action of Dynkin diagram automorphisms $σ$ on generalized Gaudin algebras, focusing in particular on the big Gaudin algebra $\mathcal{B}(\mathfrak{g}) \subset (U(\mathfrak{g}) \otimes S(\mathfrak{g}))^{\mathfrak{g}}$ and its evaluated versions $\mathcal{B}^λ(\mathfrak{g})$ and $\mathcal{B}_χ(\mathfrak{g})$. We show isomorphisms between the coinvariants of the generalized Gaudin algebras associated with $\mathfrak{g}^\vee$ and the generalized Gaudin algebras associated with $\mathfrak{g}_σ^\vee$, where $\mathfrak{g}_σ$ is the fixed point subalgebra. In particular, we get an isomorphism $\mathcal{B}^λ(\mathfrak{g}^\vee)_σ\simeq \mathcal{B}^λ(\mathfrak{g}^\vee_σ)$ for any $σ$-invariant dominant weight $λ$, which allows us to reprove Jantzen's twining formula. Our approach relies on interpreting generalized Gaudin algebras via spaces of opers, which explains the appearance of the Langlands duals in our results and in Jantzen's twining formula.

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