arXiv subjects
Publications and source records attributed to Jiuzu Hong.
Given a simply-connected simple algebraic group $G$, we determine the tangent space of any Finkelberg-Mirkovi\'c Schubert scheme at the base point of the affine Grassmannian of $G$. As a consequence, we exhibit non-reduced Finkelberg-Mirkovi\'c Schubert schemes when $G$ is of type $E_6,E_7$ and $E_8$.
In this paper we investigate line bundles on $\mathrm{Bun}_{\mathcal{G}}$ the moduli stack of parahoric Bruhat--Tits bundles over a smooth projective curve. Translating this problem into one concerning twisted conformal blocks, we are able to establish criteria that detect when line bundles on an appropriate flag variety descend to $\mathrm{Bun}_{\mathcal{G}}$. Along the way we establish a conjecture of Pappas and Rapoport which describes sections of line bundles on $\mathrm{Bun}_{\mathcal{G}}$ using representation-theoretical means. We conclude the paper with examples where our methods allow us to explicitly determine the Picard group of $\mathrm{Bun}_{\mathcal{G}}$.
Let $\mathscr{G}$ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb{C}$, excluding the absolutely special case of $A_{2\ell}^{(2)}$. Using the methods and results of Zhu, we prove a duality theorem for general $\mathscr{G}$ : there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for $\mathscr{G}$. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of $\mathscr{G}$. In particular, this confirms a conjecture of Haines and Richarz.
The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a certain union of Schubert varieties in a partial affine flag variety. In this paper, we enhance this equality of dimensions to an isomorphism of representations, which leads to interesting consequences in the setting of affine Demazure modules. Zhu's proof of coherence conjcture and our comparison theorem are established by introducing a parahoric Bruhat-Tits group scheme $\mathcal{G}$ over the affine line that is ramified at $0$. We further strengthen this comparison by equipping any line bundle on the global affine Grassmannian of $\mathcal{G}$ with a unique equivariant structure under the global jet group scheme of $\mathcal{G}$.
Let $\mathcal{G}$ be a parahoric Bruhat-Tits group schemes arising from a $Γ$-curve $C$ and a certain $Γ$-action on a simple algebraic group $G$ for some finite cyclic group $Γ$. We prove the flatness of Beilinson-Drinfeld Schubert varieties of $\mathcal{G}$, we determine the rigidified Picard group of the Beilinson-Drinfeld Grassmannian ${\rm Gr}_{\mathcal{G},C^n}$ of $\mathcal{G}$, and we establish the factorizable and equivariant structures on rigidified line bundles on ${\rm Gr}_{\mathcal{G},C^n}$. We develop an algebraic theory of global Demazure modules of twisted current algebras, and using our geometric results we prove that when $C = \mathbb{A}^1$, the spaces of global sections of line bundles on BD Schubert varieties of $\mathcal{G}$ are dual to the twisted global Demazure modules. This generalizes the work of Dumanski-Feigin-Finkelberg in the untwisted setting,
We study the spaces of twisted conformal blocks attached to a $Γ$-curve $Σ$ with marked $Γ$-orbits and an action of $Γ$ on a simple Lie algebra $\mathfrak{g}$, where $Γ$ is a finite group. We prove that if $Γ$ stabilizes a Borel subalgebra of $\mathfrak{g}$, then Propagation Theorem and Factorization Theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $Γ$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathscr{G}$ be the parahoric Bruhat-Tits group scheme on the quotient curve $Σ/Γ$ obtained via the $Γ$-invariance of Weil restriction associated to $Σ$ and the simply-connected simple algebraic group $G$ with Lie algebra $\mathfrak{g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathscr{G}$-torsors on $Σ/Γ$ when the level $c$ is divisible by $|Γ|$ (establishing a conjecture due to Pappas-Rapoport).
Let $G$ be a simple algebraic group over an algebraically closed field $k$. Let $Γ$ be a finite group acting on $G$. We classify and compute the local types of $(Γ, G)$-bundles on a smooth projective $Γ$-curve in terms of the first non-abelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in $G$. When $\text{char}(k)=0$, we prove that any generically simply-connected parahoric Bruhat--Tits group scheme can arise from a $(Γ,G_{\text{ad}})$-bundle. We also prove a local version of this theorem, i.e. parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.
The explicit Verlinde formula for the dimension of conformal blocks, attached to a marked projective curve $Σ$, a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and integrable highest weight modules of a fixed central charge of the corresponding affine Lie algebra $\hat{L}(\mathfrak{g})$ attached to the marked points, requires (among several other important ingredients) a Lie algebra cohomology vanishing result due to C. Teleman for the positive part $\hat{L}^+(\mathfrak{g})$ with coefficients in the tensor product of an integrable highest weight module with copies of finite dimensional evaluation modules. The aim of this paper is to extend this result of Teleman to a twisted setting where $\mathfrak{g}$ is endowed with a special automorphism $σ$ and the curve $Σ$ is endowed with the action of $σ$. In this general setting, the affine Lie algebra gets replaced by twisted affine Lie algebras. The crucial ingredient (as in Teleman) is to prove a certain Nakano Identity.
Let $Γ$ be a finite group acting on a simple Lie algebra $\mathfrak{g}$ and acting on a $s$-pointed projective curve $(Σ, \vec{p}=\{p_1, \dots, p_s\})$ faithfully (for $s\geq 1$). Also, let an integrable highest weight module $\mathscr{H}_c(λ_i)$ of an appropriate twisted affine Lie algebra determined by the ramification at $p_i$ with a fixed central charge $c$ is attached to each $p_i$. We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of $Γ$ acting on $\mathfrak{g}$ by diagram automorphisms and acting on a quotient of $Σ$. Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when $Γ$ acts on $\mathfrak{g}$ by diagram automorphisms and covers of $\mathbb{P}^1$ with 3 marked points. Assuming a twisted analogue of Teleman's vanishing theorem of Lie algebra homology, we derive an analogue of the Kac-Walton formula and the Verlinde formula for general $Γ$-curves (with mild restrictions on ramification types). In particular, if the Lie algebra $\mathfrak{g}$ is not of type $D_4$, there are no restrictions on ramification types.
We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar-Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces.
Let $\overline{\mathtt{X}}_λ$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_λ$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $λ$ is a coweight of $G$. We find a simple algorithm which describes the set $Ψ(λ)$ of all $\mathtt{I}$-orbits in $\overline{\mathtt{X}}_λ$ in terms of coweights. We introduce $R$-operators (associated to positive roots) on the coweight lattice of $G$, which exactly describe the closure relation of $\mathtt{I}$-orbits. These operators satisfy Braid relations generically on the coweight lattice. We also establish a duality between the set $Ψ(λ)$ and the weight system of the level one affine Demazure module $\hat{\mathscr{D}}_λ$ of $^L\tilde{\mathfrak{g}}$ indexed by $λ$, where $^L\tilde{\mathfrak{g}}$ is the affine Kac-Moody algebra dual to the affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}$ associated to the Lie algebra $\mathfrak{g}$ of $G$.
The Verlinde formula computes the dimension of conformal blocks associated to simple Lie algebras and stable pointed curves. If a simply-laced simple Lie algebra admits a nontrivial diagram automorphism, then this automorphism acts on the space of conformal blocks naturally. We prove an analogue of Verlinde formula for the trace of the diagram automorphism on the space of conformal blocks. Along the way, we get an analogue of Kac-Walton formula for the trace of the diagram automorphism. We also get a twining type formula between the conformal blocks for the pair $(sl(2n+1),sp(2n))$.
We investigate the relationship between the poles of Igusa zeta integrals and the unextendability of semi-invariant distributions. Under some algebraic conditions, we obtain an upper bound for the order of the poles of Igusa zeta integral, and by using the order of the poles we give a criterion on the unextendability of semi-invariant distributions. A key ingredient of our method is the idea of generalized semi-invariant distributions.
In this note we describe a general elementary procedure to attach a fusion ring to any Kac-Moody algebra of affine type. In the case of untwisted affine algebras, they are usual fusion rings in the literature. In the case of twisted affine algebras, they are exactly the twisted fusion rings defined by the author in [Ho2] via tracing out diagram automorphisms on conformal blocks for appropriate simply-laced Lie algebras. We also relate the fusion ring to the modular S-matrix for any Kac-Moody algebra of affine type.
We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this category (duality, projective generators, braiding etc.) in analogy with classical strict polynomial functors. We then apply the work of Hashimoto and Hayashi in this context to construct quantum Schur/Weyl functors, and use this to provide new and easy derivations of quantum $(GL_m,GL_n)$ duality, along with other results in quantum invariant theory.
In this paper we investigate some methods on calculating the spaces of generalized semi-invariant distributions on p-adic spaces. Using homological methods, we give a criterion of automatic extension of (generalized) semi-invariant distributions. Based on the meromorphic continuations of Igusa zeta integrals, we give another criteria with purely algebraic geometric conditions, on the extension of generalized semi-invariant distributions.
Let G be a connected almost simple algebraic group with a Dynkin automorphism σ. Let G_σ be the connected almost simple algebraic group associated to G and σ. We prove that the dimension of the tensor invariant space of G_σ is equal to the trace of σ on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does Gσ. As a consequence, we show that the spin group Spin(2n + 1) is of saturation property with factor 2, which strengthens the results of Belkale-Kumar and Sam in the case of type B_n.
We make an observation which doubles the exponent of distribution in certain Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and Kontorovich-Oh. As a consequence, we decrease the known bounds on the saturation numbers in these problems.