Fusion ring revisited

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.Comment: To appear in Contemporary Mathematic

Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits

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.Comment: 24 page

Conformal blocks for Galois covers of algebraic curves

We study the spaces of twisted conformal blocks attached to a $\Gamma$-curve $\Sigma$ with marked $\Gamma$-orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak{g}$, where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak{g}$, then Propagation Theorem and Factorization Theorem hold. We endow a projectively flat connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$-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 $\Sigma/\Gamma$ obtained via the $\Gamma$-invariance of Weil restriction associated to $\Sigma$ 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 $\Sigma/\Gamma$, when the level $c$ is divisible by $|\Gamma|$ (establishing a conjecture due to Pappas-Rapoport).Comment: 72 pages; This paper supersedes the original version. This is a much larger version with many more results. In particular, we confirm a conjecture by Pappas-Rapoport for the parahoric Bruhat-Tits group schemes considered in our pape

Mirkovic-Vilonen cycles and polytopes for a Symmetric pair

Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $\sigma$ be a Dynkin automorphism on $G$. In this paper, we get a bijection between the set of \st-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for G^\st, which is the fixed point subgroup of $G$; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula.Comment: 12 pages; This is a shortened versio

Quantum Polynomial Functors

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.Comment: 34 pages, final version to appear in Journal of Algebr