# Quasi-Invariants in Characteristic $p$ and Twisted Quasi-Invariants

## TL;DR

This paper explores the properties of quasi-invariant polynomials in positive characteristic fields, proposing conjectures on their Hilbert series and extending results to twisted spaces by smooth functions.

## Contribution

It introduces partial results and conjectures on the Hilbert series of quasi-invariant spaces over fields of positive characteristic and extends twisted quasi-invariant spaces to include smooth functions.

## Key findings

- Partial results on Hilbert series in positive characteristic
- Two conjectures on Hilbert series of quasi-invariant spaces
- Extension of twisted quasi-invariant spaces to smooth functions

## Abstract

The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.