Category Archives: representation-theory

A positive characteristic theory for polar representations?

Let $G$ be a split reductive algebraic group over a field $k$ of characteristic zero and $\mathfrak{g}$ it's Lie algebra. If $T\subset G$ is a maximal torus with Lie algebra $\mathfrak{t}$ and Weyl group $W$, then there is a well-known isomorphism of algebras $$k[\gfr]^G\xrightarrow{\sim}k[\mathfrak{t}]^W.$$ This is called the Chevalley restriction theorem. There are many ways […]

Automorphic representations: a short list of books

This is a short list of books to get you started on learning automorphic representations. Before I talk about them, I will first define automorphic representation, which will take a few paragraphs. To start, we need an affine algebraic $F$-group scheme $G$ where $F$ is a number field or function field. We let $\A_F$ be […]

Booker's Extension of the Selberg Class

Briefly, the Selberg class is a set of functions $F:\C\to\C$ such that $f(s)$ can be written as a Dirichlet series for $\Re(s) > 1$ and that satisfies a form of analytic continuation, a functional equation, a Ramanujan hypothesis bound on coefficients of the Dirichlet series, and an Euler product formula. Andrew Booker in [1] has […]