How to get Haar measure on a compact Lie group, given the complexification?

Question

This is the first in what may be a series of questions on the theme "a Banach algebraist/Bear Of Little Brain needs help with algebraic geometry". $\newcommand{\Cplx}{{\mathbb C}}\newcommand{\fg}{{\mathfrak g}}$

Let $\fg$ be a complex semisimple Lie algebra. I have two objects on my mind:

-- the algebraic group over $\Cplx$ obtained by exponentiating $\fg$; and

-- the compact Lie group obtained by exponentiating the compact real form of $\fg$. Wikipedia

To be honest, what I'm really interested in the matrix coefficient ring of the latter, but according to what I've swotted up on, this is naturally isomorphic in a fairly precise sense to the coordinate ring of the former, in the sense of affine algebraic varieties. Let me denote this ring by $R$.

Anyway, if I think of $R$ as an algebra of smooth functions on the compact group, then there is a pairing $R\times R \to \Cplx$ given by $(f,g)\mapsto \int f(p)g(p)\,dp$ where $dp$ denotes a fixed choice of Haar measure on the compact group. My somewhat vague and naive question is:

Q1. What is the right way to "see" this pairing when I think of $R$ as the coordinate ring of an affine algebraic variety over $\Cplx$?

Here, by "see", I mean that I want some way of getting it in principle from the algebro-geometric object $R$, which is not just a concatenation of big existence results.

This may or may not require an answer to another simple-minded question:

Q2. What is the natural/canonical way of "seeing" the compact group, up to suitable isomorphism, if we are just given $R$ + Hopf algebra structure?

Answer 1

I'll write $K$ for the compact group. A correction to start with: You write $\int_K f g$, but I assume you mean $\int_K f \bar{g}$.

Complex conjugation We need to identify the map $\sigma: R \to R$ which has the property that $\sigma(f)|_K = \overline{\sigma(f)}$. Note that this map will NOT be complex conjugation of functions on $G$; all of the elements of $R$ are holomorphic as functions on $G$. For example, if $K=U(n)$ and $G=GL_n$, then $\sigma(f)(g) = \overline{f}(\overline{g}^{-T})$.

We cannot define $\sigma$ completely canonically, because choosing a different $K$ will change $\sigma$. However, if you have a subalgebra $\mathfrak{k}$ of $\mathfrak{g}$ which you want to be the algebra of $K$, then I think that $\sigma$ is the unique $\mathbb{C}$-antilinear Hopf automorphism (commutes with both multiplication and co-multiplication) whose induced action on $\mathfrak{g}$ fixes $\mathfrak{k}$. And, of course, an $\mathbb{R}$-subalgera $L$ of $\mathfrak{g}$ is a possible candidate for $L$ iff $\mathfrak{g} = \mathfrak{k} \otimes \mathbb{C}$ and the Killing form is positive definite (or maybe its negative definite) on $L$.

Integration Integration is a special case of the Reynolds operator. In general, if $G$ is a reductive group, and $V$ is an algebraic representation of $G$, than $\rho: V \to V$ is the unique $G$-equivariant idempotent whose image is the $G$-invariants. I'm not sure of a really clean way to define it. One fairly algebraic way is the following: Let $v \in V$, and let $U \subset V$ be the sub-rep spanned by the $G$-orbit of $v$. Note that $U$ is finite dimensional. Let $\Omega$ denote the Casimir; let the characteristic polynomial of $\Omega$ acting on $U$ be $x^k g(x)$ with $g(0) \neq 0$; then $\rho|_U = g(\Omega)/g(0)$. Conceptually, $\rho = \lim_{t \to \infty} e^{- t \Omega}$ (the heat equation) but that's not algebraic.

In your case, you want the Reynolds operator for $G$ acting on $R$. Since I think it is easiest to define $\rho$ from the Cassimir, you need to know how to algebraically write down the action of $\mathfrak{g}$ on $R$. I think it is as follows: Let $\epsilon: R \to \mathbb{C}$ be evaluation at the identity (normally called counit). Let $\mathfrak{m}$ be the kernel of $\epsilon$, so $\mathfrak{m}/\mathfrak{m}^2 = \mathfrak{g}^{\vee}$. Then $\mathfrak{g}$ maps $R \to \mathbb{C}$ by $g \cdot f = \langle g, f-\epsilon(f)+\mathfrak{m}^2 \rangle$. This is the standard description of how to see elements of the Zariski tangent space as derivations.

We want to extend from a single vector to an entire left invariant vector field. I think the recipe is $R \stackrel{\Delta}{\longrightarrow} R \otimes R \stackrel{g \cdot \otimes 1}{\longrightarrow} \mathbb{C} \otimes R \cong R$, where $\Delta$ is comultiplication.

Hope that helps!