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?

simply connectedsemisimple group $G$ with a given semisimple Lie algebra $\mathfrak{g}$ over a field $k$ of char. 0. Section 3 in Hochschild's 1970 paper in vol. 34 of the Illinois Math Journal provides a commutative Hopf algebra structure on the direct limit of the duals $(U(\mathfrak{g})/J)^{\ast}$ for 2-sided ideals $J$ of finite codimension in the universal enveloping algebra, and sections 2 and 5 of his 1959 paper in the same journal show that this limit is $R$ (e.g., it really is a domain finitely generated over $k$). – user22479 Sep 1 '12 at 6:02