I need help on this one: 


In Chriss & Ginzburg book on representation theory and complex geometry I came across the following statement: 


maximal compact (in analytic topology) subgroup G\_comp of reductive group G is dense in Zariski topology of G.


If this is true, what about the case of S^1 inside C\*. Since it is closed
it doesn't look dense to me.


Answers appreciated!



Maximal compact subroup is dense in Zariski?
When talking about Zariski-topology it is important to specify which field you are working over! Over $\mathbb{C}$ the cirkel is not closed in the Zariski topology on the *one-dimensional* reductive complex algebraic group $\mathbb{C}^\*$, because it is not the zero set of a polynomial in one variable over $\mathbb{C}$. However, if we view $\mathbb{C}^\*$ as a *two-dimenional* Lie group over $\mathbb{R}$ (i.e. isomorphic to the cylinder $S^1 \times \mathbb{R}\_{>0}$) then the circle subgroup is Zariski-closed and your objection applies. So perhaps check if the book makes any assumptions about the Lie groups in question being reductive algebraic groups over $\mathbb{C}$.


