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I am looking for a reference that contains lots of calculations for specific examples of various objects one can associate to a reductive algebraic group. For example, given a (specific) linear algebraic group $\mathbb{G}$, I would like to know things like:

-a maximal compact subgroup of $\mathbb{G}(\mathbb{R})$

-the rank of $\mathbb{G}(\mathbb{R})$

-a Borel subgroup for $\mathbb{G}$

-the Lie algebra of $\mathbb{G}$

-a Cartan involution for $\mathbb{G}$

I assume these things are well known to people who study algebraic groups. I just can't find a place where specific examples are worked out (or even stated) in any detail. Thanks!

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I think these lectures are enough good

1.Real Reductive groups II, Wallach

2.ALGORITHMS FOR COMPUTATIONS IN LOCAL SYMMETRIC SPACES, JENNIFER R. FOWLER AND ALOYSIUS G. HELMINCK*

3.Lectures on Algebraic Groups, Dipendra Prasad

4.Lectures on Algebraic Groups, Alexander Kleshchev

5.LECTURES ON SHIMURA CURVES 4.5: A CRASH COURSE ON LINEAR ALGEBRAIC GROUPS , PETE L. CLARK

6.Structure Theory of Reductive Groups through Examples, Shotaro Makisumi.

7.Reductive Groups, J.S. Milne

8.INTRODUCTION TO REDUCTIVE GROUP SCHEMES OVER RINGS, P. GILLE

9.Introduction to actions of algebraic groups, Michel Brion

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  • $\begingroup$ I had a course with Michel Brion, In my openion his lectures are the best. $\endgroup$ – user21574 May 10 '14 at 18:58

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