# Reference for Unitary Group attached to $E/k$

Unitary groups are very important objects in the setting of Langland's Conjecture because of the existence of Shimura Variety ( which I don't know) and also because people know how to attach a galois representation to a automorphic representation of unitary groups in almost all cases (By the work of Taylor, Harris and many others)(Which also I don't know).

I was trying to learn about unitary groups attached to $(k,E,D,*)$ where say $k$ is a totally real field,$E$ CM field of deg 2 over $k$, $D$ is a central simple algebra of rank $n^2$ over E, and $\*$ $k$ algebra anti-involution of 2nd kind on $D$ (i.e. coinciding with the action of non-trivial element of $Gal(E/k)$ on $E$).

But I could not really find a reference for this. Essentially some authors define it a inner form of a particular quasi-split unitary group, and some authors define it as functor of points. Also it is commented that there is some sort of Global-Local patching going on. Can any one give me a reference where unitary groups is covered in some what details rather than a overview in 2 pages?

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This is pretty much the opposite of what you asked for, but from what you wrote, I think it might still have a few facts you want to know. Look at Mike Harris's notes "unitary groups and base change" at edcsm.jussieu.fr/web/cours/unitarygroups.pdf – Sean Rostami Apr 5 '10 at 0:45

## 2 Answers

Scharlau's book, Quadratic and hermitian forms, gives the complete classification in Chapter 10.

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That book is fantastic! Thanks for the tip. – Sean Rostami Apr 8 '10 at 12:43

For discussions on forms of classical groups you can look at:

-- André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. 24 (1961), 589-623 (also in Oeuvres Complètes).

-- Platonov-Rapinchuk, Algebraic groups and number theory, Ac. Press, 1994.

-- The book of involutions, AMS Coll. Publ., vol. 44, 1998.

-- Kneser, Lecture on Galois cohomology of classical groups, Tata Inst. of Fund. Research, Bombay, 1969.

I've written a summary of Weil's theorems and proofs (following Platonov-Rapinchuk). It is in French ... and only deals with local base fields.

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