# Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field

Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two. I am looking for a reference that explains how to describe the simple (finite-dimensional) representations of $O(n)$ in terms of the simple representations of $SO(n)$.

The relation between the complex representations of the corresponding compact Lie groups is explained in section VI.7 of Bröcker and tom Dieck’s Representations of Compact Lie Groups. I believe the key statements made there also hold in the above algebraic situation, and I have worked this out in a fair amount of detail, but I am currently unwilling to believe that this has not already been done somewhere in the literature. Roughly, these statements are as follows:

• When $n$ is odd, $O(n)$ is a direct product of $SO(n)$ with $\mathbb Z/2$. Thus, every simple $SO(n)$-representation can be lifted to two distinct $O(n)$-representations, and every $O(n)$-representation arises in this way. (See also this question regarding simple representations of products.)

• When $n$ is even, $O(n)$ is only a semi-direct product of $SO(n)$ with $\mathbb Z/2$. In this case, only some simple $SO(n)$-representations can be lifted. Those that can be lifted can again be lifted to two distinct $O(n)$-representations. The remaining simple $SO(n)$-representations occur in pairs whose direct sum can be lifted to a unique simple $O(n)$-representation. All simple $O(n)$-representations arise in either of these ways.

Of course, in this question I am mainly interested in references concerning the case when $n$ is even.

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For the time being, the “fair amount of detail” I am referring to may be found in Proposition 3.18 and in Section 4.2 of arXiv:1308.0796. – Communicative Algebra Aug 7 '13 at 7:49
For $k=\mathbb C$, it's explained in Section 5.5.5 of Goodman-Wallach's book "Symmetry, represenations, and invariants". – emiliocba Aug 13 '13 at 22:17
As long as the characteristic of the field of definition is good (not 2), as you assume, it doesn't seem to matter over which field the groups are defined and split. Representations will be studied over an algebraically closed field, where for $n$ even the methods will rely on standard induction/restriction. Older group representation texts for physicists probably cover orthogonal groups, while Jantzen's book on algebraic groups may be overkill. – Jim Humphreys Aug 14 '13 at 17:34
P.S. As pointed out in a comment, the Springer GTM 255 by Goodman-Wallach is likely to be a reliable source for your purpose. I don't have that 2009 edition, but the first edition titled Representations and Invariants of the Classical Groups (Cambridge, 1998) has the relevant material in sections 5.2.2 and 10.2.5. – Jim Humphreys Aug 14 '13 at 19:00
@JimHumphreys: I’m not sure what you mean by “representations will be studied over an algebraically closed field”—I want the representations to be defined over the same field as the group itself. Do I need to make this more precise in the question? – Communicative Algebra Aug 15 '13 at 7:48