A correction to EOTAM

In my book Elliptic operators, topology and asymptotic methods (both the first and the second editions) I give a discussion of the representation theory of the groups Spin and Pin which was based (as far as I can now remember) on some notes that I took when I attended Adams’ famous course on the exceptional Lie groups, as a Part III student in 1981.  I no longer seem to have those, unfortunately (although meanwhile a version of Adams’ own notes on his approach has been published by University of Chicago Press).  Meanwhile, in 2010 Darij Grinberg pointed out on Math Overflow that the argument I gave was garbled: see this link.  In this post I want to explain what is garbled and how the useful part of the argument can be recovered.

The offending proposition is 4.9 in the second edition of my book.  This claims that there is a 1:1 correspondence (given by restriction) between three kinds of representations:

1. Algebraic representations of the Clifford algebra $$\text{Cl}(k)$$.
2. Representations of the Pin group $$\text{Pin}(k)$$, which is the multiplicative subgroup of the Clifford algebra generated by all unit vectors, on which $$\nu$$, the element $$-1$$ of the Clifford algebra, acts as $$-1$$.
3. Representations of the multiplicative subgroup $$E_k$$ of $$\text{Pin}(k)$$ generated by the vectors of an orthonormal basis, on which $$\nu$$ acts as $$-1$$.

Now, as Grinberg pointed out, it is indeed clear that there are restriction maps here – every representation of type 1 gives a representation of type 2, and every representation of type 2 gives a representation of type 3.  Moreover, every representation of type 3 extends uniquely (by linearity) to a representation of type 1.  But still, that does not preclude that there might be representations of type 2 – representations of the Pin group – which do not come from representations of the Clifford algebra.  And indeed one can straightforwardly construct such representations: those coming from the Clifford algebra have a “linearity” property which is not preserved under tensor product, and using this Grinberg gives a completely explicit counterexample.

So what is supposed to be the argument here?  I think that what must have been going on in those notes of Adams (that I now no longer have) must have been a classification of irreducible representations rather than of all representations.  Indeed, because representations of types 1 and 3 match up, the construction above does tell you that an irreducible representation of the Clifford algebra is still irreducible when restricted to the group Pin. And this is really all that gets used (see 4.14 of the book).  To show that this construction exhausts all the irreducible representations of Pin would presumably have to add another counting argument.  But nothing in the book depends on that.