# Representation of complex Clifford algebra on exterior algebras when quadratic form has odd index

### Overview

This problem entails the explicit construction of representation of Clifford algebra upon the exterior algebra, using orthogonal complex structure or polarization, namely, given a polarization $V^{\otimes \mathbb{C}}=W\oplus\bar{W}$, for $\phi\in \Lambda^* V$: $$w\cdot :=w\wedge +\bar{w}\llcorner$$ where $w\in W$. A construction using almost complex structure can be yielded similarly. One could look for page 14~16 of Fock representation for details. As for polarization, page 107 of Heat kernels and Dirac Operators offer a good dissertation.

Since the aforesaid technique is constructed on real Clifford algebra $C\ell(V, Q)$ and then complexify it to get a representation of $\mathbb{C}\ell(V)$, The polarization and the orthogonal complex structure should dependent on the quadratic form in such case.

### Question

I proved that the existence of polarization is essentially equivalent to the existence of orthogonal complex structure compatible, the later of which does not exist when the index, viz. negative signs in the signature of quadratic form is odd. But when $Q$, for instance, is $(+, +, +, -)$ and $(+, +, -, -)$, both complexified to get the same complex Clifford algebra, while the second quadratic form gives a Fock representation because of the existence of orthogonal complex structure.

So my question is if there is a way to remedy this so that we can give a representation in a similar fashion for $Q$ of odd index?