Representations of $\Spinc(V)$

We obtain representations of the group $\Spinc(V)$ by restriction of the irreducible representations of $\bCl(V)$.

$$\Spinc(V) = \set{w_1w_2\dots w_{2k} : w_i \in V^\bC, w_i^*w_i = 1}.$$

We have to check whether these restrictions are irreducible or not.

Even case, $n = 2m$: $\ga$ belongs to $\Spinc(V)$ and is central there, so $c_J(\ga)$ commutes with $c_J(\Spinc(V))$. Thus the group representation reduces over $\La^\8 W_J = \La^\even W_J \oplus \La^\odd W_J$: there are two subrepresentations. Since

$$w_1 \wyw w_{2k} = \eps(w_1) \dots \eps(w_{2k}) = c_J(w_1) \dots c_J(w_{2k}) \Om,$$

we get at once that $c_J(\Spinc(V)) \Om = \La^\even W_J$: the ``even'' subrepresentation is irreducible.

If $w_1, w_2 \in W_J$ are unit vectors, then

$$c_J(w_2\bar w_1) w_1 = \eps(w_2)\iota(\bar w_1) w_1 = \eps(w_2) \Om = w_2.$$

From there we soon conclude that $c_J(\Spinc(V)) w_1 = \La^\odd W_J$: the ``odd'' subrepresentation is also irreducible.

¿Are these subrepresentations equivalent? No: for suppose $R\: \La^\even W_J \to \La^\odd W_J$ intertwines both subrepresentations. Then in particular $R c_J(\ga) = c_J(\ga) R$ means that $R(+1) = (-1)R : \La^\even W_J \to \La^\odd W_J$, so that $R = 0$.

Conclusion: The algebra representation $c_J$ of $\bCl(V)$ restricts to a group representation $c_J$ of $\Spinc(V)$ which is the direct sum of two inequivalent irreducible subrepresentations, if $\dim V$ is even.

Odd case, $n = 2m + 1$: There are two irreducible representations $c_J$ and $c'_J$ of $\bCl(V)$ on $\sF_J(U)$, but they coincide on $\bCl^0(V)$: in this case, $\ga$ is odd. Declaring $c_J(\ga)$ to be, say, $+1$ on $\sF(U)$, we get for $w_1,\dots,w_{2k+1} \in W_J$:
$$\begin{align*} w_1 \wyw w_{2k} &= c_J(w_1 \dots w_{2k}) \Om, \\ w_1 \wyw w_{2k+1} &= c_J(w_1 \dots w_{2k+1} \ga) \Om, \end{align*}$$
so that in this case, $\La^\8 W_J$ is an irreducible representation if $\dim V$ is odd.

Conclusion: The two algebra representations $c_J$ and $c'_J$ of $\bCl(V)$ restrict to the same group representation $c_J$ of $\Spinc(V)$ which is already irreducible, if $\dim V$ is odd.