$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin\_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible representations of its Lie algebra $\mathfrak{so}\_n$. $\Spin\_n$ is the universal cover of $\SO\_n$, another compact simple Lie group. But $\SO\_n$ is not simply connected so it has "fewer irreducible representations". In this [answer](https://math.stackexchange.com/a/1495187)
to [Shared representation of $SU(2)$ and $SO(3)$](https://math.stackexchange.com/questions/1495019/shared-representation-of-su2-and-so3), it is discussed which representations of $\Spin\_3 = \operatorname{SU}\_2$ "descend" to representations of $\SO\_3$. In short, it is those "labelled by even powers of the fundamental representation" of $\mathfrak{sl}\_2$.


**Question:** How does this look for higher $\SO\_n$? What are the integral positive weights of $\mathfrak{so}\_n$ that give well-defined representations of $\SO\_n$?


**Guess** Based on some basic calculations, and by looking at the question [Non-faithful irreducible representations of simple Lie groups](https://mathoverflow.net/questions/328138/non-faithful-irreducible-representations-of-simple-lie-groups), I would guess that the "descendable" representations are those with weight
$$
\lambda = \sum\_{i=1}^l a\_i \varpi\_i, ~~ \textrm{ satisfies } a\_l \in 2\mathbb{N}\_0,
$$
when $l = \operatorname{rank}(\mathfrak{so}\_n)$.



Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$
Your answer is correct for $n$ odd. For $n$ even you instead need the sum of the coefficients of the last two weights $a\_{l-1} + a\_{l}$ to be even. Either way you are effectively asking for the non-trivial element of the centre of $\mathrm{Spin}(n)$ to act trivially. Since it acts as $-1$ on the spin representation (or on each half spin representation for $n$ even), intuitively we need an "even number of them".


For example let $n$ be odd and $\Delta$ the spin representation of $\mathrm{Spin}(n)$. $\Delta$ does not have a representation of $SO(n)$ but the tensor product $\Delta \otimes \Delta$ does. As do its subrepresentations $\bigwedge^2 \Delta$, $S^2\Delta$.


