How to decompose a composition of representations?

Question

Background

I would like to know if there is some slick machinery to solve the following representation-theoretic problem.

Let $\left(V,\langle-,-\rangle\right)$ be a finite-dimensional real inner product space and let $\mathfrak{so}(V)$ denote the Lie algebra of skewsymmetric endomorphisms.

Let $\mathfrak{g}$ be a Lie algebra and $\rho: \mathfrak{g} \to \mathfrak{so}(V)$ be a Lie algebra homomorphism. This makes $V$ into an orthogonal $\mathfrak{g}$-module.

Let $\sigma: \mathfrak{so}(V) \to \operatorname{End}(\Delta)$ denote an irreducible spinor representation of $\mathfrak{so}(V)$. Then the composition $\sigma \circ \rho : \mathfrak{g} \to \operatorname{End}(\Delta)$ of the representations turns $\Delta$ into a $\mathfrak{g}$-module.

Question

Is there a nice description of this module? And in particular about its decomposition into irreducibles?

Contextualisation

This sort of problem arises often in my line of work; although usually with $\mathfrak{g}$ itself being an orthogonal Lie algebra and $V$ itself an irreducible spinor representation. In other words, I end up considering spinors of spinors and this is useful in determining the structure of representations ("supermultiplets") of Lie superalgebras appearing in supersymmetric field theories. I usually work this out "by hand" (or using LiE) in terms of roots and weights, but was wondering whether there was some more conceptual machinery available.

Answer 1

As David remarked, this is nearly a hopeless task in general, but some special cases can be computed explicitly.

1. If $\mathfrak{g}$ is a member of a skew reductive dual pair $(\mathfrak{g},\mathfrak{g'})$ in $\mathfrak{so}(V)$ then the answer is known as part of skew Howe duality. Disregarding some important technicalities related to central covers, there is an explicit bijection between simple $G$-modules and simple $G'$-modules that occur in the restriction of the spinor representation of $\mathfrak{so}(V).$ In other words, the multiplicity spaces for $\mathfrak{g}$ are (almost) irreducible as $\mathfrak{g'}$-modules and it is known which modules occur.

2. If $V$ is even-dimensional and $\rho(\mathfrak{g})$ preserves a polarization $V=U\oplus U^{*}$ then the spinor representation $S$ of $\mathfrak{so}(V)$ may be identified with the exterior algebra $\wedge U,$ and this identification is $\mathfrak{g}$-equivariant. The decomposition of $S$ into two half-spinor representations $S^{\pm}$ corresponds to the decomposition of the exterior algebra into its even and odd parts. For $\mathfrak{g}$-modules $U$ that are reasonably small, the decomposition of $\wedge U$ into irreducibles is known. Roger Howe's Schur lectures contain the full description of the cases where this decomposition is multiplicity-free.

Answer 2

I just came across this question, so apologies if it is no longer relevant to your research, but I figured I'd also point out my answer to this MO question which might be helpful: A Decomposition of the 'Spin' Representation of $\mathfrak{sl}_2$

There I describe a computational method for your question in the case of odd-dimensional irreducible representations of $\mathfrak{su}_2/\mathfrak{sl}_2$ using basic character theory. The first part of this method, wherein one lines up the weights, can be done for any orthogonal representation. The second part (using the special 'factorization' of the odd spinor representations) only works when $V$ is odd-dimensional; when $V$ is even-dimensional there is no analogous 'factorization' of the spinor irreps but nevertheless with a little work one can modify the odd-dimensional case slightly to handle this (by using the relationships between the spinor irreps of even and odd-dimensional orthogonal groups under restrictions).