Theorem. The left Killing-Clifford module is primary of type \(V_\rho \) [fcap-003G]
Theorem. The left Killing-Clifford module is primary of type \(V_\rho \) [fcap-003G]
Let \(\mathfrak g\) be a finite-dimensional complex semisimple Lie algebra of rank \(\ell \). Choose positive roots \(\Phi ^+\), put \(r=\lvert \Phi ^+\rvert \), and write \(\rho =\frac 12\sum _{\alpha >0}\alpha \). Then \[\dim \mathfrak g=\ell +2r.\] Give \(\mathcal {C}\kern -2pt\ell (Q_\kappa )\) the left action \(L\) of Proposition [fcap-003E]. As a \(\mathfrak g\)-module it is \[\mathcal {C}\kern -2pt\ell (Q_\kappa )\simeq 2^{\ell +r}V_\rho .\] Here the exponent is the multiplicity; \(\dim V_\rho =2^r\), so the total dimension is \(2^{\ell +2r}\), as required. For the zero Lie algebra this reads as the one-dimensional trivial module.
First compute the spinor constituent. The weights of the adjoint orthogonal module are zero with multiplicity \(\ell \) and the roots in opposite pairs. The exterior spinor character is therefore \[\operatorname {ch}S= 2^{\lfloor \ell/2\rfloor } \prod _{\alpha >0} \left (e^{\alpha/2}+e^{-\alpha/2}\right ).\] The Weyl character formula for highest weight \(\rho \), together with the Weyl denominator identity applied to \(2\rho \) and divided by the ordinary denominator, gives \[\operatorname {ch}V_\rho =e^\rho \prod _{\alpha >0}(1+e^{-\alpha }) =\prod _{\alpha >0} \left (e^{\alpha/2}+e^{-\alpha/2}\right ).\] Complete reducibility and independence of irreducible characters now give \[S\simeq 2^{\lfloor \ell/2\rfloor }V_\rho .\] [panyushev2001exterior, Example 2.5(1), internal preprint p. 8] records this adjoint specialization under the complex semisimple hypotheses. The character calculation above supplies the comparison in the fixed normalization rather than substituting that conclusion without proof.
It remains to pass from a spinor carrier to the full Clifford algebra. If \(n=\ell +2r\) is even, the matrix range of Theorem [fcap-002V] gives \[\mathcal {C}\kern -2pt\ell (Q_\kappa )\simeq \operatorname {End}(S)\simeq S\otimes S^*.\] Left multiplication acts on the first factor, so \(S^*\) is a trivial multiplicity space of dimension \(2^{n/2}\). If \(n\) is odd, the odd structure of Proposition [fcap-002W] gives two matrix blocks \(\operatorname {End}(S_1)\oplus \operatorname {End}(S_2)\). The two full Clifford actions differ, but their restrictions to the even algebra are equivalent; since \(\delta (\mathfrak g)\) is even, both blocks contain the same \(\mathfrak g\)-module. Their combined multiplicity-space dimension is \(2\cdot 2^{(n-1)/2}=2^{\lceil n/2\rceil }\).
In both parity cases the multiplicity of \(V_\rho \) is \[2^{\lceil n/2\rceil } 2^{\lfloor \ell/2\rfloor } =2^{\ell +r}.\] This proves the displayed decomposition. [kostant1997clifford, Theorem 39, pp. 312--313] independently proves the exact left-regular primary-type statement by a highest-weight right ideal. TauCeti formalizes the selected left action and its commuting right multiplications, but the character, matrix-block, and multiplicity synthesis here is not a packaged formal theorem.
The conclusion is a module decomposition only. It does not assert the stronger algebra factorization \(\operatorname {End}(V_\rho )\otimes \mathcal {C}\kern -2pt\ell (P)\), identify the full commutant, or invoke the Hopf--Koszul--Samelson primitive algebra. The normal-ordered \(\mathfrak {gl}_N\) calculation in the separate CAR section uses a trace shift and is not a substitute for this Killing-adjoint theorem.