Theorem. The invariant algebra is the internal commutant [fcap-003P]
Theorem. The invariant algebra is the internal commutant [fcap-003P]
Let \(\mathfrak g\) be complex semisimple of rank \(\ell \), with \(r\) positive roots and \(\rho \) their half-sum. Put \(A=\mathcal {C}\kern -2pt\ell (Q_\kappa )\), where \(Q_\kappa (x)=\kappa (x,x)\), and use the lift \(\delta \) of Proposition [fcap-003E]. Define the unital algebra \(E=\delta (U(\mathfrak g))\subseteq A^0\) and the commutator invariants \[J=\{a\in A:[\delta (x),a]=0\text { for every }x\in \mathfrak g\}.\] Then \(J=\operatorname {Cent}_A(E)\), \(E\simeq \operatorname {End}(V_\rho )\), and multiplication is an isomorphism of ordinary complex algebras \[E\otimes J\longrightarrow A,\qquad e\otimes j\longmapsto ej.\] In particular, \(\dim J=2^\ell \). These assertions include the zero Lie algebra, where \(A=E=J=\mathbb C\).
An element commutes with the generators \(\delta (\mathfrak g)\) exactly when it commutes with their unital associative algebra, proving the centralizer assertion. Set \(d=2^{\lfloor \ell /2\rfloor }\). The spinor calculation in Theorem [fcap-003G] gives \(S\simeq V_\rho \otimes \mathbb C^d\). The associative algebra generated by an irreducible complex representation is its full endomorphism algebra (Burnside's theorem). Thus the image of \(U(\mathfrak g)\) on \(S\) is \(\operatorname {End}(V_\rho )\otimes 1\). This is the primary-representation calculation in [kostant1997clifford, equations (74)--(76), p. 295].
If \(\ell \) is even, then \(\dim \mathfrak g=\ell +2r\) is even, and Theorem [fcap-002V] identifies \(A\) with \(\operatorname {End}(S)\). Matrix units in \(\operatorname {End}(V_\rho )\) show that its commutant on \(V_\rho \otimes \mathbb C^d\) is \(1\otimes \operatorname {End}(\mathbb C^d)\): commuting with diagonal units forces a block-diagonal operator, and commuting with off-diagonal units makes all its diagonal blocks equal. Consequently \[J\simeq \operatorname {End}(\mathbb C^d),\qquad E\otimes J\simeq \operatorname {End}(V_\rho \otimes \mathbb C^d)=A.\] This also gives \(\dim J=d^2=2^\ell \).
If \(\ell \) is odd, Proposition [fcap-002W] identifies \(A\) with \(\operatorname {End}(S_+)\oplus \operatorname {End}(S_-)\). The two spinor restrictions to \(A^0\) are equivalent and faithful. After identifying them, \(E\subseteq A^0\) is the diagonal copy of \(\operatorname {End}(V_\rho )\otimes 1\). The same matrix-unit argument in each block gives \[J\simeq \operatorname {End}(\mathbb C^d) \oplus \operatorname {End}(\mathbb C^d).\] Multiplication \(E\otimes J\to A\) is the preceding matrix isomorphism in each of the two summands, and \(\dim J=2d^2=2^\ell \). This proves both parity cases without assuming that the full odd-dimensional Clifford algebra is simple. [kostant1997clifford, Propositions 17--20, pp. 294--297] gives this centralizer factorization for any primary spin representation.
Under the commutator action, \(\mathfrak g\) acts trivially on \(J\) and by conjugation on \(E\). Equivariant antisymmetrization from Proposition [fcap-003E] therefore yields the exterior-module consequence \[\bigwedge \mathfrak g\simeq 2^\ell (V_\rho \otimes V_\rho ^*).\] This multiplicity concerns the commutator action; the left-action multiplicity in Theorem [fcap-003G] is \(2^{\ell +r}\). The algebra factorization and its proof here are not a packaged formal theorem.