We extend Kostant's results about $\mathfrak{ g}$-invariants in the Clifford algebra $Cl(\mathfrak{ g})$ of a complex semisimple Lie algebra $\mathfrak{ g}$ to the relative case of $\mathfrak{ k}$-invariants in the Clifford algebra $Cl(\mathfrak{ p})$, where $(\mathfrak{ g},\mathfrak{ k})$ is a classical symmetric pair and $\mathfrak{ p}$ is the $(-1)$-eigenspace of the corresponding involution. In this setup we prove the Cartan theorem for Clifford algebras, a relative transgression theorem, the Harish--Chandra isomorphism for $Cl(\mathfrak{ p})$, and a relative version of Kostant's Clifford algebra conjecture.
Categories: math.RT math.DG