Treffer: An application of the Littlewood restriction formula to the Kostant-Rallis Theorem: An application of the Littlewood restriction formula to the Kostant-Rallis theorem

Consider a symmetric pair(G,K)(G,K)of linear algebraic groups withg≅k⊕p\mathfrak {g} \cong \mathfrak {k} \oplus \mathfrak {p}, wherek\mathfrak {k}andp\mathfrak {p}are defined as the +1 and -1 eigenspaces of the involution definingKK. We view the ring of polynomial functions onp\mathfrak {p}as a representation ofKK. Moreover, setP(p)=⨁d=0∞Pd(p)\mathcal {P}(\mathfrak {p}) = \bigoplus _{d=0}^\infty \mathcal {P}^d(\mathfrak {p}), wherePd(p)\mathcal {P}^d(\mathfrak {p})is the space of homogeneous polynomial functions onp\mathfrak {p}of degreedd. This decomposition provides a gradedKK-module structure onP(p)\mathcal {P}(\mathfrak {p}). A decomposition ofPd(p)\mathcal {P}^d(\mathfrak {p})is provided for some classical families(G,K)(G,K)whenddis within a certain stable range. The stable range is defined so that the spacesPd(p)\mathcal {P}^d(\mathfrak {p})are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition ofPd(p)\mathcal {P}^d(\mathfrak {p})is interpreted as aqq–analog of the Kostant-Rallis theorem.