Spectral functions of invariant operators on skew multiplicity free spaces

This thesis extends results on spectral functions of invariant differential operators on multiplicity free spaces to the setting of skew multiplicity free spaces, which are representations of a reductive group whose exterior algebra decomposes into a direct sum of pairwise nonisomorphic irreducibles. We prove in the general skew multiplicity free case that the spectral functions satisfy a vanishing property and a transposition formula which are formally identical to those satisfied by their multiplicity free analogues. We investigate two special cases, the GLn C modules S2 Cn and &bigand;2 Cn , for which the spectral functions of invariant operators form a family of supersymmetric functions which can be identified with the factorial Schur Q functions. From this equivalence we deduce several properties of each family, giving the spectral functions a combinatorial interpretation and the factorial Schur Q functions a new representation theoretic one.