From p-adic to real Grassmannians via the quantum

Abstract

Let F be a local field. The action of GL n ( F ) on the Grassmann variety Gr ( m , n , F ) induces a continuous representation of the maximal compact subgroup of GL n ( F ) on the space of L 2 -functions on Gr ( m , n , F ) . The irreducible constituents of this representation are parameterized by the same underlying set both for Archimedean and non-Archimedean fields [G. Hill, On the nilpotent representations of GL n ( O ) , Manuscripta Math. 82 (1994) 293–311; A.T. James A.G. Constantine, Generalized Jacobi polynomials as spherical functions of the Grassmann manifold, Proc. London Math. Soc. 29(3) (1974) 174–192]. This paper connects the Archimedean and non-Archimedean theories using the quantum Grassmannian [M.S. Dijkhuizen, J.V. Stokman, Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians, Publ. Res. Inst. Math. Sci. 35 (1999) 451–500; J.V. Stokman, Multivariable big and little q-Jacobi polynomials, SIAM J. Math. Anal. 28 (1997) 452–480]. In particular, idempotents in the Hecke algebra associated to this representation are the image of the quantum zonal spherical functions after taking appropriate limits. Consequently, a correspondence is established between some irreducible representations with Archimedean and non-Archimedean origin.