On Some Geometric Representations of GL N (𝔬)

We study a family of complex representations of the group GL _n (𝔬), where 𝔬 is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL _n (F) to its maximal compact subgroup GL _n (𝔬). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis that consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite 𝔬-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.