On Some Geometric Representations of GL N (𝔬)

Abstract

Listen

Translate article

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.