Ph.D Student | Onn Uri |
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Subject | On the Grassmann Representation of GL(n) |

Department | Department of Mathematics |

Supervisor | Professor Shai Haran |

This thesis is about analogues between archimedean and non-archimedean places of a number field. Let F be a non-archimedean local field and O its ring of integers. The group GL(n,F) acts on Gr(m,n,F), the Grassmannian of m-dimensional subspaces of an n-dimensional space. This action gives rise to a complex representation of GL(n,F) on the space of square integrable functions over Gr(m,n,F) which we call the Grassmann representation. In our work we study the restriction of this representation to the maximal compact subgroup GL(n,O). We decompose this representation to irreducible representations. The multiplicity of each irreducible is one, and hence contains a unique normalized spherical function. We compute explicitly the spherical functions, and thus obtain the Fourier decomposition of the endomorphism algebra of the space. This is the non-archimedean analogue of a similar theory discovered by James and Constantine in 1974 over the real (complex) numbers and relates representations of the orthogonal (unitary) group to spectral properties of the Laplace operator of the Grassmann manifold. We make first steps in generalizing Haran's results concerning interpolation between these archimedean and non-archimedean spherical functions from GL(2) to GL(n). The main tools in this work are intertwining operators, algebraic invariants of lattices of modules (such as zeta and Mobius functions), representations of Hecke algebras and the theory of special functions.