On the Hecke Module of GLₙ(k[[z]])\GLₙ(k((z)))/GLₙ(k((z²)))

Citation

Jin, Yuhui (2024) On the Hecke Module of GLₙ(k[[z]])\GLₙ(k((z)))/GLₙ(k((z²))). Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/d0bn-5e47. https://resolver.caltech.edu/CaltechTHESIS:12082023-083025167

Abstract

[See Abstract in text of thesis for correct representation of mathematics]

Every double coset in GLₘ(k[[z]])\GLₘ(k((z)))/GLₘ(k((z²))) is uniquely represented by a block diagonal matrix with diagonal blocks in { 1,z, (11 z \\0 zⁱ \\) (i>1) } if char(k) ≠ 2 and k is a finite field. These cosets form a (spherical) Hecke module H(G,H,K) over the (spherical) Hecke algebra H(G,K) of double cosets in K\G/H, where K=GLₘ(k[[z]]) and H=GLₘ(k((z²))) and G=GLₘ(k((z))). Similarly to Hall polynomial hλ,ν^µ from the Hecke algebra H(G,K), coefficients hλ,ν^µ arise from the Hecke module. We will provide a closed formula for hλ,ν^µ, under some restrictions over λ, ν, µ.

Thesis Files