The characters of depth-zero representations of Sp¦4(F)

grantor: University of TorontoThe harmonic analysis of a p-adic group G(F) with Lie algebra g(F) studies two families of distributions on the space $C\sbsp{c}{\infty}({\bf g}({\rm F})\sb{tn})$ of Schwartz functions supported by topologically nilpotent elements: the Fourier transform of regular orbital integrals$$f\to{\rm I}\sb{G}(X, \ f) := \int\sb{T({\rm F})\\ G({\rm F})}\ \ f(Ad(x\sp{-1})X)dx$$and characters of supercuspidal representations $\pi$ of G(F)$$f\to{\rm I}\sb{G}(\pi, f \circ\log) := {\rm Trace}\ (\pi(f\circ\log)).$$ Although it is widely believed that the two finite-dimensional vector spaces spanned by these distributions are closely related, only in a few instances is it known how to write elements from one space as a linear combination of elements from the other. This thesis produces such a relationship for certain representation of Sp$\sb4$(F); specifically, for certain supercuspidal, depth-zero representations $\pi$ of Sp$\sb4$(F) we find $H\in sp\sb4$(F) and integers $c\sb{H}(\pi)$ such that$${\rm I}\sb{G}(\pi, f\circ\log)=\sum\sb{H}c\sb{H}(\pi){\rm I}\sb{G}(H, \ f)$$whenever $f\in C\sbsp{c}{\infty}({\bf g}(\rm F))$ has small support. Our technique makes use of recent work of L. Morris concerning certain filtrations of g(F) as well as the generalised Springer correspondence on ${\bf g}({\rm I\!F}\sb{q}).$ Along the way we deal with some geometric questions which refer to the study of affine Springer fibres by Bernstein, Kazhdan and Lusztig. We also study the stability of certain sums of characters and produce identities relating the values of Shalika germs at unramified and ramified points.Ph.D