# NTS ABSTRACTSpring2021

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
 Monica Nevins Interpreting the local character expansion of p-adic SL(2) The Harish-Chandra—Howe local character expansion expresses the character of an admissible representation of a p-adic group G as a linear combination with complex coefficients of the (finitely many!) Fourier transforms of nilpotent orbital integrals $$\widehat{\mu}_{\mathcal{O}}$$ --- near the identity. Approaching from another direction: we can restrict the representation to any compact open subgroup K of G, obtaining its branching rules, which also describe the representation near the identity, in a different sense. We show that for G=SL(2,k), k a nonarchimedean local field, where the branching rules to maximal compact open subgroups K are known, each of these terms $$\widehat{\mu}_{\mathcal{O}}$$ can be interpreted as the character $$\tau_{\mathcal{O}}$$ of a representation of K, up to an error term arising from the zero orbit. Moreover, the irreducible components of $$\tau_{\mathcal{O}}$$ are explicitly constructed from the K -orbits in $$\mathcal{O}$$. This work in progress offers a conjectural alternative interpretation of branching rules of admissible representations.