The multitemporal wave equation on Bruhat–Tits buildings

Abstract: The Satake isomorphism is an algebra isomorphism from the spherical Hecke algebra $H(G, K)$ of a (adjoint) semisimple group over a non-archimedean local field to $W$-invariant elements in the group ring of the coweight lattice $P$. The multitemporal wave equation on the Bruhat–Tits building, first introduced in the work of Anker–Rémy–Trojan '23, then corresponds to functions $G/K \times P \to \mathbb{C}$ such that applying an element in $H(G, K)$ to the “space variable” $G/K$ is equal to applying its image under the Satake isomorphism in the “time variable” $P$.

In this talk we shall motivate this equation, largely by focusing on the rank one case, and discuss several of its properties such as existence and uniqueness of solutions, finite speed of propagation, conservation of energy, scattering theory, and the connection with objects of central interest in representation theory such as Schur polynomials and Kazhdan–Lusztig polynomials. This is based on joint ongoing work with Jean–Philippe Anker, Bertrand Rémy, and Bartosz Trojan.

number theoryrepresentation theory

Audience: researchers in the topic

University of Utah Representation Theory / Number Theory Seminar