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Thursday, April 16, 2015

**Abstract:** In this talk we prove a relative trace formula for all pairs of connected algebraic groups $H \leq G \times G$ with $G$ a reductive group and $H$ the direct product of a reductive group and a unipotent group given that the test function satisfies simplifying hypotheses. As an application, we prove a relative analogue of the Weyl law, giving an asymptotic formula for the number of eigenfunctions of the Laplacian on a locally symmetric space associated to $G$ weighted by their $L^2$-restriction norm over a locally symmetric subspace associated to $H_0 \leq G$. Time permitting, we discuss how this relative Weyl law can be used to systematically construct families of automorphic forms with “large periods.” This is joint work with J. R. Getz and M. Lipnowski.