Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory of parity complexes in a certain 2-periodic localization of the equivariant constructible derived category D ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} . Under certain assumptions, we use this to define a functor from the category of parity sheaves on X to the category of parity sheaves on the fixed-point locus X ℤ / p ⁢ ℤ {X^{\mathbb{Z}/p\mathbb{Z}}} . This may be thought of as a categorification of Smith theory. When X is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

Motivated by global applications, we propose a theory of relative endoscopic data and transfer factors for the symmetric pair (U (2n), U (n) × U (n)) over a local field. We then formulate the smooth transfer and fundamental lemma conjectures, establish the existence of smooth transfer for many test functions, and prove the fundamental lemma for the symmetric pair (U (4), U (2) × U (2)). Contents 1. Introduction 1 2. Endoscopy 6 3. The Lie algebra of the symmetric space 7 4. Relative endoscopy for (U (W ), U (W 1 ) × U (W 2 )) 10 5. The relative fundamental lemma for (U (V 4 ), U (V 2 ) × U (V 2 )) 13 Appendix A. Comment on Torsors 21 References 22

We study a new lifting of automorphic representations using the theta representation Θ on the 4-fold cover of the symplectic group, Sp 2r (A). This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to Θ.We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to Sp 2r (A). Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of Sp 2r (A).Proof. See [29, Section7.1]. 7.2. Local Root Exchange. Suppose now that F is a non-archimedean local field, and let G be the F -rational points of a split algebraic group over F or finite cover thereof. As in the global setting, we consider unipotent subgroups C, X, Y ⊂ G, and a nontrivial characterAs before, set B = Y C, D = XC, and A = BX = DY . We also assume that these subgroups satisfy the local analogue of the six properties preceding Lemma 7.1 (For a precise statement and proof, see [13, Section 6.1]).Lemma 7.2. Suppose that π is a smooth representation of A, and extend the character ψ C trivially to ψ B on B and ψ D on D. Then we have an isomorphism of C-modules J B,ψ B (π) ∼ = J D,ψ D (π).Moreover, J C,ψ C (π) = 0 ⇐⇒ J B,ψ B (π) = 0 ⇐⇒ J D,ψ D (π) = 0.

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