Search

Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. 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 $X^{\mathbb{Z}/p\mathbb{Z}}$. 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., Comment: 34 pages;v3: substantial edits throughout, improved exposition and introduction, main application to the Frobenius-contraction functor strengthened; v4: accepted version, to appear in Journal f\"{u}r die Reine und Angewandte Mathematik

We study a new lifting of automorphic representations using the theta representation $\Theta$ on the $4$-fold cover of the symplectic group, $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. 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 $\Theta$. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. 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 $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$.

Motivated by the study of periods of automorphic forms and relative trace formulae, we develop the theory of descent necessary to study orbital integrals arising in the fundamental lemma for a general class of symmetric spaces over a $p$-adic field $F$. More precisely, we prove that a connected symmetric space over $F$ enjoys a notion of topological Jordan decomposition, which may be of independent interest, and establish a relative version of a lemma of Kazhdan that played a crucial role in the proof of the Langlands-Shelstad fundamental lemma. As our main application, we use these results to prove the endoscopic fundamental lemma for the unit element of the Hecke algebra for the symmetric space associated to unitary Friedberg-Jacquet periods., Comment: v4, accepted version, to appear in Advances in Mathematics: edits throughout, added detail on symmetric spaces over ring of integers

Formulas for the product of an irreducible character $$\chi _\lambda $$ of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal $${\mathcal {B}}(\lambda +\rho )$$ go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of p-adic groups as a sum over the canonical basis $${\mathcal {B}}(-\infty )$$ , which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirkovic–Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce resonant Mirkovic–Vilonen polytopes as the corresponding geometric context. Focusing on the exceptional group $$G_2$$ , we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over $${\mathcal {B}}(\lambda +\rho )$$ plus a geometric error term coming from finitely many crystals of resonant polytopes.

Motivated by questions in the study of relative trace formulae, we construct a generalization of Grothendieck's simultaneous resolution over the regular locus of certain symmetric pairs. We use this space to prove a relative version of results of Donagi and Gaitsgory about the automorphism sheaf of regular stabilizers. We also obtain partial results toward applications in Springer theory for symmetric spaces., Comment: 30 pages;typos fixed. final version, to appear in Algebra & Number Theory

We study calculus on an associative algebra \(\mathcal{A}\) of dimension n = 1, 2, 3, and 4 over the real numbers. Following the approach of numerous authors from Scheffers (Mathematisch-physikalische Klasse 46:120-134, 1894) to Vladimirov and Volovich (USSR Acad Sci 59(1):3-27, 1984) we take \(\mathcal{A}\)-linearity of the differential as the defining condition of \(\mathcal{A}\)-differentiability. The necessary condition for \(\mathcal{A}\)-differentiability is the generalized Cauchy Riemann equations. The Cauchy Riemann equations are seen as conditions to place the Jacobian matrix in the left regular representation of the algebra. We introduce the \(\mathcal{A}\)-Laplace equation as an n-th order partial differential equation solved by \(\mathcal{A}\)-differentiable functions. In contrast, Wagner constructed \(\frac{n(n-1)} {2}\) Laplace equations for the case of Frobenius algebras. We discuss the distinction between our approach and that of Wagner. Explicit Cauchy Riemann equations and the \(\mathcal{A}\)-Laplacian are given for each of the nine associative semisimple real algebras of dimensions 1, 2, 3, or 4.