Let G denote a connected reductive group over a nonarchimedean local field F . Let K denote a special maximal parahoric subgroup of G(F ). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F ). The key ingredient is a Cartan decomposition describing the double coset space K\G(F )/K. As an application we define a transfer homomorphism t : HK∗ (G∗) → HK(G) where G∗ is the quasi-split inner form of G. We also describe how our results relate to the treatment of Cartier [Car], where K is replaced by a special maximal compact open subgroup K ⊂ G(F ) and where a Satake isomorphism is established for the Hecke algebra H K .

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The Satake isomorphism for special maximal parahoric Hecke algebras

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.

Local models of Shimura varieties, I. Geometry and combinatorics

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature \((n-1,1)\). We construct an arithmetic theta lift from harmonic Maass forms of weight \(2-n\) to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form \(f\) a linear combination of Kudla–Rapoport divisors, equipped with the Green function given by the regularized theta lift of \(f\). Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of \(f\) against a CM cycle, and (2) the central derivative of the convolution \(L\)-function of a weight \(n\) cusp form (depending on \(f\)) and the theta function of a positive definite hermitian lattice of rank \(n-1\). When specialized to the case \(n=2\), this result can be viewed as a variant of the Gross–Zagier formula for Shimura curves associated to unitary groups of signature \((1,1)\). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

Heights of kudla-rapoport divisors and derivatives of l-functions

We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension As an application, we give a description of the nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models

The Geometric Satake Correspondence for Ramified Groups

We study the intersections of special cycles on a unitary Shimura variety of signature (n-1,1) and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.

http://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p09-p.pdf

Complex multiplication cycles and Kudla-Rapoport divisors

In this article, we study local models associated to certain Shimura varieties. In particular, we present a resoultion of their singularities. As a consequence, we are able to determine the alternating semisimple trace of the geometric Frobenius on the sheaf of nearby cycles.

Nicole Kraemer

Papers

Local models for ramified unitary groups