Let V be an affine symmetric variety defined over Q. We compute the asymptotic distribution of the angular components of the integral points in V . This distribution is described by a family of invariant measures concentrated on the Satake boundary of V . In the course of the proof, we describe the structure of the Satake compactifications for general affine symmetic varieties and compute the a...

This article, a continuation of [HRo], constructs the Satake parameter for any irreducible smooth J-spherical representation of a p-adic group, where J is any parahoric subgroup. This parametrizes such representations when J is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake...

We determine the extent to which the collection of Γ-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of Γ-EulerSatake characteristics corresponding to free or free abelian Γ and are not classified by those corresponding to any finite collection of finitely generated di...

We construct a mirabolic analogue of the geometric Satake equivalence. also prove an equivalence that relates representations supergroup with category $GL(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on affine Grassmannian $GL_N$. explain how our equivalences fit into more general framework conjectures due to Gaiotto and Ben-Zvi, Sakellaridis Venkatesh.

Given a simple algebraic group G, a web is a directed trivalent graph with edges labelled by dominant minuscule weights. There is a natural surjection of webs onto the invariant space of tensor products of minuscule representations. Following the work of Westbury, we produce a set of webs for SLn which form a basis for the invariant space via the geometric Satake correspondence. In fact, there ...

Abstract We refine the geometric Satake equivalence due to Ginzburg, Beilinson–Drinfeld, and Mirkovi?–Vilonen an between mixed Tate motives on double quotient $$L^+ G {\backslash }LG / L^+ G$$ L + G \ / </mml...

This paper is a continuation of [3] in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik’s double affine Hec...